id sid tid token lemma pos ddj-151 1 1 dna dna PROPN ddj-151 1 2 decipher decipher PROPN ddj-151 1 3 journal journal PROPN ddj-151 1 4 | | ADP ddj-151 1 5 november november PROPN ddj-151 1 6 2018 2018 NUM ddj-151 1 7 | | ADV ddj-151 1 8 volume volume NOUN ddj-151 1 9 8 8 NUM ddj-151 1 10 | | NOUN ddj-151 1 11 issue issue VERB ddj-151 1 12 3 3 NUM ddj-151 1 13 | | CCONJ ddj-151 1 14 pp pp ADJ ddj-151 1 15 . . PUNCT ddj-151 2 1 197 197 NUM ddj-151 2 2 - - SYM ddj-151 2 3 205 205 NUM ddj-151 2 4 197 197 NUM ddj-151 2 5 pitkänen pitkänen NOUN ddj-151 2 6 , , PUNCT ddj-151 2 7 m. m. NOUN ddj-151 2 8 , , PUNCT ddj-151 2 9 thoughts thought NOUN ddj-151 2 10 on on ADP ddj-151 2 11 modification modification NOUN ddj-151 2 12 of of ADP ddj-151 2 13 bio bio ADJ ddj-151 2 14 - - ADJ ddj-151 2 15 harmony harmony ADJ ddj-151 2 16 essay essay ADJ ddj-151 2 17 thoughts thought NOUN ddj-151 2 18 on on ADP ddj-151 2 19 modification modification NOUN ddj-151 2 20 of of ADP ddj-151 2 21 bio bio NOUN ddj-151 2 22 - - NOUN ddj-151 2 23 harmony harmony NOUN ddj-151 2 24 matti matti PROPN ddj-151 2 25 pitkänen pitkänen PROPN ddj-151 2 26 1 1 NUM ddj-151 2 27 abstract abstract NOUN ddj-151 2 28 i i PRON ddj-151 2 29 have have AUX ddj-151 2 30 constructed construct VERB ddj-151 2 31 a a DET ddj-151 2 32 model model NOUN ddj-151 2 33 of of ADP ddj-151 2 34 bio bio NOUN ddj-151 2 35 - - NOUN ddj-151 2 36 harmony harmony NOUN ddj-151 2 37 as as ADP ddj-151 2 38 a a DET ddj-151 2 39 fusion fusion NOUN ddj-151 2 40 of of ADP ddj-151 2 41 3 3 NUM ddj-151 2 42 icosahedral icosahedral ADJ ddj-151 2 43 harmonies harmony NOUN ddj-151 2 44 and and CCONJ ddj-151 2 45 tetrahedral tetrahedral ADJ ddj-151 2 46 harmony harmony NOUN ddj-151 2 47 . . PUNCT ddj-151 3 1 the the DET ddj-151 3 2 icosahedral icosahedral ADJ ddj-151 3 3 harmonies harmony NOUN ddj-151 3 4 are be AUX ddj-151 3 5 defined define VERB ddj-151 3 6 by by ADP ddj-151 3 7 hamiltonian hamiltonian ADJ ddj-151 3 8 cycles cycle NOUN ddj-151 3 9 at at ADP ddj-151 3 10 icosahedron icosahedron NUM ddj-151 3 11 going go VERB ddj-151 3 12 through through ADP ddj-151 3 13 every every DET ddj-151 3 14 vertex vertex NOUN ddj-151 3 15 of of ADP ddj-151 3 16 the the DET ddj-151 3 17 icosahedron icosahedron NOUN ddj-151 3 18 and and CCONJ ddj-151 3 19 therefore therefore ADV ddj-151 3 20 assigning assign VERB ddj-151 3 21 to to ADP ddj-151 3 22 each each DET ddj-151 3 23 triangular triangular NOUN ddj-151 3 24 face face VERB ddj-151 3 25 an an DET ddj-151 3 26 allowed allow VERB ddj-151 3 27 3 3 NUM ddj-151 3 28 - - NOUN ddj-151 3 29 chord chord NOUN ddj-151 3 30 of of ADP ddj-151 3 31 the the DET ddj-151 3 32 harmony harmony NOUN ddj-151 3 33 . . PUNCT ddj-151 4 1 the the DET ddj-151 4 2 fascinating fascinating ADJ ddj-151 4 3 outcome outcome NOUN ddj-151 4 4 is be AUX ddj-151 4 5 that that SCONJ ddj-151 4 6 the the DET ddj-151 4 7 model model NOUN ddj-151 4 8 can can AUX ddj-151 4 9 reproduces reproduce VERB ddj-151 4 10 genetic genetic ADJ ddj-151 4 11 code code NOUN ddj-151 4 12 . . PUNCT ddj-151 5 1 the the DET ddj-151 5 2 model model NOUN ddj-151 5 3 for for ADP ddj-151 5 4 how how SCONJ ddj-151 5 5 one one PRON ddj-151 5 6 can can AUX ddj-151 5 7 understand understand VERB ddj-151 5 8 how how SCONJ ddj-151 5 9 12 12 NUM ddj-151 5 10 - - PUNCT ddj-151 5 11 note note NOUN ddj-151 5 12 scale scale NOUN ddj-151 5 13 can can AUX ddj-151 5 14 represent represent VERB ddj-151 5 15 64 64 NUM ddj-151 5 16 genetic genetic ADJ ddj-151 5 17 codons codon NOUN ddj-151 5 18 has have VERB ddj-151 5 19 the the DET ddj-151 5 20 basic basic ADJ ddj-151 5 21 property property NOUN ddj-151 5 22 that that PRON ddj-151 5 23 each each DET ddj-151 5 24 note note NOUN ddj-151 5 25 belongs belong VERB ddj-151 5 26 to to ADP ddj-151 5 27 16 16 NUM ddj-151 5 28 chords chord NOUN ddj-151 5 29 . . PUNCT ddj-151 6 1 the the DET ddj-151 6 2 reason reason NOUN ddj-151 6 3 is be AUX ddj-151 6 4 that that SCONJ ddj-151 6 5 there there PRON ddj-151 6 6 are be VERB ddj-151 6 7 3 3 NUM ddj-151 6 8 disjoint disjoint NOUN ddj-151 6 9 sets set NOUN ddj-151 6 10 of of ADP ddj-151 6 11 notes note NOUN ddj-151 6 12 and and CCONJ ddj-151 6 13 given give VERB ddj-151 6 14 3chord 3chord NUM ddj-151 6 15 is be AUX ddj-151 6 16 obtained obtain VERB ddj-151 6 17 by by ADP ddj-151 6 18 taking take VERB ddj-151 6 19 1 1 NUM ddj-151 6 20 note note NOUN ddj-151 6 21 from from ADP ddj-151 6 22 each each DET ddj-151 6 23 set set NOUN ddj-151 6 24 . . PUNCT ddj-151 7 1 for for ADP ddj-151 7 2 bio bio NOUN ddj-151 7 3 - - NOUN ddj-151 7 4 harmony harmony NOUN ddj-151 7 5 obtained obtain VERB ddj-151 7 6 as as ADP ddj-151 7 7 union union NOUN ddj-151 7 8 of of ADP ddj-151 7 9 3 3 NUM ddj-151 7 10 icosahedral icosahedral ADJ ddj-151 7 11 harmonies harmony NOUN ddj-151 7 12 and and CCONJ ddj-151 7 13 tetrahedral tetrahedral ADJ ddj-151 7 14 harmony harmony NOUN ddj-151 7 15 note note NOUN ddj-151 7 16 typically typically ADV ddj-151 7 17 belongs belong VERB ddj-151 7 18 to to ADP ddj-151 7 19 15 15 NUM ddj-151 7 20 chords chord NOUN ddj-151 7 21 . . PUNCT ddj-151 8 1 the the DET ddj-151 8 2 representation representation NOUN ddj-151 8 3 in in ADP ddj-151 8 4 terms term NOUN ddj-151 8 5 of of ADP ddj-151 8 6 frequencies frequency NOUN ddj-151 8 7 however however ADV ddj-151 8 8 requires require VERB ddj-151 8 9 16 16 NUM ddj-151 8 10 chords chord NOUN ddj-151 8 11 per per ADP ddj-151 8 12 note note NOUN ddj-151 8 13 . . PUNCT ddj-151 9 1 consistency consistency NOUN ddj-151 9 2 a a DET ddj-151 9 3 modification modification NOUN ddj-151 9 4 of of ADP ddj-151 9 5 the the DET ddj-151 9 6 model model NOUN ddj-151 9 7 of of ADP ddj-151 9 8 icosahedral icosahedral ADJ ddj-151 9 9 harmony harmony NOUN ddj-151 9 10 . . PUNCT ddj-151 10 1 the the DET ddj-151 10 2 necessity necessity NOUN ddj-151 10 3 to to PART ddj-151 10 4 introduce introduce VERB ddj-151 10 5 tetrahedron tetrahedron NOUN ddj-151 10 6 for for ADP ddj-151 10 7 one one NUM ddj-151 10 8 of of ADP ddj-151 10 9 the the DET ddj-151 10 10 3 3 NUM ddj-151 10 11 fused fuse VERB ddj-151 10 12 harmonies harmony NOUN ddj-151 10 13 is be AUX ddj-151 10 14 indeed indeed ADV ddj-151 10 15 an an DET ddj-151 10 16 ugly ugly ADJ ddj-151 10 17 looking look VERB ddj-151 10 18 feature feature NOUN ddj-151 10 19 of of ADP ddj-151 10 20 the the DET ddj-151 10 21 model model NOUN ddj-151 10 22 . . PUNCT ddj-151 11 1 the the DET ddj-151 11 2 question question NOUN ddj-151 11 3 is be AUX ddj-151 11 4 whether whether SCONJ ddj-151 11 5 one one NUM ddj-151 11 6 of of ADP ddj-151 11 7 the the DET ddj-151 11 8 harmonies harmony NOUN ddj-151 11 9 could could AUX ddj-151 11 10 be be AUX ddj-151 11 11 replaced replace VERB ddj-151 11 12 with with ADP ddj-151 11 13 some some DET ddj-151 11 14 other other ADJ ddj-151 11 15 harmony harmony NOUN ddj-151 11 16 with with ADP ddj-151 11 17 12 12 NUM ddj-151 11 18 notes note NOUN ddj-151 11 19 and and CCONJ ddj-151 11 20 24 24 NUM ddj-151 11 21 chords chord NOUN ddj-151 11 22 . . PUNCT ddj-151 12 1 if if SCONJ ddj-151 12 2 this this PRON ddj-151 12 3 would would AUX ddj-151 12 4 work work VERB ddj-151 12 5 one one PRON ddj-151 12 6 would would AUX ddj-151 12 7 have have VERB ddj-151 12 8 64 64 NUM ddj-151 12 9 chords chord NOUN ddj-151 12 10 equal equal ADJ ddj-151 12 11 to to ADP ddj-151 12 12 the the DET ddj-151 12 13 number number NOUN ddj-151 12 14 of of ADP ddj-151 12 15 genetic genetic ADJ ddj-151 12 16 codons codon NOUN ddj-151 12 17 and and CCONJ ddj-151 12 18 5 5 NUM ddj-151 12 19 + + SYM ddj-151 12 20 5 5 NUM ddj-151 12 21 + + SYM ddj-151 12 22 6 6 NUM ddj-151 12 23 = = SYM ddj-151 12 24 16 16 NUM ddj-151 12 25 chords chord NOUN ddj-151 12 26 per per ADP ddj-151 12 27 note note NOUN ddj-151 12 28 . . PUNCT ddj-151 13 1 one one PRON ddj-151 13 2 can can AUX ddj-151 13 3 imagine imagine VERB ddj-151 13 4 toric toric ADJ ddj-151 13 5 variants variant NOUN ddj-151 13 6 of of ADP ddj-151 13 7 harmonies harmony NOUN ddj-151 13 8 realized realize VERB ddj-151 13 9 in in ADP ddj-151 13 10 terms term NOUN ddj-151 13 11 of of ADP ddj-151 13 12 hamiltonian hamiltonian ADJ ddj-151 13 13 cycles cycle NOUN ddj-151 13 14 and and CCONJ ddj-151 13 15 one one NOUN ddj-151 13 16 indeed indeed ADV ddj-151 13 17 obtains obtain VERB ddj-151 13 18 a a DET ddj-151 13 19 toric toric ADJ ddj-151 13 20 harmony harmony NOUN ddj-151 13 21 with with ADP ddj-151 13 22 12 12 NUM ddj-151 13 23 notes note NOUN ddj-151 13 24 and and CCONJ ddj-151 13 25 24 24 NUM ddj-151 13 26 3 3 NUM ddj-151 13 27 - - PUNCT ddj-151 13 28 chords chord NOUN ddj-151 13 29 . . PUNCT ddj-151 14 1 bio bio NOUN ddj-151 14 2 - - NOUN ddj-151 14 3 harmony harmony NOUN ddj-151 14 4 could could AUX ddj-151 14 5 correspond correspond VERB ddj-151 14 6 to to ADP ddj-151 14 7 the the DET ddj-151 14 8 fusion fusion NOUN ddj-151 14 9 of of ADP ddj-151 14 10 2 2 NUM ddj-151 14 11 icosahedral icosahedral ADJ ddj-151 14 12 harmonies harmony NOUN ddj-151 14 13 with with ADP ddj-151 14 14 20 20 NUM ddj-151 14 15 chords chord NOUN ddj-151 14 16 and and CCONJ ddj-151 14 17 toric toric ADJ ddj-151 14 18 harmony harmony NOUN ddj-151 14 19 with with ADP ddj-151 14 20 24 24 NUM ddj-151 14 21 chords chord NOUN ddj-151 14 22 having have VERB ddj-151 14 23 therefore therefore ADV ddj-151 14 24 64 64 NUM ddj-151 14 25 chords chord NOUN ddj-151 14 26 . . PUNCT ddj-151 15 1 whether whether SCONJ ddj-151 15 2 the the DET ddj-151 15 3 predictions prediction NOUN ddj-151 15 4 for for ADP ddj-151 15 5 the the DET ddj-151 15 6 numbers number NOUN ddj-151 15 7 of of ADP ddj-151 15 8 codons codon NOUN ddj-151 15 9 coding code VERB ddj-151 15 10 for for ADP ddj-151 15 11 given give VERB ddj-151 15 12 amino amino NOUN ddj-151 15 13 - - PUNCT ddj-151 15 14 acids acid NOUN ddj-151 15 15 come come VERB ddj-151 15 16 out out ADP ddj-151 15 17 correctly correctly ADV ddj-151 15 18 for for ADP ddj-151 15 19 some some DET ddj-151 15 20 choices choice NOUN ddj-151 15 21 of of ADP ddj-151 15 22 hamiltonian hamiltonian ADJ ddj-151 15 23 cycles cycle NOUN ddj-151 15 24 is be AUX ddj-151 15 25 still still ADV ddj-151 15 26 unclear unclear ADJ ddj-151 15 27 . . PUNCT ddj-151 16 1 keywords keyword NOUN ddj-151 16 2 : : PUNCT ddj-151 16 3 bio bio NOUN ddj-151 16 4 - - NOUN ddj-151 16 5 harmony harmony ADJ ddj-151 16 6 , , PUNCT ddj-151 16 7 modification modification NOUN ddj-151 16 8 , , PUNCT ddj-151 16 9 genetic genetic ADJ ddj-151 16 10 code code NOUN ddj-151 16 11 , , PUNCT ddj-151 16 12 tgd tgd PROPN ddj-151 16 13 framework framework NOUN ddj-151 16 14 . . PUNCT ddj-151 17 1 1 1 NUM ddj-151 17 2 introduction introduction NOUN ddj-151 17 3 i i PRON ddj-151 17 4 have have AUX ddj-151 17 5 developed develop VERB ddj-151 17 6 a a DET ddj-151 17 7 rather rather ADV ddj-151 17 8 detailed detailed ADJ ddj-151 17 9 model model NOUN ddj-151 17 10 of of ADP ddj-151 17 11 bio bio NOUN ddj-151 17 12 - - NOUN ddj-151 17 13 harmony harmony NOUN ddj-151 17 14 as as ADP ddj-151 17 15 a a DET ddj-151 17 16 fusion fusion NOUN ddj-151 17 17 of of ADP ddj-151 17 18 3 3 NUM ddj-151 17 19 icosahedral icosahedral ADJ ddj-151 17 20 harmonies harmony NOUN ddj-151 17 21 and and CCONJ ddj-151 17 22 tetrahedral tetrahedral ADJ ddj-151 17 23 harmony harmony NOUN ddj-151 17 24 [ [ X ddj-151 17 25 3 3 NUM ddj-151 17 26 , , PUNCT ddj-151 17 27 4](see 4](see NUM ddj-151 17 28 http://tinyurl.com/yad4tqwl http://tinyurl.com/yad4tqwl NOUN ddj-151 17 29 and and CCONJ ddj-151 17 30 http://tinyurl.com/y8njuctq http://tinyurl.com/y8njuctq NOUN ddj-151 17 31 ) ) PUNCT ddj-151 17 32 . . PUNCT ddj-151 18 1 the the DET ddj-151 18 2 icosahedral icosahedral ADJ ddj-151 18 3 harmonies harmony NOUN ddj-151 18 4 are be AUX ddj-151 18 5 defined define VERB ddj-151 18 6 by by ADP ddj-151 18 7 hamiltonian hamiltonian ADJ ddj-151 18 8 cycles cycle NOUN ddj-151 18 9 at at ADP ddj-151 18 10 icosahedron icosahedron NUM ddj-151 18 11 going go VERB ddj-151 18 12 through through ADP ddj-151 18 13 every every DET ddj-151 18 14 vertex vertex NOUN ddj-151 18 15 of of ADP ddj-151 18 16 the the DET ddj-151 18 17 icosahedron icosahedron NOUN ddj-151 18 18 and and CCONJ ddj-151 18 19 therefore therefore ADV ddj-151 18 20 assigning assign VERB ddj-151 18 21 to to ADP ddj-151 18 22 each each DET ddj-151 18 23 triangular triangular NOUN ddj-151 18 24 face face VERB ddj-151 18 25 an an DET ddj-151 18 26 allowed allow VERB ddj-151 18 27 3 3 NUM ddj-151 18 28 - - NOUN ddj-151 18 29 chord chord NOUN ddj-151 18 30 of of ADP ddj-151 18 31 the the DET ddj-151 18 32 harmony harmony NOUN ddj-151 18 33 . . PUNCT ddj-151 19 1 the the DET ddj-151 19 2 surprising surprising ADJ ddj-151 19 3 outcome outcome NOUN ddj-151 19 4 is be AUX ddj-151 19 5 that that SCONJ ddj-151 19 6 the the DET ddj-151 19 7 model model NOUN ddj-151 19 8 can can AUX ddj-151 19 9 reproduces reproduce VERB ddj-151 19 10 genetic genetic ADJ ddj-151 19 11 code code NOUN ddj-151 19 12 . . PUNCT ddj-151 20 1 the the DET ddj-151 20 2 model model NOUN ddj-151 20 3 for for ADP ddj-151 20 4 how how SCONJ ddj-151 20 5 one one PRON ddj-151 20 6 can can AUX ddj-151 20 7 understand understand VERB ddj-151 20 8 how how SCONJ ddj-151 20 9 12 12 NUM ddj-151 20 10 - - PUNCT ddj-151 20 11 note note NOUN ddj-151 20 12 scale scale NOUN ddj-151 20 13 can can AUX ddj-151 20 14 represent represent VERB ddj-151 20 15 64 64 NUM ddj-151 20 16 genetic genetic ADJ ddj-151 20 17 codons codon NOUN ddj-151 20 18 has have VERB ddj-151 20 19 the the DET ddj-151 20 20 basic basic ADJ ddj-151 20 21 property property NOUN ddj-151 20 22 that that PRON ddj-151 20 23 each each DET ddj-151 20 24 note note NOUN ddj-151 20 25 belongs belong VERB ddj-151 20 26 to to ADP ddj-151 20 27 16 16 NUM ddj-151 20 28 chords chord NOUN ddj-151 20 29 . . PUNCT ddj-151 21 1 the the DET ddj-151 21 2 reason reason NOUN ddj-151 21 3 is be AUX ddj-151 21 4 that that SCONJ ddj-151 21 5 there there PRON ddj-151 21 6 are be VERB ddj-151 21 7 3 3 NUM ddj-151 21 8 disjoint disjoint NOUN ddj-151 21 9 sets set NOUN ddj-151 21 10 of of ADP ddj-151 21 11 notes note NOUN ddj-151 21 12 and and CCONJ ddj-151 21 13 given give VERB ddj-151 21 14 3 3 NUM ddj-151 21 15 - - PUNCT ddj-151 21 16 chord chord NOUN ddj-151 21 17 is be AUX ddj-151 21 18 obtained obtain VERB ddj-151 21 19 by by ADP ddj-151 21 20 taking take VERB ddj-151 21 21 1 1 NUM ddj-151 21 22 note note NOUN ddj-151 21 23 from from ADP ddj-151 21 24 each each DET ddj-151 21 25 set set NOUN ddj-151 21 26 . . PUNCT ddj-151 22 1 for for ADP ddj-151 22 2 bio bio NOUN ddj-151 22 3 - - NOUN ddj-151 22 4 harmony harmony NOUN ddj-151 22 5 obtained obtain VERB ddj-151 22 6 as as ADP ddj-151 22 7 union union NOUN ddj-151 22 8 of of ADP ddj-151 22 9 3 3 NUM ddj-151 22 10 icosahedral icosahedral ADJ ddj-151 22 11 harmonies harmony NOUN ddj-151 22 12 and and CCONJ ddj-151 22 13 tetrahedral tetrahedral ADJ ddj-151 22 14 harmony harmony NOUN ddj-151 22 15 note note NOUN ddj-151 22 16 typically typically ADV ddj-151 22 17 belongs belong VERB ddj-151 22 18 to to ADP ddj-151 22 19 15 15 NUM ddj-151 22 20 chords chord NOUN ddj-151 22 21 . . PUNCT ddj-151 23 1 the the DET ddj-151 23 2 representation representation NOUN ddj-151 23 3 in in ADP ddj-151 23 4 terms term NOUN ddj-151 23 5 of of ADP ddj-151 23 6 frequencies frequency NOUN ddj-151 23 7 requires require VERB ddj-151 23 8 16 16 NUM ddj-151 23 9 chords chord NOUN ddj-151 23 10 per per ADP ddj-151 23 11 note note NOUN ddj-151 23 12 . . PUNCT ddj-151 24 1 if if SCONJ ddj-151 24 2 one one PRON ddj-151 24 3 wants want VERB ddj-151 24 4 consistency consistency NOUN ddj-151 24 5 one one PRON ddj-151 24 6 must must AUX ddj-151 24 7 somehow somehow ADV ddj-151 24 8 modify modify VERB ddj-151 24 9 the the DET ddj-151 24 10 model model NOUN ddj-151 24 11 of of ADP ddj-151 24 12 icosahedral icosahedral ADJ ddj-151 24 13 harmony harmony NOUN ddj-151 24 14 . . PUNCT ddj-151 25 1 the the DET ddj-151 25 2 necessity necessity NOUN ddj-151 25 3 to to PART ddj-151 25 4 introduce introduce VERB ddj-151 25 5 tetrahedron tetrahedron NOUN ddj-151 25 6 for for ADP ddj-151 25 7 one one NUM ddj-151 25 8 of of ADP ddj-151 25 9 the the DET ddj-151 25 10 3 3 NUM ddj-151 25 11 fused fuse VERB ddj-151 25 12 harmonies harmony NOUN ddj-151 25 13 is be AUX ddj-151 25 14 indeed indeed ADV ddj-151 25 15 an an DET ddj-151 25 16 ugly ugly ADJ ddj-151 25 17 looking look VERB ddj-151 25 18 feature feature NOUN ddj-151 25 19 of of ADP ddj-151 25 20 the the DET ddj-151 25 21 model model NOUN ddj-151 25 22 . . PUNCT ddj-151 26 1 the the DET ddj-151 26 2 question question NOUN ddj-151 26 3 is be AUX ddj-151 26 4 whether whether SCONJ ddj-151 26 5 one one NUM ddj-151 26 6 of of ADP ddj-151 26 7 the the DET ddj-151 26 8 harmonies harmony NOUN ddj-151 26 9 could could AUX ddj-151 26 10 be be AUX ddj-151 26 11 replaced replace VERB ddj-151 26 12 with with ADP ddj-151 26 13 some some DET ddj-151 26 14 other other ADJ ddj-151 26 15 harmony harmony NOUN ddj-151 26 16 with with ADP ddj-151 26 17 12 12 NUM ddj-151 26 18 notes note NOUN ddj-151 26 19 and and CCONJ ddj-151 26 20 24 24 NUM ddj-151 26 21 chords chord NOUN ddj-151 26 22 . . PUNCT ddj-151 27 1 if if SCONJ ddj-151 27 2 this this PRON ddj-151 27 3 would would AUX ddj-151 27 4 work work VERB ddj-151 27 5 one one PRON ddj-151 27 6 would would AUX ddj-151 27 7 have have VERB ddj-151 27 8 64 64 NUM ddj-151 27 9 chords chord NOUN ddj-151 27 10 equal equal ADJ ddj-151 27 11 to to ADP ddj-151 27 12 the the DET ddj-151 27 13 number number NOUN ddj-151 27 14 of of ADP ddj-151 27 15 genetic genetic ADJ ddj-151 27 16 codons codon NOUN ddj-151 27 17 and and CCONJ ddj-151 27 18 5 5 NUM ddj-151 27 19 + + SYM ddj-151 27 20 5 5 NUM ddj-151 27 21 + + SYM ddj-151 27 22 6 6 NUM ddj-151 27 23 = = SYM ddj-151 27 24 16 16 NUM ddj-151 27 25 chords chord NOUN ddj-151 27 26 per per ADP ddj-151 27 27 note note NOUN ddj-151 27 28 . . PUNCT ddj-151 28 1 the the DET ddj-151 28 2 addition addition NOUN ddj-151 28 3 of of ADP ddj-151 28 4 tetrahedron tetrahedron NOUN ddj-151 28 5 would would AUX ddj-151 28 6 not not PART ddj-151 28 7 be be AUX ddj-151 28 8 needed need VERB ddj-151 28 9 . . PUNCT ddj-151 29 1 one one PRON ddj-151 29 2 can can AUX ddj-151 29 3 imagine imagine VERB ddj-151 29 4 toric toric ADJ ddj-151 29 5 variants variant NOUN ddj-151 29 6 of of ADP ddj-151 29 7 harmonies harmony NOUN ddj-151 29 8 realized realize VERB ddj-151 29 9 in in ADP ddj-151 29 10 terms term NOUN ddj-151 29 11 of of ADP ddj-151 29 12 hamiltonian hamiltonian ADJ ddj-151 29 13 cycles cycle NOUN ddj-151 29 14 and and CCONJ ddj-151 29 15 one one NOUN ddj-151 29 16 indeed indeed ADV ddj-151 29 17 obtains obtain VERB ddj-151 29 18 a a DET ddj-151 29 19 toric toric ADJ ddj-151 29 20 harmony harmony NOUN ddj-151 29 21 with with ADP ddj-151 29 22 12 12 NUM ddj-151 29 23 notes note NOUN ddj-151 29 24 and and CCONJ ddj-151 29 25 24 24 NUM ddj-151 29 26 3 3 NUM ddj-151 29 27 - - PUNCT ddj-151 29 28 chords chord NOUN ddj-151 29 29 . . PUNCT ddj-151 30 1 bio bio NOUN ddj-151 30 2 - - NOUN ddj-151 30 3 harmony harmony NOUN ddj-151 30 4 could could AUX ddj-151 30 5 correspond correspond VERB ddj-151 30 6 to to ADP ddj-151 30 7 the the DET ddj-151 30 8 fusion fusion NOUN ddj-151 30 9 of of ADP ddj-151 30 10 2 2 NUM ddj-151 30 11 icosahedral icosahedral ADJ ddj-151 30 12 harmonies harmony NOUN ddj-151 30 13 with with ADP ddj-151 30 14 20 20 NUM ddj-151 30 15 chords chord NOUN ddj-151 30 16 and and CCONJ ddj-151 30 17 toric toric ADJ ddj-151 30 18 harmony harmony NOUN ddj-151 30 19 with with ADP ddj-151 30 20 24 24 NUM ddj-151 30 21 chords chord NOUN ddj-151 30 22 having have VERB ddj-151 30 23 therefore therefore ADV ddj-151 30 24 64 64 NUM ddj-151 30 25 chords chord NOUN ddj-151 30 26 . . PUNCT ddj-151 31 1 whether whether SCONJ ddj-151 31 2 the the DET ddj-151 31 3 predictions prediction NOUN ddj-151 31 4 for for ADP ddj-151 31 5 the the DET ddj-151 31 6 numbers number NOUN ddj-151 31 7 of of ADP ddj-151 31 8 codons codon NOUN ddj-151 31 9 coding code VERB ddj-151 31 10 for for ADP ddj-151 31 11 given give VERB ddj-151 31 12 amino amino NOUN ddj-151 31 13 - - PUNCT ddj-151 31 14 acids acid NOUN ddj-151 31 15 come come VERB ddj-151 31 16 out out ADP ddj-151 31 17 correctly correctly ADV ddj-151 31 18 for for ADP ddj-151 31 19 1correspondence 1correspondence NUM ddj-151 31 20 : : PUNCT ddj-151 31 21 matti matti PROPN ddj-151 31 22 pitkänen pitkänen PROPN ddj-151 31 23 http://tgdtheory.fi/. http://tgdtheory.fi/. PROPN ddj-151 31 24 address address NOUN ddj-151 31 25 : : PUNCT ddj-151 31 26 rinnekatu rinnekatu ADJ ddj-151 31 27 2 2 NUM ddj-151 31 28 - - SYM ddj-151 31 29 4 4 NUM ddj-151 31 30 8a 8a NUM ddj-151 31 31 , , PUNCT ddj-151 31 32 03620 03620 NUM ddj-151 31 33 , , PUNCT ddj-151 31 34 karkkila karkkila PROPN ddj-151 31 35 , , PUNCT ddj-151 31 36 finland finland PROPN ddj-151 31 37 . . PUNCT ddj-151 32 1 email email NOUN ddj-151 32 2 : : PUNCT ddj-151 32 3 matpitka6@gmail.com matpitka6@gmail.com X ddj-151 32 4 . . PUNCT ddj-151 33 1 issn issn PROPN ddj-151 33 2 : : PUNCT ddj-151 33 3 2159 2159 NUM ddj-151 33 4 - - PUNCT ddj-151 33 5 046x 046x NOUN ddj-151 33 6 dna dna NOUN ddj-151 33 7 decipher decipher PROPN ddj-151 33 8 journal journal PROPN ddj-151 33 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 33 10 published publish VERB ddj-151 33 11 by by ADP ddj-151 33 12 quantumdream quantumdream PROPN ddj-151 33 13 , , PUNCT ddj-151 33 14 inc inc PROPN ddj-151 33 15 . . PROPN ddj-151 33 16 http://tinyurl.com/yad4tqwl http://tinyurl.com/yad4tqwl VERB ddj-151 33 17 http://tinyurl.com/y8njuctq http://tinyurl.com/y8njuctq PROPN ddj-151 33 18 http://tgdtheory.fi/ http://tgdtheory.fi/ ADJ ddj-151 33 19 mailto:matpitka6@gmail.com mailto:matpitka6@gmail.com PROPN ddj-151 33 20 dna dna PROPN ddj-151 33 21 decipher decipher PROPN ddj-151 33 22 journal journal PROPN ddj-151 33 23 | | ADP ddj-151 33 24 november november PROPN ddj-151 33 25 2018 2018 NUM ddj-151 33 26 | | ADV ddj-151 33 27 volume volume NOUN ddj-151 33 28 8 8 NUM ddj-151 34 1 | | NOUN ddj-151 34 2 issue issue VERB ddj-151 34 3 3 3 NUM ddj-151 34 4 | | CCONJ ddj-151 34 5 pp pp ADJ ddj-151 34 6 . . PUNCT ddj-151 35 1 197 197 NUM ddj-151 35 2 - - SYM ddj-151 35 3 205 205 NUM ddj-151 35 4 198 198 NUM ddj-151 35 5 pitkänen pitkänen NOUN ddj-151 35 6 , , PUNCT ddj-151 35 7 m. m. NOUN ddj-151 35 8 , , PUNCT ddj-151 35 9 thoughts thought NOUN ddj-151 35 10 on on ADP ddj-151 35 11 modification modification NOUN ddj-151 35 12 of of ADP ddj-151 35 13 bio bio NOUN ddj-151 35 14 - - NOUN ddj-151 35 15 harmony harmony NOUN ddj-151 35 16 some some DET ddj-151 35 17 choices choice NOUN ddj-151 35 18 of of ADP ddj-151 35 19 hamiltonian hamiltonian ADJ ddj-151 35 20 cycles cycle NOUN ddj-151 35 21 is be AUX ddj-151 35 22 still still ADV ddj-151 35 23 unclear unclear ADJ ddj-151 35 24 . . PUNCT ddj-151 36 1 this this PRON ddj-151 36 2 would would AUX ddj-151 36 3 require require VERB ddj-151 36 4 an an DET ddj-151 36 5 explicit explicit ADJ ddj-151 36 6 construction construction NOUN ddj-151 36 7 of of ADP ddj-151 36 8 toric toric ADJ ddj-151 36 9 hamiltonian hamiltonian ADJ ddj-151 36 10 cycles cycle NOUN ddj-151 36 11 . . PUNCT ddj-151 37 1 before before ADP ddj-151 37 2 discussing discuss VERB ddj-151 37 3 the the DET ddj-151 37 4 possible possible ADJ ddj-151 37 5 role role NOUN ddj-151 37 6 of of ADP ddj-151 37 7 toric toric ADJ ddj-151 37 8 harmonies harmony NOUN ddj-151 37 9 some some DET ddj-151 37 10 previous previous ADJ ddj-151 37 11 results result NOUN ddj-151 37 12 will will AUX ddj-151 37 13 be be AUX ddj-151 37 14 summarized summarize VERB ddj-151 37 15 . . PUNCT ddj-151 38 1 1.1 1.1 NUM ddj-151 38 2 icosahedral icosahedral ADJ ddj-151 38 3 bio bio NOUN ddj-151 38 4 - - NOUN ddj-151 38 5 harmonies harmonie VERB ddj-151 38 6 the the DET ddj-151 38 7 model model NOUN ddj-151 38 8 of of ADP ddj-151 38 9 bio bio NOUN ddj-151 38 10 - - NOUN ddj-151 38 11 harmony harmony NOUN ddj-151 38 12 [ [ X ddj-151 38 13 3 3 NUM ddj-151 38 14 ] ] PUNCT ddj-151 38 15 starts start VERB ddj-151 38 16 from from ADP ddj-151 38 17 a a DET ddj-151 38 18 model model NOUN ddj-151 38 19 for for ADP ddj-151 38 20 music music NOUN ddj-151 38 21 harmony harmony NOUN ddj-151 38 22 as as ADP ddj-151 38 23 a a DET ddj-151 38 24 hamiltonian hamiltonian ADJ ddj-151 38 25 cycle cycle NOUN ddj-151 38 26 at at ADP ddj-151 38 27 icosahedron icosahedron PROPN ddj-151 38 28 having have VERB ddj-151 38 29 12 12 NUM ddj-151 38 30 vertices vertex NOUN ddj-151 38 31 identified identify VERB ddj-151 38 32 as as ADP ddj-151 38 33 12 12 NUM ddj-151 38 34 notes note NOUN ddj-151 38 35 and and CCONJ ddj-151 38 36 20 20 NUM ddj-151 38 37 triangular triangular NOUN ddj-151 38 38 faces face NOUN ddj-151 38 39 defining define VERB ddj-151 38 40 the the DET ddj-151 38 41 allowed allow VERB ddj-151 38 42 chords chord NOUN ddj-151 38 43 of of ADP ddj-151 38 44 the the DET ddj-151 38 45 harmony harmony NOUN ddj-151 38 46 . . PUNCT ddj-151 39 1 the the DET ddj-151 39 2 identification identification NOUN ddj-151 39 3 is be AUX ddj-151 39 4 determined determine VERB ddj-151 39 5 by by ADP ddj-151 39 6 a a DET ddj-151 39 7 hamiltonian hamiltonian ADJ ddj-151 39 8 cycle cycle NOUN ddj-151 39 9 going go VERB ddj-151 39 10 once once ADV ddj-151 39 11 through through ADP ddj-151 39 12 each each DET ddj-151 39 13 vertex vertex NOUN ddj-151 39 14 of of ADP ddj-151 39 15 icosahedron icosahedron NOUN ddj-151 39 16 and and CCONJ ddj-151 39 17 consisting consisting NOUN ddj-151 39 18 of of ADP ddj-151 39 19 edges edge NOUN ddj-151 39 20 of of ADP ddj-151 39 21 the the DET ddj-151 39 22 icosahedral icosahedral ADJ ddj-151 39 23 tesselation tesselation NOUN ddj-151 39 24 of of ADP ddj-151 39 25 sphere sphere NOUN ddj-151 39 26 ( ( PUNCT ddj-151 39 27 analog analog NOUN ddj-151 39 28 of of ADP ddj-151 39 29 lattice lattice NOUN ddj-151 39 30 ): ): PUNCT ddj-151 39 31 each each DET ddj-151 39 32 edge edge NOUN ddj-151 39 33 corresponds correspond VERB ddj-151 39 34 to to ADP ddj-151 39 35 quint quint NOUN ddj-151 39 36 that that PRON ddj-151 39 37 is be AUX ddj-151 39 38 scaling scale VERB ddj-151 39 39 of of ADP ddj-151 39 40 the the DET ddj-151 39 41 frequency frequency NOUN ddj-151 39 42 of of ADP ddj-151 39 43 the the DET ddj-151 39 44 note note NOUN ddj-151 39 45 by by ADP ddj-151 39 46 factor factor NOUN ddj-151 39 47 3/2 3/2 NUM ddj-151 39 48 ( ( PUNCT ddj-151 39 49 or or CCONJ ddj-151 39 50 by by ADP ddj-151 39 51 factor factor NOUN ddj-151 39 52 27/12 27/12 NUM ddj-151 39 53 in in ADP ddj-151 39 54 well well ADV ddj-151 39 55 - - PUNCT ddj-151 39 56 tempered temper VERB ddj-151 39 57 scale scale NOUN ddj-151 39 58 ) ) PUNCT ddj-151 39 59 . . PUNCT ddj-151 40 1 this this DET ddj-151 40 2 identification identification NOUN ddj-151 40 3 assigns assign NOUN ddj-151 40 4 to to ADP ddj-151 40 5 each each DET ddj-151 40 6 triangle triangle NOUN ddj-151 40 7 of of ADP ddj-151 40 8 the the DET ddj-151 40 9 icosahedron icosahedron PROPN ddj-151 40 10 a a DET ddj-151 40 11 3 3 NUM ddj-151 40 12 - - NOUN ddj-151 40 13 chord chord NOUN ddj-151 40 14 . . PUNCT ddj-151 41 1 the the DET ddj-151 41 2 20 20 NUM ddj-151 41 3 faces face NOUN ddj-151 41 4 of of ADP ddj-151 41 5 icosahedron icosahedron X ddj-151 41 6 define define NOUN ddj-151 41 7 therefore therefore ADV ddj-151 41 8 the the DET ddj-151 41 9 allowed allowed ADJ ddj-151 41 10 3 3 NUM ddj-151 41 11 - - NOUN ddj-151 41 12 chords chord NOUN ddj-151 41 13 of of ADP ddj-151 41 14 the the DET ddj-151 41 15 harmony harmony NOUN ddj-151 41 16 . . PUNCT ddj-151 42 1 there there PRON ddj-151 42 2 exists exist VERB ddj-151 42 3 quite quite DET ddj-151 42 4 a a DET ddj-151 42 5 large large ADJ ddj-151 42 6 number number NOUN ddj-151 42 7 of of ADP ddj-151 42 8 icosahedral icosahedral ADJ ddj-151 42 9 hamiltonian hamiltonian ADJ ddj-151 42 10 cycles cycle NOUN ddj-151 42 11 and and CCONJ ddj-151 42 12 thus thus ADV ddj-151 42 13 harmonies harmony NOUN ddj-151 42 14 . . PUNCT ddj-151 43 1 the the DET ddj-151 43 2 fact fact NOUN ddj-151 43 3 that that SCONJ ddj-151 43 4 the the DET ddj-151 43 5 number number NOUN ddj-151 43 6 of of ADP ddj-151 43 7 chords chord NOUN ddj-151 43 8 is be AUX ddj-151 43 9 20 20 NUM ddj-151 43 10 the the DET ddj-151 43 11 number number NOUN ddj-151 43 12 of of ADP ddj-151 43 13 amino amino NOUN ddj-151 43 14 - - PUNCT ddj-151 43 15 acids acid NOUN ddj-151 43 16 leads lead NOUN ddj-151 43 17 to to ADP ddj-151 43 18 the the DET ddj-151 43 19 question question NOUN ddj-151 43 20 whether whether SCONJ ddj-151 43 21 one one PRON ddj-151 43 22 might might AUX ddj-151 43 23 somehow somehow ADV ddj-151 43 24 understand understand VERB ddj-151 43 25 genetic genetic ADJ ddj-151 43 26 code code NOUN ddj-151 43 27 and and CCONJ ddj-151 43 28 64 64 NUM ddj-151 43 29 dna dna NOUN ddj-151 43 30 codons codon NOUN ddj-151 43 31 in in ADP ddj-151 43 32 this this DET ddj-151 43 33 framework framework NOUN ddj-151 43 34 . . PUNCT ddj-151 44 1 by by ADP ddj-151 44 2 combining combine VERB ddj-151 44 3 3 3 NUM ddj-151 44 4 icosahedral icosahedral ADJ ddj-151 44 5 harmonies harmony NOUN ddj-151 44 6 with with ADP ddj-151 44 7 different different ADJ ddj-151 44 8 symmetry symmetry NOUN ddj-151 44 9 groups group NOUN ddj-151 44 10 identified identify VERB ddj-151 44 11 as as ADP ddj-151 44 12 subgroups subgroup NOUN ddj-151 44 13 of of ADP ddj-151 44 14 the the DET ddj-151 44 15 icosahedral icosahedral ADJ ddj-151 44 16 group group NOUN ddj-151 44 17 , , PUNCT ddj-151 44 18 one one PRON ddj-151 44 19 obtains obtain VERB ddj-151 44 20 harmonies harmony NOUN ddj-151 44 21 with with ADP ddj-151 44 22 60 60 NUM ddj-151 44 23 3 3 NUM ddj-151 44 24 - - PUNCT ddj-151 44 25 chords chord NOUN ddj-151 44 26 . . PUNCT ddj-151 45 1 the the DET ddj-151 45 2 dna dna PROPN ddj-151 45 3 codons codon NOUN ddj-151 45 4 coding code VERB ddj-151 45 5 for for ADP ddj-151 45 6 given give VERB ddj-151 45 7 amino amino NOUN ddj-151 45 8 - - PUNCT ddj-151 45 9 acid acid NOUN ddj-151 45 10 are be AUX ddj-151 45 11 identified identify VERB ddj-151 45 12 as as ADP ddj-151 45 13 triangles triangle NOUN ddj-151 45 14 ( ( PUNCT ddj-151 45 15 3 3 NUM ddj-151 45 16 - - PUNCT ddj-151 45 17 chords chord NOUN ddj-151 45 18 ) ) PUNCT ddj-151 45 19 at at ADP ddj-151 45 20 the the DET ddj-151 45 21 orbit orbit NOUN ddj-151 45 22 of of ADP ddj-151 45 23 triangle triangle NOUN ddj-151 45 24 representing represent VERB ddj-151 45 25 the the DET ddj-151 45 26 amino amino NOUN ddj-151 45 27 - - PUNCT ddj-151 45 28 acid acid NOUN ddj-151 45 29 under under ADP ddj-151 45 30 the the DET ddj-151 45 31 symmetry symmetry NOUN ddj-151 45 32 group group NOUN ddj-151 45 33 of of ADP ddj-151 45 34 the the DET ddj-151 45 35 hamiltonian hamiltonian ADJ ddj-151 45 36 cycle cycle NOUN ddj-151 45 37 . . PUNCT ddj-151 46 1 the the DET ddj-151 46 2 predictions prediction NOUN ddj-151 46 3 for for ADP ddj-151 46 4 the the DET ddj-151 46 5 numbers number NOUN ddj-151 46 6 of of ADP ddj-151 46 7 dnas dna NOUN ddj-151 46 8 coding code VERB ddj-151 46 9 given give VERB ddj-151 46 10 amino amino NOUN ddj-151 46 11 - - PUNCT ddj-151 46 12 acid acid NOUN ddj-151 46 13 are be AUX ddj-151 46 14 highly highly ADV ddj-151 46 15 suggestive suggestive ADJ ddj-151 46 16 for for ADP ddj-151 46 17 the the DET ddj-151 46 18 vertebrate vertebrate ADJ ddj-151 46 19 genetic genetic ADJ ddj-151 46 20 code code NOUN ddj-151 46 21 . . PUNCT ddj-151 47 1 by by ADP ddj-151 47 2 gluing glue VERB ddj-151 47 3 to to ADP ddj-151 47 4 the the DET ddj-151 47 5 icosahedron icosahedron ADJ ddj-151 47 6 tetrahedron tetrahedron NOUN ddj-151 47 7 along along ADP ddj-151 47 8 common common ADJ ddj-151 47 9 face face NOUN ddj-151 47 10 one one NUM ddj-151 47 11 obtains obtain VERB ddj-151 47 12 4 4 NUM ddj-151 47 13 more more ADJ ddj-151 47 14 codons codon NOUN ddj-151 47 15 and and CCONJ ddj-151 47 16 two two NUM ddj-151 47 17 slightly slightly ADV ddj-151 47 18 different different ADJ ddj-151 47 19 codes code NOUN ddj-151 47 20 are be AUX ddj-151 47 21 the the DET ddj-151 47 22 outcome outcome NOUN ddj-151 47 23 . . PUNCT ddj-151 48 1 also also ADV ddj-151 48 2 the the DET ddj-151 48 3 2 2 NUM ddj-151 48 4 amino amino NOUN ddj-151 48 5 - - PUNCT ddj-151 48 6 acids acids NOUN ddj-151 48 7 pyl pyl NOUN ddj-151 48 8 and and CCONJ ddj-151 48 9 sec sec PROPN ddj-151 48 10 can can AUX ddj-151 48 11 be be AUX ddj-151 48 12 understood understand VERB ddj-151 48 13 . . PUNCT ddj-151 49 1 one one PRON ddj-151 49 2 can can AUX ddj-151 49 3 also also ADV ddj-151 49 4 regard regard VERB ddj-151 49 5 the the DET ddj-151 49 6 tetrahedral tetrahedral ADJ ddj-151 49 7 4 4 NUM ddj-151 49 8 chord chord NOUN ddj-151 49 9 harmony harmony NOUN ddj-151 49 10 as as ADP ddj-151 49 11 additional additional ADJ ddj-151 49 12 harmony harmony NOUN ddj-151 49 13 so so SCONJ ddj-151 49 14 that that SCONJ ddj-151 49 15 one one PRON ddj-151 49 16 would would AUX ddj-151 49 17 have have VERB ddj-151 49 18 fusion fusion NOUN ddj-151 49 19 of of ADP ddj-151 49 20 four four NUM ddj-151 49 21 harmonies harmony NOUN ddj-151 49 22 . . PUNCT ddj-151 50 1 one one NUM ddj-151 50 2 can can AUX ddj-151 50 3 of of ADP ddj-151 50 4 course course NOUN ddj-151 50 5 criticize criticize VERB ddj-151 50 6 the the DET ddj-151 50 7 addition addition NOUN ddj-151 50 8 of of ADP ddj-151 50 9 tetrahedron tetrahedron NOUN ddj-151 50 10 as as ADP ddj-151 50 11 a a DET ddj-151 50 12 dirty dirty ADJ ddj-151 50 13 trick trick NOUN ddj-151 50 14 to to PART ddj-151 50 15 get get VERB ddj-151 50 16 genetic genetic ADJ ddj-151 50 17 code code NOUN ddj-151 50 18 . . PUNCT ddj-151 51 1 the the DET ddj-151 51 2 explicit explicit ADJ ddj-151 51 3 study study NOUN ddj-151 51 4 of of ADP ddj-151 51 5 the the DET ddj-151 51 6 chords chord NOUN ddj-151 51 7 of of ADP ddj-151 51 8 bio bio NOUN ddj-151 51 9 - - NOUN ddj-151 51 10 harmony harmony NOUN ddj-151 51 11 however however ADV ddj-151 51 12 shows show VERB ddj-151 51 13 that that SCONJ ddj-151 51 14 the the DET ddj-151 51 15 chords chord NOUN ddj-151 51 16 do do AUX ddj-151 51 17 not not PART ddj-151 51 18 contain contain VERB ddj-151 51 19 the the DET ddj-151 51 20 3 3 NUM ddj-151 51 21 - - PUNCT ddj-151 51 22 chords chord NOUN ddj-151 51 23 of of ADP ddj-151 51 24 the the DET ddj-151 51 25 standard standard ADJ ddj-151 51 26 harmonies harmony NOUN ddj-151 51 27 familiar familiar ADJ ddj-151 51 28 from from ADP ddj-151 51 29 classical classical ADJ ddj-151 51 30 music music NOUN ddj-151 51 31 ( ( PUNCT ddj-151 51 32 say say VERB ddj-151 51 33 major major ADJ ddj-151 51 34 and and CCONJ ddj-151 51 35 minor minor ADJ ddj-151 51 36 scale scale NOUN ddj-151 51 37 and and CCONJ ddj-151 51 38 corresponding correspond VERB ddj-151 51 39 chords chord NOUN ddj-151 51 40 ) ) PUNCT ddj-151 51 41 . . PUNCT ddj-151 52 1 garage garage NOUN ddj-151 52 2 band band NOUN ddj-151 52 3 experimentation experimentation NOUN ddj-151 52 4 with with ADP ddj-151 52 5 random random ADJ ddj-151 52 6 sequences sequence NOUN ddj-151 52 7 of of ADP ddj-151 52 8 chords chord NOUN ddj-151 52 9 requiring require VERB ddj-151 52 10 conservability conservability NOUN ddj-151 52 11 that that SCONJ ddj-151 52 12 two two NUM ddj-151 52 13 subsequent subsequent ADJ ddj-151 52 14 chords chord NOUN ddj-151 52 15 have have VERB ddj-151 52 16 at at ADV ddj-151 52 17 least least ADV ddj-151 52 18 one one NUM ddj-151 52 19 common common ADJ ddj-151 52 20 note note NOUN ddj-151 52 21 however however ADV ddj-151 52 22 shows show VERB ddj-151 52 23 that that SCONJ ddj-151 52 24 these these DET ddj-151 52 25 harmonies harmony NOUN ddj-151 52 26 are be AUX ddj-151 52 27 at at ADP ddj-151 52 28 least least ADJ ddj-151 52 29 to to ADP ddj-151 52 30 my my PRON ddj-151 52 31 opinion opinion NOUN ddj-151 52 32 aesthetically aesthetically ADV ddj-151 52 33 feasible feasible ADJ ddj-151 52 34 although although SCONJ ddj-151 52 35 somewhat somewhat ADV ddj-151 52 36 boring boring ADJ ddj-151 52 37 . . PUNCT ddj-151 53 1 1.2 1.2 NUM ddj-151 53 2 explanation explanation NOUN ddj-151 53 3 for for ADP ddj-151 53 4 the the DET ddj-151 53 5 number number NOUN ddj-151 53 6 12 12 NUM ddj-151 53 7 of of ADP ddj-151 53 8 notes note NOUN ddj-151 53 9 of of ADP ddj-151 53 10 12 12 NUM ddj-151 53 11 - - PUNCT ddj-151 53 12 note note NOUN ddj-151 53 13 scale scale NOUN ddj-151 53 14 one one PRON ddj-151 53 15 also also ADV ddj-151 53 16 ends end VERB ddj-151 53 17 up up ADP ddj-151 53 18 to to ADP ddj-151 53 19 an an DET ddj-151 53 20 argument argument NOUN ddj-151 53 21 explaining explain VERB ddj-151 53 22 the the DET ddj-151 53 23 number number NOUN ddj-151 53 24 12 12 NUM ddj-151 53 25 for for ADP ddj-151 53 26 the the DET ddj-151 53 27 notes note NOUN ddj-151 53 28 of of ADP ddj-151 53 29 the the DET ddj-151 53 30 12 12 NUM ddj-151 53 31 - - PUNCT ddj-151 53 32 note note NOUN ddj-151 53 33 scale scale NOUN ddj-151 53 34 [ [ X ddj-151 53 35 3 3 NUM ddj-151 53 36 ] ] PUNCT ddj-151 53 37 . . PUNCT ddj-151 54 1 there there PRON ddj-151 54 2 is be VERB ddj-151 54 3 also also ADV ddj-151 54 4 second second ADJ ddj-151 54 5 representation representation NOUN ddj-151 54 6 of of ADP ddj-151 54 7 genetic genetic ADJ ddj-151 54 8 code code NOUN ddj-151 54 9 provided provide VERB ddj-151 54 10 by by ADP ddj-151 54 11 dark dark ADJ ddj-151 54 12 proton proton NOUN ddj-151 54 13 triplets triplet NOUN ddj-151 54 14 . . PUNCT ddj-151 55 1 the the DET ddj-151 55 2 dark dark ADJ ddj-151 55 3 proton proton NOUN ddj-151 55 4 triplets triplet NOUN ddj-151 55 5 representing represent VERB ddj-151 55 6 dark dark ADJ ddj-151 55 7 genetic genetic ADJ ddj-151 55 8 codons codon NOUN ddj-151 55 9 are be AUX ddj-151 55 10 in in ADP ddj-151 55 11 one one NUM ddj-151 55 12 - - PUNCT ddj-151 55 13 one one NUM ddj-151 55 14 correspondence correspondence NOUN ddj-151 55 15 with with ADP ddj-151 55 16 ordinary ordinary ADJ ddj-151 55 17 dna dna PROPN ddj-151 55 18 codons codon NOUN ddj-151 55 19 . . PUNCT ddj-151 56 1 also also ADV ddj-151 56 2 aminoacids aminoacid NOUN ddj-151 56 3 , , PUNCT ddj-151 56 4 rna rna NOUN ddj-151 56 5 and and CCONJ ddj-151 56 6 trna trna PROPN ddj-151 56 7 have have VERB ddj-151 56 8 analogs analog NOUN ddj-151 56 9 as as ADP ddj-151 56 10 states state NOUN ddj-151 56 11 of of ADP ddj-151 56 12 3 3 NUM ddj-151 56 13 dark dark ADJ ddj-151 56 14 protons proton NOUN ddj-151 56 15 . . PUNCT ddj-151 57 1 the the DET ddj-151 57 2 number number NOUN ddj-151 57 3 of of ADP ddj-151 57 4 trnas trnas NOUN ddj-151 57 5 is be AUX ddj-151 57 6 predicted predict VERB ddj-151 57 7 to to PART ddj-151 57 8 be be AUX ddj-151 57 9 40 40 NUM ddj-151 57 10 . . PUNCT ddj-151 58 1 the the DET ddj-151 58 2 dark dark ADJ ddj-151 58 3 codons codon NOUN ddj-151 58 4 represent represent VERB ddj-151 58 5 entangled entangle VERB ddj-151 58 6 states state NOUN ddj-151 58 7 of of ADP ddj-151 58 8 protons proton NOUN ddj-151 58 9 and and CCONJ ddj-151 58 10 one one PRON ddj-151 58 11 can can AUX ddj-151 58 12 not not PART ddj-151 58 13 decompose decompose VERB ddj-151 58 14 them they PRON ddj-151 58 15 into into ADP ddj-151 58 16 a a DET ddj-151 58 17 product product NOUN ddj-151 58 18 state state NOUN ddj-151 58 19 . . PUNCT ddj-151 59 1 the the DET ddj-151 59 2 only only ADJ ddj-151 59 3 manner manner NOUN ddj-151 59 4 to to PART ddj-151 59 5 assign assign VERB ddj-151 59 6 to to ADP ddj-151 59 7 the the DET ddj-151 59 8 3 3 NUM ddj-151 59 9 - - PUNCT ddj-151 59 10 chord chord NOUN ddj-151 59 11 representing represent VERB ddj-151 59 12 the the DET ddj-151 59 13 triplet triplet NOUN ddj-151 59 14 ordinary ordinary ADJ ddj-151 59 15 dna dna PROPN ddj-151 59 16 codon codon VERB ddj-151 59 17 such such DET ddj-151 59 18 that that SCONJ ddj-151 59 19 each each DET ddj-151 59 20 letter letter NOUN ddj-151 59 21 in in ADP ddj-151 59 22 { { PUNCT ddj-151 59 23 a a PRON ddj-151 59 24 , , PUNCT ddj-151 59 25 t t PROPN ddj-151 59 26 , , PUNCT ddj-151 59 27 c c X ddj-151 59 28 , , PUNCT ddj-151 59 29 g g NOUN ddj-151 59 30 } } PUNCT ddj-151 59 31 corresponds correspond VERB ddj-151 59 32 to to ADP ddj-151 59 33 a a DET ddj-151 59 34 frequency frequency NOUN ddj-151 59 35 is be AUX ddj-151 59 36 to to PART ddj-151 59 37 assume assume VERB ddj-151 59 38 that that SCONJ ddj-151 59 39 the the DET ddj-151 59 40 frequency frequency NOUN ddj-151 59 41 depends depend VERB ddj-151 59 42 on on ADP ddj-151 59 43 the the DET ddj-151 59 44 position position NOUN ddj-151 59 45 of of ADP ddj-151 59 46 the the DET ddj-151 59 47 letter letter NOUN ddj-151 59 48 in in ADP ddj-151 59 49 the the DET ddj-151 59 50 codon codon NOUN ddj-151 59 51 . . PUNCT ddj-151 60 1 one one NOUN ddj-151 60 2 has have VERB ddj-151 60 3 altogether altogether ADV ddj-151 60 4 3×4 3×4 NUM ddj-151 60 5 = = SYM ddj-151 60 6 12 12 NUM ddj-151 60 7 frequencies frequency NOUN ddj-151 60 8 corresponding correspond VERB ddj-151 60 9 to to ADP ddj-151 60 10 3 3 NUM ddj-151 60 11 positions position NOUN ddj-151 60 12 for for ADP ddj-151 60 13 given give VERB ddj-151 60 14 letter letter NOUN ddj-151 60 15 selected select VERB ddj-151 60 16 from from ADP ddj-151 60 17 four four NUM ddj-151 60 18 letters letter NOUN ddj-151 60 19 . . PUNCT ddj-151 61 1 without without ADP ddj-151 61 2 additional additional ADJ ddj-151 61 3 conditions condition NOUN ddj-151 61 4 any any DET ddj-151 61 5 decomposition decomposition NOUN ddj-151 61 6 of of ADP ddj-151 61 7 12 12 NUM ddj-151 61 8 notes note NOUN ddj-151 61 9 of of ADP ddj-151 61 10 the the DET ddj-151 61 11 scale scale NOUN ddj-151 61 12 to to ADP ddj-151 61 13 3 3 NUM ddj-151 61 14 disjoint disjoint NOUN ddj-151 61 15 groups group NOUN ddj-151 61 16 of of ADP ddj-151 61 17 4 4 NUM ddj-151 61 18 notes note NOUN ddj-151 61 19 is be AUX ddj-151 61 20 possible possible ADJ ddj-151 61 21 and and CCONJ ddj-151 61 22 possible possible ADJ ddj-151 61 23 chords chord NOUN ddj-151 61 24 are be AUX ddj-151 61 25 obtained obtain VERB ddj-151 61 26 by by ADP ddj-151 61 27 choosing choose VERB ddj-151 61 28 one one NUM ddj-151 61 29 note note NOUN ddj-151 61 30 from from ADP ddj-151 61 31 each each DET ddj-151 61 32 group group NOUN ddj-151 61 33 . . PUNCT ddj-151 62 1 the the DET ddj-151 62 2 most most ADV ddj-151 62 3 symmetric symmetric ADJ ddj-151 62 4 choice choice NOUN ddj-151 62 5 assigns assign NOUN ddj-151 62 6 to to ADP ddj-151 62 7 the the DET ddj-151 62 8 4 4 NUM ddj-151 62 9 letters letter NOUN ddj-151 62 10 the the DET ddj-151 62 11 notes note NOUN ddj-151 62 12 { { PUNCT ddj-151 62 13 c c NOUN ddj-151 62 14 , , PUNCT ddj-151 62 15 c],d c],d X ddj-151 62 16 , , PUNCT ddj-151 62 17 d d X ddj-151 62 18 ] ] X ddj-151 62 19 } } PUNCT ddj-151 62 20 in in ADP ddj-151 62 21 the the DET ddj-151 62 22 first first ADJ ddj-151 62 23 position position NOUN ddj-151 62 24 , , PUNCT ddj-151 62 25 { { PUNCT ddj-151 62 26 e e NOUN ddj-151 62 27 , , PUNCT ddj-151 62 28 f f PROPN ddj-151 62 29 , , PUNCT ddj-151 62 30 f f X ddj-151 62 31 ] ] X ddj-151 62 32 , , PUNCT ddj-151 62 33 g g NOUN ddj-151 62 34 } } PUNCT ddj-151 62 35 in in ADP ddj-151 62 36 the the DET ddj-151 62 37 second second ADJ ddj-151 62 38 position position NOUN ddj-151 62 39 , , PUNCT ddj-151 62 40 and and CCONJ ddj-151 62 41 { { PUNCT ddj-151 62 42 g],a g],a NOUN ddj-151 62 43 , , PUNCT ddj-151 62 44 b[,b b[,b PROPN ddj-151 62 45 } } PUNCT ddj-151 62 46 in in ADP ddj-151 62 47 the the DET ddj-151 62 48 third third ADJ ddj-151 62 49 position position NOUN ddj-151 62 50 . . PUNCT ddj-151 63 1 the the DET ddj-151 63 2 codons codon NOUN ddj-151 63 3 of of ADP ddj-151 63 4 type type NOUN ddj-151 63 5 xxx xxx NOUN ddj-151 63 6 would would AUX ddj-151 63 7 correspond correspond VERB ddj-151 63 8 to to ADP ddj-151 63 9 ceg ceg VERB ddj-151 63 10 ] ] PUNCT ddj-151 63 11 or or CCONJ ddj-151 63 12 its its PRON ddj-151 63 13 transpose transpose NOUN ddj-151 63 14 . . PUNCT ddj-151 64 1 one one PRON ddj-151 64 2 can can AUX ddj-151 64 3 transpose transpose VERB ddj-151 64 4 this this DET ddj-151 64 5 proposal proposal NOUN ddj-151 64 6 and and CCONJ ddj-151 64 7 there there PRON ddj-151 64 8 are be VERB ddj-151 64 9 4 4 NUM ddj-151 64 10 non non ADJ ddj-151 64 11 - - ADJ ddj-151 64 12 quivalent quivalent ADJ ddj-151 64 13 transposes transpose VERB ddj-151 64 14 , , PUNCT ddj-151 64 15 which which PRON ddj-151 64 16 could could AUX ddj-151 64 17 be be AUX ddj-151 64 18 seen see VERB ddj-151 64 19 as as ADP ddj-151 64 20 analogs analog NOUN ddj-151 64 21 of of ADP ddj-151 64 22 music music NOUN ddj-151 64 23 keys key NOUN ddj-151 64 24 . . PUNCT ddj-151 65 1 remark remark NOUN ddj-151 65 2 : : PUNCT ddj-151 66 1 ceg ceg X ddj-151 66 2 ] ] PUNCT ddj-151 66 3 between between ADP ddj-151 66 4 c c NOUN ddj-151 66 5 - - PUNCT ddj-151 66 6 major major ADJ ddj-151 66 7 and and CCONJ ddj-151 66 8 a a DET ddj-151 66 9 - - PUNCT ddj-151 66 10 minor minor ADJ ddj-151 66 11 very very ADV ddj-151 66 12 often often ADV ddj-151 66 13 finishes finish VERB ddj-151 66 14 finnish finnish ADJ ddj-151 66 15 tango tango NOUN ddj-151 66 16 : : PUNCT ddj-151 66 17 something something PRON ddj-151 66 18 neither neither CCONJ ddj-151 66 19 sad sad ADJ ddj-151 66 20 nor nor CCONJ ddj-151 66 21 glad glad ADJ ddj-151 66 22 ! ! PUNCT ddj-151 67 1 issn issn PROPN ddj-151 67 2 : : PUNCT ddj-151 67 3 2159 2159 NUM ddj-151 67 4 - - PUNCT ddj-151 67 5 046x 046x NOUN ddj-151 67 6 dna dna NOUN ddj-151 67 7 decipher decipher PROPN ddj-151 67 8 journal journal PROPN ddj-151 67 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 67 10 published publish VERB ddj-151 67 11 by by ADP ddj-151 67 12 quantumdream quantumdream PROPN ddj-151 67 13 , , PUNCT ddj-151 67 14 inc inc PROPN ddj-151 67 15 . . PROPN ddj-151 67 16 dna dna PROPN ddj-151 67 17 decipher decipher PROPN ddj-151 67 18 journal journal PROPN ddj-151 67 19 | | ADP ddj-151 67 20 november november PROPN ddj-151 67 21 2018 2018 NUM ddj-151 67 22 | | ADV ddj-151 67 23 volume volume NOUN ddj-151 67 24 8 8 NUM ddj-151 67 25 | | NOUN ddj-151 67 26 issue issue VERB ddj-151 67 27 3 3 NUM ddj-151 67 28 | | CCONJ ddj-151 67 29 pp pp ADJ ddj-151 67 30 . . PUNCT ddj-151 68 1 197 197 NUM ddj-151 68 2 - - SYM ddj-151 68 3 205 205 NUM ddj-151 68 4 199 199 NUM ddj-151 68 5 pitkänen pitkänen NOUN ddj-151 68 6 , , PUNCT ddj-151 68 7 m. m. NOUN ddj-151 68 8 , , PUNCT ddj-151 68 9 thoughts thought NOUN ddj-151 68 10 on on ADP ddj-151 68 11 modification modification NOUN ddj-151 68 12 of of ADP ddj-151 68 13 bio bio NOUN ddj-151 68 14 - - NOUN ddj-151 68 15 harmony harmony NOUN ddj-151 68 16 one one NOUN ddj-151 68 17 can can AUX ddj-151 68 18 look look VERB ddj-151 68 19 what what PRON ddj-151 68 20 kind kind NOUN ddj-151 68 21 of of ADP ddj-151 68 22 chords chord NOUN ddj-151 68 23 one one NUM ddj-151 68 24 obtains obtain NOUN ddj-151 68 25 . . PUNCT ddj-151 69 1 1 1 X ddj-151 69 2 . . X ddj-151 69 3 chords chord NOUN ddj-151 69 4 containing contain VERB ddj-151 69 5 notes note NOUN ddj-151 69 6 associated associate VERB ddj-151 69 7 with with ADP ddj-151 69 8 the the DET ddj-151 69 9 same same ADJ ddj-151 69 10 position position NOUN ddj-151 69 11 in in ADP ddj-151 69 12 codon codon NOUN ddj-151 69 13 are be AUX ddj-151 69 14 not not PART ddj-151 69 15 possible possible ADJ ddj-151 69 16 . . PUNCT ddj-151 70 1 2 2 X ddj-151 70 2 . . PUNCT ddj-151 70 3 given give VERB ddj-151 70 4 note note NOUN ddj-151 70 5 belongs belong VERB ddj-151 70 6 to to ADP ddj-151 70 7 6 6 NUM ddj-151 70 8 chords chord NOUN ddj-151 70 9 . . PUNCT ddj-151 71 1 in in ADP ddj-151 71 2 the the DET ddj-151 71 3 icosahedral icosahedral ADJ ddj-151 71 4 harmony harmony NOUN ddj-151 71 5 with with ADP ddj-151 71 6 20 20 NUM ddj-151 71 7 chords chord NOUN ddj-151 71 8 given give VERB ddj-151 71 9 note note NOUN ddj-151 71 10 belongs belong VERB ddj-151 71 11 to to ADP ddj-151 71 12 5 5 NUM ddj-151 71 13 chords chord NOUN ddj-151 71 14 ( ( PUNCT ddj-151 71 15 there there PRON ddj-151 71 16 are be VERB ddj-151 71 17 5 5 NUM ddj-151 71 18 triangles triangle NOUN ddj-151 71 19 containing contain VERB ddj-151 71 20 given give VERB ddj-151 71 21 vertex vertex NOUN ddj-151 71 22 ) ) PUNCT ddj-151 71 23 . . PUNCT ddj-151 72 1 therefore therefore ADV ddj-151 72 2 the the DET ddj-151 72 3 harmony harmony NOUN ddj-151 72 4 in in ADP ddj-151 72 5 question question NOUN ddj-151 72 6 can can AUX ddj-151 72 7 not not PART ddj-151 72 8 be be AUX ddj-151 72 9 equivalent equivalent ADJ ddj-151 72 10 with with ADP ddj-151 72 11 20 20 NUM ddj-151 72 12 - - PUNCT ddj-151 72 13 chord chord NOUN ddj-151 72 14 icosahedral icosahedral ADJ ddj-151 72 15 harmony harmony NOUN ddj-151 72 16 . . PUNCT ddj-151 73 1 neither neither CCONJ ddj-151 73 2 can can AUX ddj-151 73 3 the the DET ddj-151 73 4 bio bio NOUN ddj-151 73 5 - - NOUN ddj-151 73 6 harmony harmony NOUN ddj-151 73 7 with with ADP ddj-151 73 8 64 64 NUM ddj-151 73 9 chords chord NOUN ddj-151 73 10 satisfy satisfy VERB ddj-151 73 11 the the DET ddj-151 73 12 condition condition NOUN ddj-151 73 13 that that SCONJ ddj-151 73 14 given give VERB ddj-151 73 15 note note NOUN ddj-151 73 16 is be AUX ddj-151 73 17 contained contain VERB ddj-151 73 18 by by ADP ddj-151 73 19 6 6 NUM ddj-151 73 20 3 3 NUM ddj-151 73 21 - - PUNCT ddj-151 73 22 chords chord NOUN ddj-151 73 23 . . PUNCT ddj-151 74 1 3 3 X ddj-151 74 2 . . X ddj-151 74 3 first first ADJ ddj-151 74 4 and and CCONJ ddj-151 74 5 second second ADJ ddj-151 74 6 notes note NOUN ddj-151 74 7 of of ADP ddj-151 74 8 the the DET ddj-151 74 9 chords chord NOUN ddj-151 74 10 are be AUX ddj-151 74 11 separated separate VERB ddj-151 74 12 by by ADP ddj-151 74 13 at at ADV ddj-151 74 14 least least ADJ ddj-151 74 15 major major ADJ ddj-151 74 16 third third ADV ddj-151 74 17 as as ADP ddj-151 74 18 also also ADV ddj-151 74 19 those those DET ddj-151 74 20 second second ADJ ddj-151 74 21 and and CCONJ ddj-151 74 22 third third ADJ ddj-151 74 23 notes note NOUN ddj-151 74 24 . . PUNCT ddj-151 75 1 the the DET ddj-151 75 2 chords chord NOUN ddj-151 75 3 satisfy satisfy VERB ddj-151 75 4 however however ADV ddj-151 75 5 octave octave NOUN ddj-151 75 6 equivalence equivalence NOUN ddj-151 75 7 so so SCONJ ddj-151 75 8 that that SCONJ ddj-151 75 9 the the DET ddj-151 75 10 distance distance NOUN ddj-151 75 11 between between ADP ddj-151 75 12 the the DET ddj-151 75 13 first first ADJ ddj-151 75 14 and and CCONJ ddj-151 75 15 third third ADJ ddj-151 75 16 notes note NOUN ddj-151 75 17 can can AUX ddj-151 75 18 be be AUX ddj-151 75 19 smaller small ADJ ddj-151 75 20 even even ADV ddj-151 75 21 half half ADJ ddj-151 75 22 step step NOUN ddj-151 75 23 and and CCONJ ddj-151 75 24 one one NUM ddj-151 75 25 finds find VERB ddj-151 75 26 that that SCONJ ddj-151 75 27 one one PRON ddj-151 75 28 can can AUX ddj-151 75 29 get get VERB ddj-151 75 30 the the DET ddj-151 75 31 basic basic ADJ ddj-151 75 32 chords chord NOUN ddj-151 75 33 a a DET ddj-151 75 34 - - PUNCT ddj-151 75 35 minor minor ADJ ddj-151 75 36 scale scale NOUN ddj-151 75 37 : : PUNCT ddj-151 75 38 am be AUX ddj-151 75 39 , , PUNCT ddj-151 75 40 dm dm PROPN ddj-151 75 41 , , PUNCT ddj-151 75 42 e7 e7 PROPN ddj-151 75 43 , , PUNCT ddj-151 75 44 and and CCONJ ddj-151 75 45 also also ADV ddj-151 75 46 g g PROPN ddj-151 75 47 and and CCONJ ddj-151 75 48 f. f. PROPN ddj-151 75 49 also also ADV ddj-151 75 50 the the DET ddj-151 75 51 basic basic ADJ ddj-151 75 52 chords chord NOUN ddj-151 75 53 of of ADP ddj-151 75 54 f f NOUN ddj-151 75 55 - - PUNCT ddj-151 75 56 major major ADJ ddj-151 75 57 scale scale NOUN ddj-151 75 58 can can AUX ddj-151 75 59 be be AUX ddj-151 75 60 represented represent VERB ddj-151 75 61 . . PUNCT ddj-151 76 1 also also ADV ddj-151 76 2 the the DET ddj-151 76 3 transposes transpose NOUN ddj-151 76 4 of of ADP ddj-151 76 5 these these DET ddj-151 76 6 scales scale NOUN ddj-151 76 7 by by ADP ddj-151 76 8 2 2 NUM ddj-151 76 9 whole whole ADJ ddj-151 76 10 steps step NOUN ddj-151 76 11 can can AUX ddj-151 76 12 be be AUX ddj-151 76 13 represented represent VERB ddj-151 76 14 so so SCONJ ddj-151 76 15 that that SCONJ ddj-151 76 16 one one PRON ddj-151 76 17 obtains obtain VERB ddj-151 76 18 am be AUX ddj-151 76 19 , , PUNCT ddj-151 76 20 c]m c]m VERB ddj-151 76 21 , , PUNCT ddj-151 76 22 fm fm PROPN ddj-151 76 23 and and CCONJ ddj-151 76 24 corresponding correspond VERB ddj-151 76 25 major major ADJ ddj-151 76 26 scales scale NOUN ddj-151 76 27 . . PUNCT ddj-151 77 1 these these DET ddj-151 77 2 harmonies harmony NOUN ddj-151 77 3 could could AUX ddj-151 77 4 allow allow VERB ddj-151 77 5 the the DET ddj-151 77 6 harmonies harmony NOUN ddj-151 77 7 of of ADP ddj-151 77 8 classical classical ADJ ddj-151 77 9 and and CCONJ ddj-151 77 10 popular popular ADJ ddj-151 77 11 music music NOUN ddj-151 77 12 . . PUNCT ddj-151 78 1 these these DET ddj-151 78 2 observations observation NOUN ddj-151 78 3 encourage encourage VERB ddj-151 78 4 to to PART ddj-151 78 5 ask ask VERB ddj-151 78 6 whether whether SCONJ ddj-151 78 7 a a DET ddj-151 78 8 representation representation NOUN ddj-151 78 9 of of ADP ddj-151 78 10 the the DET ddj-151 78 11 new new ADJ ddj-151 78 12 harmonies harmony NOUN ddj-151 78 13 as as ADP ddj-151 78 14 hamiltonian hamiltonian ADJ ddj-151 78 15 cycles cycle NOUN ddj-151 78 16 of of ADP ddj-151 78 17 some some DET ddj-151 78 18 tesselation tesselation NOUN ddj-151 78 19 could could AUX ddj-151 78 20 exist exist VERB ddj-151 78 21 . . PUNCT ddj-151 79 1 the the DET ddj-151 79 2 tesselation tesselation NOUN ddj-151 79 3 should should AUX ddj-151 79 4 be be AUX ddj-151 79 5 such such ADJ ddj-151 79 6 that that SCONJ ddj-151 79 7 6 6 NUM ddj-151 79 8 triangles triangle NOUN ddj-151 79 9 meet meet VERB ddj-151 79 10 at at ADP ddj-151 79 11 given give VERB ddj-151 79 12 vertex vertex NOUN ddj-151 79 13 . . PUNCT ddj-151 80 1 triangular triangular NOUN ddj-151 80 2 tesselation tesselation NOUN ddj-151 80 3 of of ADP ddj-151 80 4 torus torus NOUN ddj-151 80 5 having have VERB ddj-151 80 6 interpretation interpretation NOUN ddj-151 80 7 in in ADP ddj-151 80 8 terms term NOUN ddj-151 80 9 of of ADP ddj-151 80 10 a a DET ddj-151 80 11 planar planar ADJ ddj-151 80 12 parallelogram parallelogram NOUN ddj-151 80 13 ( ( PUNCT ddj-151 80 14 or or CCONJ ddj-151 80 15 perhaps perhaps ADV ddj-151 80 16 more more ADV ddj-151 80 17 general general ADJ ddj-151 80 18 planar planar ADJ ddj-151 80 19 region region NOUN ddj-151 80 20 ) ) PUNCT ddj-151 80 21 with with ADP ddj-151 80 22 edges edge NOUN ddj-151 80 23 at at ADP ddj-151 80 24 the the DET ddj-151 80 25 boundary boundary ADJ ddj-151 80 26 suitable suitable ADJ ddj-151 80 27 identified identify VERB ddj-151 80 28 to to PART ddj-151 80 29 obtain obtain VERB ddj-151 80 30 torus torus NOUN ddj-151 80 31 topology topology NOUN ddj-151 80 32 seems seem VERB ddj-151 80 33 to to PART ddj-151 80 34 be be AUX ddj-151 80 35 the the DET ddj-151 80 36 natural natural ADJ ddj-151 80 37 option option NOUN ddj-151 80 38 . . PUNCT ddj-151 81 1 clearly clearly ADV ddj-151 81 2 this this DET ddj-151 81 3 region region NOUN ddj-151 81 4 would would AUX ddj-151 81 5 correspond correspond VERB ddj-151 81 6 to to ADP ddj-151 81 7 a a DET ddj-151 81 8 planar planar ADJ ddj-151 81 9 lattice lattice NOUN ddj-151 81 10 with with ADP ddj-151 81 11 periodic periodic ADJ ddj-151 81 12 boundary boundary ADJ ddj-151 81 13 conditions condition NOUN ddj-151 81 14 . . PUNCT ddj-151 82 1 2 2 NUM ddj-151 82 2 is be AUX ddj-151 82 3 it it PRON ddj-151 82 4 possible possible ADJ ddj-151 82 5 to to PART ddj-151 82 6 have have VERB ddj-151 82 7 toric toric ADJ ddj-151 82 8 harmonies harmony NOUN ddj-151 82 9 ? ? PUNCT ddj-151 83 1 the the DET ddj-151 83 2 basic basic ADJ ddj-151 83 3 question question NOUN ddj-151 83 4 is be AUX ddj-151 83 5 whether whether SCONJ ddj-151 83 6 one one PRON ddj-151 83 7 can can AUX ddj-151 83 8 have have VERB ddj-151 83 9 a a DET ddj-151 83 10 representation representation NOUN ddj-151 83 11 of of ADP ddj-151 83 12 the the DET ddj-151 83 13 new new ADJ ddj-151 83 14 candidate candidate NOUN ddj-151 83 15 for for ADP ddj-151 83 16 harmonies harmony NOUN ddj-151 83 17 in in ADP ddj-151 83 18 terms term NOUN ddj-151 83 19 of of ADP ddj-151 83 20 a a DET ddj-151 83 21 tesselation tesselation NOUN ddj-151 83 22 of of ADP ddj-151 83 23 torus torus NOUN ddj-151 83 24 having have VERB ddj-151 83 25 v v NOUN ddj-151 83 26 = = SYM ddj-151 83 27 12 12 NUM ddj-151 83 28 vertices vertex NOUN ddj-151 83 29 and and CCONJ ddj-151 83 30 f f NOUN ddj-151 83 31 = = SYM ddj-151 83 32 20 20 NUM ddj-151 83 33 triangular triangular NOUN ddj-151 83 34 faces face VERB ddj-151 83 35 . . PUNCT ddj-151 84 1 the the DET ddj-151 84 2 reading reading NOUN ddj-151 84 3 of of ADP ddj-151 84 4 the the DET ddj-151 84 5 article article NOUN ddj-151 84 6 ” " PUNCT ddj-151 84 7 equivelar equivelar ADJ ddj-151 84 8 maps map NOUN ddj-151 84 9 on on ADP ddj-151 84 10 the the DET ddj-151 84 11 torus torus NOUN ddj-151 84 12 ” " PUNCT ddj-151 85 1 [ [ X ddj-151 85 2 1 1 NUM ddj-151 85 3 ] ] PUNCT ddj-151 85 4 ( ( PUNCT ddj-151 85 5 see see VERB ddj-151 85 6 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 85 7 ) ) PUNCT ddj-151 85 8 discussing discuss VERB ddj-151 85 9 toric toric ADJ ddj-151 85 10 tesselations tesselation NOUN ddj-151 85 11 makes make VERB ddj-151 85 12 clear clear ADJ ddj-151 85 13 that that SCONJ ddj-151 85 14 this this PRON ddj-151 85 15 is be AUX ddj-151 85 16 impossible impossible ADJ ddj-151 85 17 . . PUNCT ddj-151 86 1 one one NUM ddj-151 86 2 however however ADV ddj-151 86 3 have have VERB ddj-151 86 4 ( ( PUNCT ddj-151 86 5 v v NOUN ddj-151 86 6 , , PUNCT ddj-151 86 7 f f PROPN ddj-151 86 8 ) ) PUNCT ddj-151 86 9 = = PUNCT ddj-151 87 1 ( ( PUNCT ddj-151 87 2 12 12 NUM ddj-151 87 3 , , PUNCT ddj-151 87 4 24 24 NUM ddj-151 87 5 ) ) PUNCT ddj-151 87 6 ( ( PUNCT ddj-151 87 7 see see VERB ddj-151 87 8 http://tinyurl.com/ http://tinyurl.com/ PROPN ddj-151 87 9 y7xfromc y7xfromc PROPN ddj-151 87 10 ) ) PUNCT ddj-151 87 11 . . PUNCT ddj-151 88 1 a a DET ddj-151 88 2 rather rather ADV ddj-151 88 3 promising promising ADJ ddj-151 88 4 realization realization NOUN ddj-151 88 5 of of ADP ddj-151 88 6 the the DET ddj-151 88 7 genetic genetic ADJ ddj-151 88 8 code code NOUN ddj-151 88 9 in in ADP ddj-151 88 10 terms term NOUN ddj-151 88 11 of of ADP ddj-151 88 12 bio bio NOUN ddj-151 88 13 - - NOUN ddj-151 88 14 harmony harmony NOUN ddj-151 88 15 would would AUX ddj-151 88 16 be be AUX ddj-151 88 17 as as ADP ddj-151 88 18 a a DET ddj-151 88 19 fusion fusion NOUN ddj-151 88 20 of of ADP ddj-151 88 21 two two NUM ddj-151 88 22 icosahedral icosahedral ADJ ddj-151 88 23 harmonies harmony NOUN ddj-151 88 24 and and CCONJ ddj-151 88 25 toric toric ADJ ddj-151 88 26 harmony harmony NOUN ddj-151 88 27 with with ADP ddj-151 88 28 ( ( PUNCT ddj-151 88 29 v v NOUN ddj-151 88 30 , , PUNCT ddj-151 88 31 f f NOUN ddj-151 88 32 ) ) PUNCT ddj-151 89 1 = = PUNCT ddj-151 89 2 ( ( PUNCT ddj-151 89 3 12 12 NUM ddj-151 89 4 , , PUNCT ddj-151 89 5 24 24 NUM ddj-151 89 6 ) ) PUNCT ddj-151 89 7 . . PUNCT ddj-151 90 1 this this PRON ddj-151 90 2 in in ADP ddj-151 90 3 principle principle NOUN ddj-151 90 4 allows allow VERB ddj-151 90 5 also also ADV ddj-151 90 6 to to PART ddj-151 90 7 have have VERB ddj-151 90 8 24 24 NUM ddj-151 90 9 3 3 NUM ddj-151 90 10 - - PUNCT ddj-151 90 11 chords chord NOUN ddj-151 90 12 which which PRON ddj-151 90 13 can can AUX ddj-151 90 14 realize realize VERB ddj-151 90 15 classical classical ADJ ddj-151 90 16 harmony harmony NOUN ddj-151 90 17 ( ( PUNCT ddj-151 90 18 major major ADJ ddj-151 90 19 / / SYM ddj-151 90 20 minor minor ADJ ddj-151 90 21 scale scale NOUN ddj-151 90 22 ) ) PUNCT ddj-151 90 23 . . PUNCT ddj-151 91 1 1 1 X ddj-151 91 2 . . X ddj-151 92 1 the the DET ddj-151 92 2 local local ADJ ddj-151 92 3 properties property NOUN ddj-151 92 4 of of ADP ddj-151 92 5 the the DET ddj-151 92 6 tesselations tesselation NOUN ddj-151 92 7 for for ADP ddj-151 92 8 any any DET ddj-151 92 9 topology topology NOUN ddj-151 92 10 are be AUX ddj-151 92 11 characterized characterize VERB ddj-151 92 12 by by ADP ddj-151 92 13 a a DET ddj-151 92 14 pair pair NOUN ddj-151 92 15 ( ( PUNCT ddj-151 92 16 m m NOUN ddj-151 92 17 , , PUNCT ddj-151 92 18 n n CCONJ ddj-151 92 19 ) ) PUNCT ddj-151 92 20 of of ADP ddj-151 92 21 positive positive ADJ ddj-151 92 22 integers integer NOUN ddj-151 92 23 . . PUNCT ddj-151 93 1 m m PROPN ddj-151 93 2 is be AUX ddj-151 93 3 the the DET ddj-151 93 4 number number NOUN ddj-151 93 5 of of ADP ddj-151 93 6 edges edge NOUN ddj-151 93 7 meeting meeting NOUN ddj-151 93 8 in in ADP ddj-151 93 9 given give VERB ddj-151 93 10 vertex vertex NOUN ddj-151 93 11 ( ( PUNCT ddj-151 93 12 valence valence NOUN ddj-151 93 13 ) ) PUNCT ddj-151 93 14 and and CCONJ ddj-151 93 15 n n PRON ddj-151 93 16 is be AUX ddj-151 93 17 the the DET ddj-151 93 18 number number NOUN ddj-151 93 19 of of ADP ddj-151 93 20 edges edge NOUN ddj-151 93 21 and and CCONJ ddj-151 93 22 vertices vertex NOUN ddj-151 93 23 for for ADP ddj-151 93 24 the the DET ddj-151 93 25 face face NOUN ddj-151 93 26 . . PUNCT ddj-151 94 1 now now ADV ddj-151 94 2 one one PRON ddj-151 94 3 has have VERB ddj-151 94 4 ( ( PUNCT ddj-151 94 5 m m PROPN ddj-151 94 6 , , PUNCT ddj-151 94 7 n n CCONJ ddj-151 94 8 ) ) PUNCT ddj-151 94 9 = = SYM ddj-151 94 10 ( ( PUNCT ddj-151 94 11 6 6 NUM ddj-151 94 12 , , PUNCT ddj-151 94 13 3 3 NUM ddj-151 94 14 ) ) PUNCT ddj-151 94 15 . . PUNCT ddj-151 95 1 the the DET ddj-151 95 2 dual dual ADJ ddj-151 95 3 of of ADP ddj-151 95 4 this this DET ddj-151 95 5 tesselation tesselation NOUN ddj-151 95 6 is be AUX ddj-151 95 7 hexagonal hexagonal ADJ ddj-151 95 8 tesselation tesselation NOUN ddj-151 95 9 ( ( PUNCT ddj-151 95 10 m m NOUN ddj-151 95 11 , , PUNCT ddj-151 95 12 n n CCONJ ddj-151 95 13 ) ) PUNCT ddj-151 95 14 = = SYM ddj-151 95 15 ( ( PUNCT ddj-151 95 16 3 3 NUM ddj-151 95 17 , , PUNCT ddj-151 95 18 6 6 NUM ddj-151 95 19 ) ) PUNCT ddj-151 95 20 obtained obtain VERB ddj-151 95 21 by by ADP ddj-151 95 22 defining define VERB ddj-151 95 23 vertices vertex NOUN ddj-151 95 24 as as ADP ddj-151 95 25 centers center NOUN ddj-151 95 26 of of ADP ddj-151 95 27 the the DET ddj-151 95 28 triangles triangle NOUN ddj-151 95 29 so so SCONJ ddj-151 95 30 that that SCONJ ddj-151 95 31 faces faces AUX ddj-151 95 32 become become VERB ddj-151 95 33 vertices vertex NOUN ddj-151 95 34 and and CCONJ ddj-151 95 35 vice vice ADV ddj-151 95 36 versa versa ADV ddj-151 95 37 . . PUNCT ddj-151 96 1 2 2 X ddj-151 96 2 . . X ddj-151 96 3 the the DET ddj-151 96 4 rule rule NOUN ddj-151 96 5 v v ADP ddj-151 96 6 −e+f −e+f PROPN ddj-151 96 7 = = SYM ddj-151 96 8 2(1−g)−h 2(1−g)−h NOUN ddj-151 96 9 , , PUNCT ddj-151 96 10 where where SCONJ ddj-151 96 11 v v NOUN ddj-151 96 12 , , PUNCT ddj-151 96 13 e e PROPN ddj-151 96 14 and and CCONJ ddj-151 96 15 f f PROPN ddj-151 96 16 are be AUX ddj-151 96 17 the the DET ddj-151 96 18 numbers number NOUN ddj-151 96 19 of of ADP ddj-151 96 20 vertices vertex NOUN ddj-151 96 21 , , PUNCT ddj-151 96 22 edges edge NOUN ddj-151 96 23 , , PUNCT ddj-151 96 24 and and CCONJ ddj-151 96 25 faces face NOUN ddj-151 96 26 , , PUNCT ddj-151 96 27 relates relate VERB ddj-151 96 28 v v ADP ddj-151 96 29 −e−f −e−f NOUN ddj-151 96 30 to to ADP ddj-151 96 31 the the DET ddj-151 96 32 topology topology NOUN ddj-151 96 33 of of ADP ddj-151 96 34 the the DET ddj-151 96 35 graph graph NOUN ddj-151 96 36 , , PUNCT ddj-151 96 37 which which PRON ddj-151 96 38 in in ADP ddj-151 96 39 the the DET ddj-151 96 40 recent recent ADJ ddj-151 96 41 case case NOUN ddj-151 96 42 is be AUX ddj-151 96 43 triangular triangular NOUN ddj-151 96 44 tesselation tesselation NOUN ddj-151 96 45 . . PUNCT ddj-151 97 1 g g PROPN ddj-151 97 2 is be AUX ddj-151 97 3 the the DET ddj-151 97 4 genus genus NOUN ddj-151 97 5 of of ADP ddj-151 97 6 the the DET ddj-151 97 7 surface surface NOUN ddj-151 97 8 at at ADP ddj-151 97 9 which which PRON ddj-151 97 10 the the DET ddj-151 97 11 triangulation triangulation NOUN ddj-151 97 12 is be AUX ddj-151 97 13 i i PRON ddj-151 97 14 m m VERB ddj-151 97 15 eded eded ADJ ddj-151 97 16 and and CCONJ ddj-151 97 17 h h NOUN ddj-151 97 18 is be AUX ddj-151 97 19 the the DET ddj-151 97 20 number number NOUN ddj-151 97 21 of of ADP ddj-151 97 22 holes hole NOUN ddj-151 97 23 in in ADP ddj-151 97 24 it it PRON ddj-151 97 25 . . PUNCT ddj-151 98 1 in in ADP ddj-151 98 2 case case NOUN ddj-151 98 3 of of ADP ddj-151 98 4 torus torus NOUN ddj-151 98 5 one one PRON ddj-151 98 6 would would AUX ddj-151 98 7 have have VERB ddj-151 98 8 e e NOUN ddj-151 98 9 = = SYM ddj-151 98 10 v v PROPN ddj-151 98 11 + + CCONJ ddj-151 98 12 f f NOUN ddj-151 98 13 giving give VERB ddj-151 98 14 in in ADP ddj-151 98 15 the the DET ddj-151 98 16 recent recent ADJ ddj-151 98 17 case case NOUN ddj-151 98 18 e e X ddj-151 98 19 = = NOUN ddj-151 98 20 36 36 NUM ddj-151 98 21 for for ADP ddj-151 98 22 ( ( PUNCT ddj-151 98 23 v v NOUN ddj-151 98 24 , , PUNCT ddj-151 98 25 f f NOUN ddj-151 98 26 ) ) PUNCT ddj-151 99 1 = = PUNCT ddj-151 99 2 ( ( PUNCT ddj-151 99 3 12 12 NUM ddj-151 99 4 , , PUNCT ddj-151 99 5 24 24 NUM ddj-151 99 6 ) ) PUNCT ddj-151 99 7 ( ( PUNCT ddj-151 99 8 see see VERB ddj-151 99 9 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 99 10 ) ) PUNCT ddj-151 99 11 whereas whereas SCONJ ddj-151 99 12 in in ADP ddj-151 99 13 the the DET ddj-151 99 14 icosahedral icosahedral ADJ ddj-151 99 15 case case NOUN ddj-151 99 16 one one PRON ddj-151 99 17 has have VERB ddj-151 99 18 e e NOUN ddj-151 99 19 = = SYM ddj-151 99 20 32 32 NUM ddj-151 99 21 . . PUNCT ddj-151 100 1 3 3 X ddj-151 100 2 . . X ddj-151 101 1 this this DET ddj-151 101 2 kind kind NOUN ddj-151 101 3 of of ADP ddj-151 101 4 tesselations tesselation NOUN ddj-151 101 5 are be AUX ddj-151 101 6 obtained obtain VERB ddj-151 101 7 by by ADP ddj-151 101 8 applying apply VERB ddj-151 101 9 periodic periodic ADJ ddj-151 101 10 boundary boundary ADJ ddj-151 101 11 conditions condition NOUN ddj-151 101 12 to to PART ddj-151 101 13 triangular triangular VERB ddj-151 101 14 lattices lattice NOUN ddj-151 101 15 in in ADP ddj-151 101 16 plane plane NOUN ddj-151 101 17 defining define VERB ddj-151 101 18 parallelogram parallelogram NOUN ddj-151 101 19 . . PUNCT ddj-151 102 1 the the DET ddj-151 102 2 intuitive intuitive ADJ ddj-151 102 3 expectation expectation NOUN ddj-151 102 4 is be AUX ddj-151 102 5 that that SCONJ ddj-151 102 6 this this DET ddj-151 102 7 lattices lattice NOUN ddj-151 102 8 can can AUX ddj-151 102 9 be be AUX ddj-151 102 10 labelled label VERB ddj-151 102 11 by by ADP ddj-151 102 12 two two NUM ddj-151 102 13 integers integer NOUN ddj-151 102 14 ( ( PUNCT ddj-151 102 15 m m NOUN ddj-151 102 16 , , PUNCT ddj-151 102 17 n n CCONJ ddj-151 102 18 ) ) PUNCT ddj-151 102 19 characterizing characterize VERB ddj-151 102 20 the the DET ddj-151 102 21 lengths length NOUN ddj-151 102 22 of of ADP ddj-151 102 23 the the DET ddj-151 102 24 sides side NOUN ddj-151 102 25 of of ADP ddj-151 102 26 the the DET ddj-151 102 27 parallelogram parallelogram NOUN ddj-151 102 28 plus plus CCONJ ddj-151 102 29 angle angle NOUN ddj-151 102 30 between between ADP ddj-151 102 31 two two NUM ddj-151 102 32 sides side NOUN ddj-151 102 33 : : PUNCT ddj-151 102 34 this this DET ddj-151 102 35 angle angle NOUN ddj-151 102 36 defines define VERB ddj-151 102 37 the the DET ddj-151 102 38 conformal conformal ADJ ddj-151 102 39 equivalence equivalence NOUN ddj-151 102 40 class class NOUN ddj-151 102 41 of of ADP ddj-151 102 42 torus torus NOUN ddj-151 102 43 . . PUNCT ddj-151 103 1 one one NUM ddj-151 103 2 can can AUX ddj-151 103 3 also also ADV ddj-151 103 4 introduce introduce VERB ddj-151 103 5 two two NUM ddj-151 103 6 unit unit NOUN ddj-151 103 7 vectors vector NOUN ddj-151 103 8 e1 e1 VERB ddj-151 103 9 and and CCONJ ddj-151 103 10 e2 e2 NOUN ddj-151 103 11 characterizing characterize VERB ddj-151 103 12 the the DET ddj-151 103 13 conformal conformal ADJ ddj-151 103 14 equivalence equivalence NOUN ddj-151 103 15 class class NOUN ddj-151 103 16 of of ADP ddj-151 103 17 torus torus PROPN ddj-151 103 18 . . PUNCT ddj-151 104 1 issn issn PROPN ddj-151 104 2 : : PUNCT ddj-151 104 3 2159 2159 NUM ddj-151 104 4 - - PUNCT ddj-151 104 5 046x 046x NOUN ddj-151 104 6 dna dna NOUN ddj-151 104 7 decipher decipher PROPN ddj-151 104 8 journal journal PROPN ddj-151 104 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 104 10 published publish VERB ddj-151 104 11 by by ADP ddj-151 104 12 quantumdream quantumdream PROPN ddj-151 104 13 , , PUNCT ddj-151 104 14 inc inc PROPN ddj-151 104 15 . . PUNCT ddj-151 105 1 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 105 2 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 105 3 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 105 4 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 105 5 dna dna PROPN ddj-151 105 6 decipher decipher PROPN ddj-151 105 7 journal journal PROPN ddj-151 105 8 | | ADP ddj-151 105 9 november november PROPN ddj-151 105 10 2018 2018 NUM ddj-151 105 11 | | ADV ddj-151 105 12 volume volume NOUN ddj-151 105 13 8 8 NUM ddj-151 105 14 | | NOUN ddj-151 105 15 issue issue VERB ddj-151 105 16 3 3 NUM ddj-151 105 17 | | CCONJ ddj-151 105 18 pp pp ADJ ddj-151 105 19 . . PUNCT ddj-151 106 1 197 197 NUM ddj-151 106 2 - - SYM ddj-151 106 3 205 205 NUM ddj-151 106 4 200 200 NUM ddj-151 106 5 pitkänen pitkänen NOUN ddj-151 106 6 , , PUNCT ddj-151 106 7 m. m. NOUN ddj-151 106 8 , , PUNCT ddj-151 106 9 thoughts thought NOUN ddj-151 106 10 on on ADP ddj-151 106 11 modification modification NOUN ddj-151 106 12 of of ADP ddj-151 106 13 bio bio NOUN ddj-151 106 14 - - ADJ ddj-151 106 15 harmony harmony NOUN ddj-151 106 16 second second ADJ ddj-151 106 17 naive naive ADJ ddj-151 106 18 expectation expectation NOUN ddj-151 106 19 is be AUX ddj-151 106 20 that that SCONJ ddj-151 106 21 m m VERB ddj-151 106 22 × × NOUN ddj-151 106 23 n n PRON ddj-151 106 24 × × NOUN ddj-151 106 25 sin(θ sin(θ ADJ ddj-151 106 26 ) ) PUNCT ddj-151 106 27 represents represent VERB ddj-151 106 28 the the DET ddj-151 106 29 area area NOUN ddj-151 106 30 of of ADP ddj-151 106 31 the the DET ddj-151 106 32 parallelogram parallelogram NOUN ddj-151 106 33 . . PUNCT ddj-151 107 1 sin(θ sin(θ PROPN ddj-151 107 2 ) ) PUNCT ddj-151 107 3 equals equal VERB ddj-151 107 4 to to ADP ddj-151 107 5 the the DET ddj-151 107 6 length length NOUN ddj-151 107 7 of of ADP ddj-151 107 8 the the DET ddj-151 107 9 exterior exterior ADJ ddj-151 107 10 product product NOUN ddj-151 107 11 |e1 |e1 NOUN ddj-151 107 12 × × PROPN ddj-151 107 13 e2| e2| PROPN ddj-151 107 14 = = PUNCT ddj-151 107 15 sin(θ sin(θ PROPN ddj-151 107 16 ) ) PUNCT ddj-151 107 17 representing represent VERB ddj-151 107 18 twice twice DET ddj-151 107 19 the the DET ddj-151 107 20 area area NOUN ddj-151 107 21 of of ADP ddj-151 107 22 the the DET ddj-151 107 23 triangle triangle NOUN ddj-151 107 24 so so SCONJ ddj-151 107 25 that that SCONJ ddj-151 107 26 there there PRON ddj-151 107 27 would would AUX ddj-151 107 28 be be AUX ddj-151 107 29 2m×n 2m×n NUM ddj-151 107 30 triangular triangular NOUN ddj-151 107 31 faces face NOUN ddj-151 107 32 . . PUNCT ddj-151 108 1 the the DET ddj-151 108 2 division division NOUN ddj-151 108 3 of of ADP ddj-151 108 4 the the DET ddj-151 108 5 planar planar ADJ ddj-151 108 6 lattice lattice NOUN ddj-151 108 7 by by ADP ddj-151 108 8 group group NOUN ddj-151 108 9 generated generate VERB ddj-151 108 10 by by ADP ddj-151 108 11 pe1 pe1 PROPN ddj-151 108 12 + + PROPN ddj-151 108 13 qe2 qe2 PROPN ddj-151 108 14 defines define VERB ddj-151 108 15 boundary boundary ADJ ddj-151 108 16 conditions condition NOUN ddj-151 108 17 . . PUNCT ddj-151 109 1 besides besides SCONJ ddj-151 109 2 this this PRON ddj-151 109 3 the the DET ddj-151 109 4 rotation rotation NOUN ddj-151 109 5 group group NOUN ddj-151 109 6 z6 z6 PROPN ddj-151 109 7 acts act VERB ddj-151 109 8 as as ADP ddj-151 109 9 analog analog NOUN ddj-151 109 10 for for ADP ddj-151 109 11 the the DET ddj-151 109 12 symmetries symmetry NOUN ddj-151 109 13 of of ADP ddj-151 109 14 a a DET ddj-151 109 15 unit unit NOUN ddj-151 109 16 cell cell NOUN ddj-151 109 17 in in ADP ddj-151 109 18 lattice lattice NOUN ddj-151 109 19 . . PUNCT ddj-151 110 1 this this DET ddj-151 110 2 naive naive ADJ ddj-151 110 3 expectation expectation NOUN ddj-151 110 4 need need VERB ddj-151 110 5 not not PART ddj-151 110 6 of of ADV ddj-151 110 7 course course NOUN ddj-151 110 8 be be AUX ddj-151 110 9 strictly strictly ADV ddj-151 110 10 correct correct ADJ ddj-151 110 11 . . PUNCT ddj-151 111 1 4 4 X ddj-151 111 2 . . X ddj-151 111 3 as as SCONJ ddj-151 111 4 noticed notice VERB ddj-151 111 5 , , PUNCT ddj-151 111 6 it it PRON ddj-151 111 7 is be AUX ddj-151 111 8 not not PART ddj-151 111 9 possible possible ADJ ddj-151 111 10 to to PART ddj-151 111 11 have have VERB ddj-151 111 12 triangular triangular NOUN ddj-151 111 13 toric toric ADJ ddj-151 111 14 tesselations tesselation NOUN ddj-151 111 15 with with ADP ddj-151 111 16 ( ( PUNCT ddj-151 111 17 v v NOUN ddj-151 111 18 , , PUNCT ddj-151 111 19 e e NOUN ddj-151 111 20 , , PUNCT ddj-151 111 21 f f X ddj-151 111 22 ) ) PUNCT ddj-151 112 1 = = PUNCT ddj-151 112 2 ( ( PUNCT ddj-151 112 3 12 12 NUM ddj-151 112 4 , , PUNCT ddj-151 112 5 30 30 NUM ddj-151 112 6 , , PUNCT ddj-151 112 7 20 20 NUM ddj-151 112 8 ) ) PUNCT ddj-151 112 9 . . PUNCT ddj-151 113 1 torus torus PROPN ddj-151 113 2 however however ADV ddj-151 113 3 has have VERB ddj-151 113 4 a a DET ddj-151 113 5 triangular triangular NOUN ddj-151 113 6 tesselation tesselation NOUN ddj-151 113 7 with with ADP ddj-151 113 8 ( ( PUNCT ddj-151 113 9 v v NOUN ddj-151 113 10 , , PUNCT ddj-151 113 11 e e NOUN ddj-151 113 12 , , PUNCT ddj-151 113 13 f f X ddj-151 113 14 ) ) PUNCT ddj-151 113 15 = = PUNCT ddj-151 114 1 ( ( PUNCT ddj-151 114 2 12 12 NUM ddj-151 114 3 , , PUNCT ddj-151 114 4 36 36 NUM ddj-151 114 5 , , PUNCT ddj-151 114 6 24 24 NUM ddj-151 114 7 ) ) PUNCT ddj-151 114 8 . . PUNCT ddj-151 115 1 an an DET ddj-151 115 2 illustration illustration NOUN ddj-151 115 3 of of ADP ddj-151 115 4 the the DET ddj-151 115 5 tesselation tesselation NOUN ddj-151 115 6 can can AUX ddj-151 115 7 be be AUX ddj-151 115 8 found find VERB ddj-151 115 9 at at ADP ddj-151 115 10 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 115 11 . . PUNCT ddj-151 116 1 it it PRON ddj-151 116 2 allows allow VERB ddj-151 116 3 to to PART ddj-151 116 4 count count VERB ddj-151 116 5 visually visually ADV ddj-151 116 6 the the DET ddj-151 116 7 numbers number NOUN ddj-151 116 8 v v ADP ddj-151 116 9 , , PUNCT ddj-151 116 10 e e NOUN ddj-151 116 11 , , PUNCT ddj-151 116 12 f f PROPN ddj-151 116 13 , , PUNCT ddj-151 116 14 and and CCONJ ddj-151 116 15 the the DET ddj-151 116 16 identifications identification NOUN ddj-151 116 17 of of ADP ddj-151 116 18 the the DET ddj-151 116 19 boundary boundary ADJ ddj-151 116 20 edges edge NOUN ddj-151 116 21 and and CCONJ ddj-151 116 22 vertices vertex NOUN ddj-151 116 23 . . PUNCT ddj-151 117 1 with with ADP ddj-151 117 2 good good ADJ ddj-151 117 3 visual visual ADJ ddj-151 117 4 imagination imagination NOUN ddj-151 117 5 one one PRON ddj-151 117 6 might might AUX ddj-151 117 7 even even ADV ddj-151 117 8 try try VERB ddj-151 117 9 to to PART ddj-151 117 10 guess guess VERB ddj-151 117 11 what what PRON ddj-151 117 12 hamiltonian hamiltonian ADJ ddj-151 117 13 cycles cycle NOUN ddj-151 117 14 look look VERB ddj-151 117 15 like like ADP ddj-151 117 16 . . PUNCT ddj-151 118 1 the the DET ddj-151 118 2 triangular triangular NOUN ddj-151 118 3 tesselations tesselation NOUN ddj-151 118 4 and and CCONJ ddj-151 118 5 their their PRON ddj-151 118 6 hexagonal hexagonal ADJ ddj-151 118 7 duals dual NOUN ddj-151 118 8 are be AUX ddj-151 118 9 characterized characterize VERB ddj-151 118 10 partially partially ADV ddj-151 118 11 by by ADP ddj-151 118 12 a a DET ddj-151 118 13 pair pair NOUN ddj-151 118 14 of of ADP ddj-151 118 15 integers integer NOUN ddj-151 118 16 ( ( PUNCT ddj-151 118 17 a a DET ddj-151 118 18 , , PUNCT ddj-151 118 19 b b NOUN ddj-151 118 20 ) ) PUNCT ddj-151 118 21 and and CCONJ ddj-151 118 22 ( ( PUNCT ddj-151 118 23 b b NOUN ddj-151 118 24 , , PUNCT ddj-151 118 25 a a PRON ddj-151 118 26 ) ) PUNCT ddj-151 118 27 . . PUNCT ddj-151 119 1 a a PRON ddj-151 119 2 and and CCONJ ddj-151 119 3 b b NOUN ddj-151 119 4 must must AUX ddj-151 119 5 both both CCONJ ddj-151 119 6 even even ADV ddj-151 119 7 or or CCONJ ddj-151 119 8 odd odd ADJ ddj-151 119 9 ( ( PUNCT ddj-151 119 10 see see VERB ddj-151 119 11 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 119 12 ) ) PUNCT ddj-151 119 13 . . PUNCT ddj-151 120 1 the the DET ddj-151 120 2 number number NOUN ddj-151 120 3 of of ADP ddj-151 120 4 faces face NOUN ddj-151 120 5 is be AUX ddj-151 120 6 f f PROPN ddj-151 120 7 = = PUNCT ddj-151 120 8 ( ( PUNCT ddj-151 120 9 a2 a2 PROPN ddj-151 120 10 + + NOUN ddj-151 120 11 3b2)/2 3b2)/2 NUM ddj-151 120 12 . . PUNCT ddj-151 121 1 for for ADP ddj-151 121 2 ( ( PUNCT ddj-151 121 3 a a DET ddj-151 121 4 , , PUNCT ddj-151 121 5 b b NOUN ddj-151 121 6 ) ) PUNCT ddj-151 121 7 = = SYM ddj-151 121 8 ( ( PUNCT ddj-151 121 9 6 6 NUM ddj-151 121 10 , , PUNCT ddj-151 121 11 2 2 NUM ddj-151 121 12 ) ) PUNCT ddj-151 121 13 one one NOUN ddj-151 121 14 indeed indeed ADV ddj-151 121 15 has have VERB ddj-151 121 16 v v NOUN ddj-151 121 17 = = SYM ddj-151 121 18 12 12 NUM ddj-151 121 19 and and CCONJ ddj-151 121 20 f f NOUN ddj-151 121 21 = = SYM ddj-151 121 22 24 24 NUM ddj-151 121 23 . . PUNCT ddj-151 122 1 from from ADP ddj-151 122 2 the the DET ddj-151 122 3 article article NOUN ddj-151 122 4 [ [ X ddj-151 122 5 1 1 X ddj-151 122 6 ] ] PUNCT ddj-151 122 7 ( ( PUNCT ddj-151 122 8 see see VERB ddj-151 122 9 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 122 10 ) ) PUNCT ddj-151 122 11 one one PRON ddj-151 122 12 learns learn VERB ddj-151 122 13 that that SCONJ ddj-151 122 14 the the DET ddj-151 122 15 number number NOUN ddj-151 122 16 of of ADP ddj-151 122 17 triangles triangle NOUN ddj-151 122 18 satisfies satisfie NOUN ddj-151 122 19 f f PROPN ddj-151 122 20 = = SYM ddj-151 122 21 2v 2v PROPN ddj-151 122 22 for for ADP ddj-151 122 23 p p NOUN ddj-151 122 24 = = X ddj-151 122 25 q q X ddj-151 122 26 at at ADV ddj-151 122 27 least least ADJ ddj-151 122 28 . . PUNCT ddj-151 123 1 if if SCONJ ddj-151 123 2 f f PROPN ddj-151 123 3 = = SYM ddj-151 123 4 2v 2v PROPN ddj-151 123 5 holds hold VERB ddj-151 123 6 true true ADJ ddj-151 123 7 more more ADV ddj-151 123 8 generally generally ADV ddj-151 123 9 one one NUM ddj-151 123 10 has have VERB ddj-151 123 11 v v NOUN ddj-151 123 12 = = SYM ddj-151 123 13 ( ( PUNCT ddj-151 123 14 a2 a2 PROPN ddj-151 123 15 + + CCONJ ddj-151 123 16 3b2)/8 3b2)/8 NUM ddj-151 123 17 , , PUNCT ddj-151 123 18 giving give VERB ddj-151 123 19 a a DET ddj-151 123 20 tight tight ADJ ddj-151 123 21 constraints constraint NOUN ddj-151 123 22 on on ADP ddj-151 123 23 a a DET ddj-151 123 24 and and CCONJ ddj-151 123 25 b. b. PROPN ddj-151 123 26 remark remark PROPN ddj-151 123 27 : : PUNCT ddj-151 123 28 the the DET ddj-151 123 29 conventions convention NOUN ddj-151 123 30 for for ADP ddj-151 123 31 the the DET ddj-151 123 32 labelling labelling NOUN ddj-151 123 33 of of ADP ddj-151 123 34 torus torus PROPN ddj-151 123 35 tesselation tesselation NOUN ddj-151 123 36 vary vary NOUN ddj-151 123 37 . . PUNCT ddj-151 124 1 the the DET ddj-151 124 2 above above ADJ ddj-151 124 3 convention convention NOUN ddj-151 124 4 based base VERB ddj-151 124 5 on on ADP ddj-151 124 6 integers integer NOUN ddj-151 124 7 ( ( PUNCT ddj-151 124 8 a a DET ddj-151 124 9 , , PUNCT ddj-151 124 10 b b NOUN ddj-151 124 11 ) ) PUNCT ddj-151 124 12 used use VERB ddj-151 124 13 in in ADP ddj-151 124 14 the the DET ddj-151 124 15 illustrations illustration NOUN ddj-151 124 16 at at ADP ddj-151 124 17 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 124 18 is be AUX ddj-151 124 19 different different ADJ ddj-151 124 20 from from ADP ddj-151 124 21 the the DET ddj-151 124 22 convention convention NOUN ddj-151 124 23 based base VERB ddj-151 124 24 on on ADP ddj-151 124 25 integer integer NOUN ddj-151 124 26 pair pair NOUN ddj-151 124 27 ( ( PUNCT ddj-151 124 28 p p X ddj-151 124 29 , , PUNCT ddj-151 124 30 q q NOUN ddj-151 124 31 ) ) PUNCT ddj-151 124 32 used use VERB ddj-151 124 33 in in ADP ddj-151 124 34 [ [ X ddj-151 124 35 1 1 NUM ddj-151 124 36 ] ] PUNCT ddj-151 124 37 . . PUNCT ddj-151 125 1 in in ADP ddj-151 125 2 this this DET ddj-151 125 3 notation notation NOUN ddj-151 125 4 torus torus PROPN ddj-151 125 5 tesselation tesselation NOUN ddj-151 125 6 with with ADP ddj-151 125 7 ( ( PUNCT ddj-151 125 8 v v NOUN ddj-151 125 9 , , PUNCT ddj-151 125 10 f f NOUN ddj-151 125 11 ) ) PUNCT ddj-151 125 12 = = PUNCT ddj-151 125 13 ( ( PUNCT ddj-151 125 14 12 12 NUM ddj-151 125 15 , , PUNCT ddj-151 125 16 24 24 NUM ddj-151 125 17 ) ) PUNCT ddj-151 125 18 corresponds correspond VERB ddj-151 125 19 to to ADP ddj-151 125 20 ( ( PUNCT ddj-151 125 21 p p X ddj-151 125 22 , , PUNCT ddj-151 125 23 q q NOUN ddj-151 125 24 ) ) PUNCT ddj-151 125 25 = = SYM ddj-151 125 26 ( ( PUNCT ddj-151 125 27 2 2 NUM ddj-151 125 28 , , PUNCT ddj-151 125 29 2 2 NUM ddj-151 125 30 ) ) PUNCT ddj-151 125 31 instead instead ADV ddj-151 125 32 of of ADP ddj-151 125 33 ( ( PUNCT ddj-151 125 34 a a DET ddj-151 125 35 , , PUNCT ddj-151 125 36 b b NOUN ddj-151 125 37 ) ) PUNCT ddj-151 125 38 = = SYM ddj-151 125 39 ( ( PUNCT ddj-151 125 40 6 6 NUM ddj-151 125 41 , , PUNCT ddj-151 125 42 2 2 NUM ddj-151 125 43 ) ) PUNCT ddj-151 125 44 . . PUNCT ddj-151 126 1 this this PRON ddj-151 126 2 requires require VERB ddj-151 126 3 ( ( PUNCT ddj-151 126 4 a a DET ddj-151 126 5 , , PUNCT ddj-151 126 6 b b NOUN ddj-151 126 7 ) ) PUNCT ddj-151 126 8 = = SYM ddj-151 126 9 ( ( PUNCT ddj-151 126 10 3p 3p NUM ddj-151 126 11 , , PUNCT ddj-151 126 12 q q NOUN ddj-151 126 13 ) ) PUNCT ddj-151 126 14 . . PUNCT ddj-151 127 1 with with ADP ddj-151 127 2 these these DET ddj-151 127 3 conventions convention NOUN ddj-151 127 4 one one PRON ddj-151 127 5 has have VERB ddj-151 127 6 v v NOUN ddj-151 127 7 = = SYM ddj-151 127 8 p2 p2 PROPN ddj-151 127 9 + + CCONJ ddj-151 127 10 q2 q2 NOUN ddj-151 127 11 + + CCONJ ddj-151 128 1 pq pq NOUN ddj-151 128 2 . . NOUN ddj-151 128 3 2.1 2.1 NUM ddj-151 128 4 the the DET ddj-151 128 5 number number NOUN ddj-151 128 6 of of ADP ddj-151 128 7 triangles triangle NOUN ddj-151 128 8 in in ADP ddj-151 128 9 the the DET ddj-151 128 10 12 12 NUM ddj-151 128 11 - - PUNCT ddj-151 128 12 vertex vertex NOUN ddj-151 128 13 tesselation tesselation NOUN ddj-151 128 14 is be AUX ddj-151 128 15 24 24 NUM ddj-151 128 16 : : PUNCT ddj-151 128 17 curse curse NOUN ddj-151 128 18 or or CCONJ ddj-151 128 19 blessing blessing NOUN ddj-151 128 20 ? ? PUNCT ddj-151 129 1 one one PRON ddj-151 129 2 could could AUX ddj-151 129 3 see see VERB ddj-151 129 4 as as ADP ddj-151 129 5 a a DET ddj-151 129 6 problem problem NOUN ddj-151 129 7 that that SCONJ ddj-151 129 8 one one PRON ddj-151 129 9 has have AUX ddj-151 129 10 f f NOUN ddj-151 129 11 = = SYM ddj-151 129 12 24 24 NUM ddj-151 129 13 > > SYM ddj-151 129 14 20 20 NUM ddj-151 129 15 ? ? PUNCT ddj-151 129 16 or or CCONJ ddj-151 129 17 is be AUX ddj-151 129 18 this this PRON ddj-151 129 19 a a DET ddj-151 129 20 problem problem NOUN ddj-151 129 21 ? ? PUNCT ddj-151 130 1 1 1 X ddj-151 130 2 . . PUNCT ddj-151 130 3 by by ADP ddj-151 130 4 fusing fuse VERB ddj-151 130 5 two two NUM ddj-151 130 6 icosahedral icosahedral ADJ ddj-151 130 7 harmonies harmony NOUN ddj-151 130 8 and and CCONJ ddj-151 130 9 one one NUM ddj-151 130 10 toric toric ADJ ddj-151 130 11 harmony harmony NOUN ddj-151 130 12 one one PRON ddj-151 130 13 would would AUX ddj-151 130 14 obtain obtain VERB ddj-151 130 15 a a DET ddj-151 130 16 harmony harmony NOUN ddj-151 130 17 with with ADP ddj-151 130 18 20 20 NUM ddj-151 130 19 + + SYM ddj-151 130 20 20 20 NUM ddj-151 130 21 + + SYM ddj-151 130 22 24 24 NUM ddj-151 130 23 = = SYM ddj-151 130 24 64 64 NUM ddj-151 130 25 chords chord NOUN ddj-151 130 26 , , PUNCT ddj-151 130 27 the the DET ddj-151 130 28 number number NOUN ddj-151 130 29 of of ADP ddj-151 130 30 dna dna PROPN ddj-151 130 31 codons codon NOUN ddj-151 130 32 ! ! PUNCT ddj-151 131 1 one one PRON ddj-151 131 2 would would AUX ddj-151 131 3 replace replace VERB ddj-151 131 4 the the DET ddj-151 131 5 fusion fusion NOUN ddj-151 131 6 of of ADP ddj-151 131 7 3 3 NUM ddj-151 131 8 icosahedral icosahedral ADJ ddj-151 131 9 harmonies harmony NOUN ddj-151 131 10 and and CCONJ ddj-151 131 11 tetrahedral tetrahedral ADJ ddj-151 131 12 harmony harmony NOUN ddj-151 131 13 with with ADP ddj-151 131 14 a a DET ddj-151 131 15 fusion fusion NOUN ddj-151 131 16 of of ADP ddj-151 131 17 2 2 NUM ddj-151 131 18 icosahedral icosahedral ADJ ddj-151 131 19 harmonies harmony NOUN ddj-151 131 20 and and CCONJ ddj-151 131 21 toric toric ADJ ddj-151 131 22 harmony harmony NOUN ddj-151 131 23 . . PUNCT ddj-151 132 1 icosahedral icosahedral ADJ ddj-151 132 2 symmetry symmetry NOUN ddj-151 132 3 with with ADP ddj-151 132 4 toric toric ADJ ddj-151 132 5 symmetry symmetry NOUN ddj-151 132 6 associated associate VERB ddj-151 132 7 with with ADP ddj-151 132 8 the the DET ddj-151 132 9 third third ADJ ddj-151 132 10 harmony harmony NOUN ddj-151 132 11 would would AUX ddj-151 132 12 be be AUX ddj-151 132 13 replaced replace VERB ddj-151 132 14 with with ADP ddj-151 132 15 a a DET ddj-151 132 16 smaller small ADJ ddj-151 132 17 toric toric ADJ ddj-151 132 18 symmetry symmetry NOUN ddj-151 132 19 . . PUNCT ddj-151 133 1 note note VERB ddj-151 133 2 however however ADV ddj-151 133 3 that that SCONJ ddj-151 133 4 the the DET ddj-151 133 5 attachment attachment NOUN ddj-151 133 6 of of ADP ddj-151 133 7 tetrahedron tetrahedron NOUN ddj-151 133 8 to to ADP ddj-151 133 9 a a DET ddj-151 133 10 fixed fix VERB ddj-151 133 11 icosahedral icosahedral ADJ ddj-151 133 12 face face NOUN ddj-151 133 13 also also ADV ddj-151 133 14 breaks break VERB ddj-151 133 15 icosahedral icosahedral ADJ ddj-151 133 16 symmetry symmetry NOUN ddj-151 133 17 . . PUNCT ddj-151 134 1 this this PRON ddj-151 134 2 raises raise VERB ddj-151 134 3 questions question NOUN ddj-151 134 4 . . PUNCT ddj-151 135 1 could could AUX ddj-151 135 2 the the DET ddj-151 135 3 presence presence NOUN ddj-151 135 4 of of ADP ddj-151 135 5 the the DET ddj-151 135 6 toric toric ADJ ddj-151 135 7 harmony harmony NOUN ddj-151 135 8 somehow somehow ADV ddj-151 135 9 relate relate VERB ddj-151 135 10 to to ADP ddj-151 135 11 the the DET ddj-151 135 12 almost almost ADV ddj-151 135 13 exact exact ADJ ddj-151 135 14 u u PRON ddj-151 135 15 ↔ ↔ PROPN ddj-151 135 16 c c NOUN ddj-151 135 17 and and CCONJ ddj-151 135 18 a a DET ddj-151 135 19 ↔ ↔ PROPN ddj-151 135 20 g g NOUN ddj-151 135 21 symmetries symmetry NOUN ddj-151 135 22 of of ADP ddj-151 135 23 the the DET ddj-151 135 24 third third ADJ ddj-151 135 25 letter letter NOUN ddj-151 135 26 of of ADP ddj-151 135 27 codons codon NOUN ddj-151 135 28 . . PUNCT ddj-151 136 1 this this PRON ddj-151 136 2 does do AUX ddj-151 136 3 not not PART ddj-151 136 4 of of ADV ddj-151 136 5 course course NOUN ddj-151 136 6 mean mean VERB ddj-151 136 7 that that SCONJ ddj-151 136 8 one one PRON ddj-151 136 9 could could AUX ddj-151 136 10 associated associate VERB ddj-151 136 11 the the DET ddj-151 136 12 toric toric ADJ ddj-151 136 13 harmony harmony NOUN ddj-151 136 14 with with ADP ddj-151 136 15 the the DET ddj-151 136 16 third third ADJ ddj-151 136 17 letter letter NOUN ddj-151 136 18 . . PUNCT ddj-151 137 1 note note VERB ddj-151 137 2 that that SCONJ ddj-151 137 3 in in ADP ddj-151 137 4 the the DET ddj-151 137 5 icosa icosa NOUN ddj-151 137 6 - - PUNCT ddj-151 137 7 tetrahedral tetrahedral ADJ ddj-151 137 8 model model NOUN ddj-151 137 9 the the DET ddj-151 137 10 three three NUM ddj-151 137 11 harmonies harmony NOUN ddj-151 137 12 are be AUX ddj-151 137 13 assumed assume VERB ddj-151 137 14 to to PART ddj-151 137 15 have have VERB ddj-151 137 16 no no DET ddj-151 137 17 common common ADJ ddj-151 137 18 chords chord NOUN ddj-151 137 19 . . PUNCT ddj-151 138 1 same same ADJ ddj-151 138 2 non non ADJ ddj-151 138 3 - - ADJ ddj-151 138 4 trivial trivial ADJ ddj-151 138 5 assumption assumption NOUN ddj-151 138 6 is be AUX ddj-151 138 7 needed need VERB ddj-151 138 8 also also ADV ddj-151 138 9 now now ADV ddj-151 138 10 in in ADP ddj-151 138 11 order order NOUN ddj-151 138 12 to to PART ddj-151 138 13 obtain obtain VERB ddj-151 138 14 64 64 NUM ddj-151 138 15 codons codon NOUN ddj-151 138 16 . . PUNCT ddj-151 139 1 2 2 X ddj-151 139 2 . . X ddj-151 139 3 what what PRON ddj-151 139 4 about about ADP ddj-151 139 5 the the DET ddj-151 139 6 number number NOUN ddj-151 139 7 of of ADP ddj-151 139 8 amino amino NOUN ddj-151 139 9 - - PUNCT ddj-151 139 10 acids acid NOUN ddj-151 139 11 : : PUNCT ddj-151 139 12 could could AUX ddj-151 139 13 it it PRON ddj-151 139 14 be be AUX ddj-151 139 15 24 24 NUM ddj-151 139 16 corresponding correspond VERB ddj-151 139 17 ordinary ordinary ADJ ddj-151 139 18 aminoacids aminoacid NOUN ddj-151 139 19 , , PUNCT ddj-151 139 20 stopping stop VERB ddj-151 139 21 sign sign NOUN ddj-151 139 22 plus plus CCONJ ddj-151 139 23 3 3 NUM ddj-151 139 24 additional additional ADJ ddj-151 139 25 exotic exotic ADJ ddj-151 139 26 amino amino NOUN ddj-151 139 27 - - PUNCT ddj-151 139 28 acids acid NOUN ddj-151 139 29 . . PUNCT ddj-151 140 1 the the DET ddj-151 140 2 20 20 NUM ddj-151 140 3 icosahedral icosahedral ADJ ddj-151 140 4 triangles triangle NOUN ddj-151 140 5 can can AUX ddj-151 140 6 corresponds correspond VERB ddj-151 140 7 to to ADP ddj-151 140 8 aminoacids aminoacid NOUN ddj-151 140 9 but but CCONJ ddj-151 140 10 not not PART ddj-151 140 11 to to ADP ddj-151 140 12 stopping stop VERB ddj-151 140 13 sign sign NOUN ddj-151 140 14 . . PUNCT ddj-151 141 1 could could AUX ddj-151 141 2 it it PRON ddj-151 141 3 be be AUX ddj-151 141 4 that that SCONJ ddj-151 141 5 one one NUM ddj-151 141 6 of of ADP ddj-151 141 7 the the DET ddj-151 141 8 additional additional ADJ ddj-151 141 9 codons codon NOUN ddj-151 141 10 in in ADP ddj-151 141 11 24 24 NUM ddj-151 141 12 corresponds correspond NOUN ddj-151 141 13 to to ADP ddj-151 141 14 stopping stop VERB ddj-151 141 15 sign sign NOUN ddj-151 141 16 and and CCONJ ddj-151 141 17 two two NUM ddj-151 141 18 exotic exotic ADJ ddj-151 141 19 amino amino NOUN ddj-151 141 20 - - PUNCT ddj-151 141 21 acids acids NOUN ddj-151 141 22 pyl pyl NOUN ddj-151 141 23 and and CCONJ ddj-151 141 24 sec sec PROPN ddj-151 141 25 appearing appear VERB ddj-151 141 26 in in ADP ddj-151 141 27 biosystems biosystem NOUN ddj-151 141 28 explained explain VERB ddj-151 141 29 by by ADP ddj-151 141 30 the the DET ddj-151 141 31 icosahedral icosahedral ADJ ddj-151 141 32 model model NOUN ddj-151 141 33 in in ADP ddj-151 141 34 terms term NOUN ddj-151 141 35 of of ADP ddj-151 141 36 a a DET ddj-151 141 37 variant variant NOUN ddj-151 141 38 of of ADP ddj-151 141 39 the the DET ddj-151 141 40 genetic genetic ADJ ddj-151 141 41 code code NOUN ddj-151 141 42 . . PUNCT ddj-151 142 1 there there PRON ddj-151 142 2 indeed indeed ADV ddj-151 142 3 exists exist VERB ddj-151 142 4 even even ADV ddj-151 142 5 third third ADJ ddj-151 142 6 exotic exotic ADJ ddj-151 142 7 amino amino NOUN ddj-151 142 8 - - PUNCT ddj-151 142 9 acid acid NOUN ddj-151 142 10 ! ! PUNCT ddj-151 143 1 n n CCONJ ddj-151 143 2 - - PUNCT ddj-151 143 3 formylmethionine formylmethionine NOUN ddj-151 143 4 ( ( PUNCT ddj-151 143 5 see see VERB ddj-151 143 6 http://tinyurl.com/jsphvgt http://tinyurl.com/jsphvgt PRON ddj-151 143 7 ) ) PUNCT ddj-151 143 8 but but CCONJ ddj-151 143 9 is be AUX ddj-151 143 10 usually usually ADV ddj-151 143 11 regarded regard VERB ddj-151 143 12 as as ADP ddj-151 143 13 as as ADP ddj-151 143 14 a a DET ddj-151 143 15 form form NOUN ddj-151 143 16 of of ADP ddj-151 143 17 methionine methionine NOUN ddj-151 143 18 rather rather ADV ddj-151 143 19 than than ADP ddj-151 143 20 as as ADP ddj-151 143 21 a a DET ddj-151 143 22 separate separate ADJ ddj-151 143 23 proteinogenic proteinogenic ADJ ddj-151 143 24 amino amino NOUN ddj-151 143 25 - - PUNCT ddj-151 143 26 acid acid NOUN ddj-151 143 27 . . PUNCT ddj-151 144 1 3 3 X ddj-151 144 2 . . X ddj-151 144 3 recall recall VERB ddj-151 144 4 that that SCONJ ddj-151 144 5 the the DET ddj-151 144 6 problem problem NOUN ddj-151 144 7 related relate VERB ddj-151 144 8 to to ADP ddj-151 144 9 the the DET ddj-151 144 10 icosa icosa NOUN ddj-151 144 11 - - PUNCT ddj-151 144 12 tetrahedral tetrahedral ADJ ddj-151 144 13 harmony harmony NOUN ddj-151 144 14 is be AUX ddj-151 144 15 that that SCONJ ddj-151 144 16 it it PRON ddj-151 144 17 does do AUX ddj-151 144 18 not not PART ddj-151 144 19 contains contain VERB ddj-151 144 20 the the DET ddj-151 144 21 chords chord NOUN ddj-151 144 22 of of ADP ddj-151 144 23 what what PRON ddj-151 144 24 might might AUX ddj-151 144 25 be be AUX ddj-151 144 26 called call VERB ddj-151 144 27 classical classical ADJ ddj-151 144 28 harmonies harmony NOUN ddj-151 144 29 ( ( PUNCT ddj-151 144 30 the the DET ddj-151 144 31 chordds chordd NOUN ddj-151 144 32 assignable assignable ADJ ddj-151 144 33 to to ADP ddj-151 144 34 major major ADJ ddj-151 144 35 and and CCONJ ddj-151 144 36 minor minor ADJ ddj-151 144 37 issn issn NOUN ddj-151 144 38 : : PUNCT ddj-151 144 39 2159 2159 NUM ddj-151 144 40 - - PUNCT ddj-151 144 41 046x 046x NOUN ddj-151 144 42 dna dna NOUN ddj-151 144 43 decipher decipher PROPN ddj-151 144 44 journal journal PROPN ddj-151 144 45 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 144 46 published publish VERB ddj-151 144 47 by by ADP ddj-151 144 48 quantumdream quantumdream PROPN ddj-151 144 49 , , PUNCT ddj-151 144 50 inc inc PROPN ddj-151 144 51 . . PROPN ddj-151 144 52 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 144 53 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 144 54 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 144 55 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 144 56 http://tinyurl.com/jsphvgt http://tinyurl.com/jsphvgt PROPN ddj-151 144 57 dna dna PROPN ddj-151 144 58 decipher decipher PROPN ddj-151 144 59 journal journal NOUN ddj-151 144 60 | | ADP ddj-151 144 61 november november PROPN ddj-151 144 62 2018 2018 NUM ddj-151 144 63 | | ADV ddj-151 144 64 volume volume NOUN ddj-151 144 65 8 8 NUM ddj-151 145 1 | | NOUN ddj-151 145 2 issue issue VERB ddj-151 145 3 3 3 NUM ddj-151 145 4 | | CCONJ ddj-151 145 5 pp pp ADJ ddj-151 145 6 . . PUNCT ddj-151 146 1 197 197 NUM ddj-151 146 2 - - SYM ddj-151 146 3 205 205 NUM ddj-151 146 4 201 201 NUM ddj-151 146 5 pitkänen pitkänen NOUN ddj-151 146 6 , , PUNCT ddj-151 146 7 m. m. NOUN ddj-151 146 8 , , PUNCT ddj-151 146 9 thoughts thought NOUN ddj-151 146 10 on on ADP ddj-151 146 11 modification modification NOUN ddj-151 146 12 of of ADP ddj-151 146 13 bio bio NOUN ddj-151 146 14 - - ADJ ddj-151 146 15 harmony harmony NOUN ddj-151 146 16 scales scale NOUN ddj-151 146 17 ) ) PUNCT ddj-151 146 18 . . PUNCT ddj-151 147 1 if if SCONJ ddj-151 147 2 24 24 NUM ddj-151 147 3 chords chord NOUN ddj-151 147 4 of of ADP ddj-151 147 5 bio bio NOUN ddj-151 147 6 - - NOUN ddj-151 147 7 harmony harmony NOUN ddj-151 147 8 correspond correspond NOUN ddj-151 147 9 to to PART ddj-151 147 10 toric toric VERB ddj-151 147 11 harmony harmony NOUN ddj-151 147 12 , , PUNCT ddj-151 147 13 one one PRON ddj-151 147 14 could could AUX ddj-151 147 15 obtain obtain VERB ddj-151 147 16 these these DET ddj-151 147 17 chords chord NOUN ddj-151 147 18 if if SCONJ ddj-151 147 19 the the DET ddj-151 147 20 chords chord NOUN ddj-151 147 21 in in ADP ddj-151 147 22 question question NOUN ddj-151 147 23 are be AUX ddj-151 147 24 chords chord NOUN ddj-151 147 25 obtainable obtainable ADJ ddj-151 147 26 by by ADP ddj-151 147 27 the the DET ddj-151 147 28 proposed propose VERB ddj-151 147 29 construction construction NOUN ddj-151 147 30 . . PUNCT ddj-151 148 1 but but CCONJ ddj-151 148 2 is be AUX ddj-151 148 3 this this DET ddj-151 148 4 construction construction NOUN ddj-151 148 5 consistent consistent ADJ ddj-151 148 6 with with ADP ddj-151 148 7 the the DET ddj-151 148 8 representation representation NOUN ddj-151 148 9 of of ADP ddj-151 148 10 64 64 NUM ddj-151 148 11 chords chord NOUN ddj-151 148 12 by by ADP ddj-151 148 13 taking take VERB ddj-151 148 14 to to ADP ddj-151 148 15 each each DET ddj-151 148 16 chord chord NOUN ddj-151 148 17 one one NUM ddj-151 148 18 note note NOUN ddj-151 148 19 from from ADP ddj-151 148 20 3 3 NUM ddj-151 148 21 disjoint disjoint NOUN ddj-151 148 22 groups group NOUN ddj-151 148 23 of of ADP ddj-151 148 24 4 4 NUM ddj-151 148 25 notes note NOUN ddj-151 148 26 in in ADP ddj-151 148 27 which which PRON ddj-151 148 28 each each DET ddj-151 148 29 note note NOUN ddj-151 148 30 belongs belong VERB ddj-151 148 31 to to ADP ddj-151 148 32 16 16 NUM ddj-151 148 33 chords chord NOUN ddj-151 148 34 . . PUNCT ddj-151 149 1 the the DET ddj-151 149 2 maximum maximum ADJ ddj-151 149 3 number number NOUN ddj-151 149 4 of of ADP ddj-151 149 5 chords chord NOUN ddj-151 149 6 that that PRON ddj-151 149 7 note note VERB ddj-151 149 8 can can AUX ddj-151 149 9 belong belong VERB ddj-151 149 10 to to PART ddj-151 149 11 would would AUX ddj-151 149 12 be be AUX ddj-151 149 13 5 5 NUM ddj-151 149 14 + + NOUN ddj-151 149 15 5 5 NUM ddj-151 149 16 + + NOUN ddj-151 149 17 6=16 6=16 NUM ddj-151 149 18 as as SCONJ ddj-151 149 19 desired desire VERB ddj-151 149 20 . . PUNCT ddj-151 150 1 if if SCONJ ddj-151 150 2 there there PRON ddj-151 150 3 are be VERB ddj-151 150 4 no no DET ddj-151 150 5 common common ADJ ddj-151 150 6 chords chord NOUN ddj-151 150 7 between between ADP ddj-151 150 8 the the DET ddj-151 150 9 3 3 NUM ddj-151 150 10 harmonies harmonie VERB ddj-151 150 11 the the DET ddj-151 150 12 conditions condition NOUN ddj-151 150 13 is be AUX ddj-151 150 14 satisfied satisfied ADJ ddj-151 150 15 . . PUNCT ddj-151 151 1 using use VERB ddj-151 151 2 for for ADP ddj-151 151 3 instance instance NOUN ddj-151 151 4 3 3 NUM ddj-151 151 5 toric toric ADJ ddj-151 151 6 representations representation NOUN ddj-151 151 7 the the DET ddj-151 151 8 number number NOUN ddj-151 151 9 would would AUX ddj-151 151 10 be be AUX ddj-151 151 11 6 6 NUM ddj-151 151 12 + + NOUN ddj-151 151 13 6 6 NUM ddj-151 151 14 + + NOUN ddj-151 151 15 6=18 6=18 NUM ddj-151 151 16 and and CCONJ ddj-151 151 17 would would AUX ddj-151 151 18 require require VERB ddj-151 151 19 dropping drop VERB ddj-151 151 20 some some DET ddj-151 151 21 chords chord NOUN ddj-151 151 22 . . PUNCT ddj-151 152 1 4 4 X ddj-151 152 2 . . X ddj-151 152 3 the the DET ddj-151 152 4 earlier early ADJ ddj-151 152 5 model model NOUN ddj-151 152 6 for for ADP ddj-151 152 7 trna trna PROPN ddj-151 152 8 as as ADP ddj-151 152 9 fusion fusion NOUN ddj-151 152 10 of of ADP ddj-151 152 11 two two NUM ddj-151 152 12 icosahedral icosahedral ADJ ddj-151 152 13 codes code NOUN ddj-151 152 14 predicting predict VERB ddj-151 152 15 20 20 NUM ddj-151 152 16 + + SYM ddj-151 152 17 20=40 20=40 ADJ ddj-151 152 18 trna trna ADJ ddj-151 152 19 codons codon NOUN ddj-151 152 20 . . PUNCT ddj-151 153 1 now now ADV ddj-151 153 2 trnas trnas NOUN ddj-151 153 3 as as ADP ddj-151 153 4 fusion fusion NOUN ddj-151 153 5 of of ADP ddj-151 153 6 two two NUM ddj-151 153 7 harmonies harmony NOUN ddj-151 153 8 allows allow VERB ddj-151 153 9 two two NUM ddj-151 153 10 basic basic ADJ ddj-151 153 11 options option NOUN ddj-151 153 12 depending depend VERB ddj-151 153 13 on on ADP ddj-151 153 14 whether whether SCONJ ddj-151 153 15 both both DET ddj-151 153 16 harmonies harmony NOUN ddj-151 153 17 are be AUX ddj-151 153 18 icosahedral icosahedral ADJ ddj-151 153 19 or or CCONJ ddj-151 153 20 whether whether SCONJ ddj-151 153 21 second second ADJ ddj-151 153 22 harmony harmony NOUN ddj-151 153 23 is be AUX ddj-151 153 24 toric toric ADJ ddj-151 153 25 . . PUNCT ddj-151 154 1 these these DET ddj-151 154 2 options option NOUN ddj-151 154 3 would would AUX ddj-151 154 4 give give VERB ddj-151 154 5 20 20 NUM ddj-151 154 6 + + NOUN ddj-151 154 7 20=40 20=40 NUM ddj-151 154 8 or or CCONJ ddj-151 154 9 20 20 NUM ddj-151 154 10 + + NOUN ddj-151 154 11 24=44 24=44 NUM ddj-151 154 12 trnas trnas NOUN ddj-151 154 13 . . PUNCT ddj-151 155 1 wikipedia wikipedia PROPN ddj-151 155 2 tells tell VERB ddj-151 155 3 that that PRON ddj-151 155 4 maximum maximum ADJ ddj-151 155 5 number number NOUN ddj-151 155 6 is be AUX ddj-151 155 7 41 41 NUM ddj-151 155 8 . . PUNCT ddj-151 156 1 some some DET ddj-151 156 2 sources source NOUN ddj-151 156 3 however however ADV ddj-151 156 4 tell tell VERB ddj-151 156 5 that that SCONJ ddj-151 156 6 there there PRON ddj-151 156 7 are be VERB ddj-151 156 8 20 20 NUM ddj-151 156 9 - - SYM ddj-151 156 10 40 40 NUM ddj-151 156 11 different different ADJ ddj-151 156 12 trnas trnas NOUN ddj-151 156 13 in in ADP ddj-151 156 14 bacterial bacterial ADJ ddj-151 156 15 cells cell NOUN ddj-151 156 16 and and CCONJ ddj-151 156 17 as as ADV ddj-151 156 18 many many ADJ ddj-151 156 19 as as ADP ddj-151 156 20 50 50 NUM ddj-151 156 21 - - SYM ddj-151 156 22 100 100 NUM ddj-151 156 23 in in ADP ddj-151 156 24 plant plant NOUN ddj-151 156 25 and and CCONJ ddj-151 156 26 animal animal NOUN ddj-151 156 27 cells cell NOUN ddj-151 156 28 . . PUNCT ddj-151 157 1 2.2 2.2 NUM ddj-151 157 2 a a DET ddj-151 157 3 more more ADV ddj-151 157 4 detailed detailed ADJ ddj-151 157 5 model model NOUN ddj-151 157 6 for for ADP ddj-151 157 7 toric toric ADJ ddj-151 157 8 harmonies harmony NOUN ddj-151 157 9 one one PRON ddj-151 157 10 can can AUX ddj-151 157 11 consider consider VERB ddj-151 157 12 also also ADV ddj-151 157 13 more more ADV ddj-151 157 14 detailed detailed ADJ ddj-151 157 15 model model NOUN ddj-151 157 16 for for ADP ddj-151 157 17 toric toric ADJ ddj-151 157 18 harmonies harmony NOUN ddj-151 157 19 . . PUNCT ddj-151 158 1 1 1 X ddj-151 158 2 . . X ddj-151 158 3 the the DET ddj-151 158 4 above above ADJ ddj-151 158 5 discussed discuss VERB ddj-151 158 6 representation representation NOUN ddj-151 158 7 in in ADP ddj-151 158 8 terms term NOUN ddj-151 158 9 of of ADP ddj-151 158 10 frequencies frequency NOUN ddj-151 158 11 assigned assign VERB ddj-151 158 12 with with ADP ddj-151 158 13 nucleotides nucleotide NOUN ddj-151 158 14 depending depend VERB ddj-151 158 15 on on ADP ddj-151 158 16 their their PRON ddj-151 158 17 position position NOUN ddj-151 158 18 requires require VERB ddj-151 158 19 the the DET ddj-151 158 20 decomposition decomposition NOUN ddj-151 158 21 of of ADP ddj-151 158 22 the the DET ddj-151 158 23 notes note NOUN ddj-151 158 24 to to ADP ddj-151 158 25 3 3 NUM ddj-151 158 26 disjoint disjoint NOUN ddj-151 158 27 groups group NOUN ddj-151 158 28 of of ADP ddj-151 158 29 4 4 NUM ddj-151 158 30 notes note NOUN ddj-151 158 31 . . PUNCT ddj-151 159 1 this this PRON ddj-151 159 2 means mean VERB ddj-151 159 3 decomposition decomposition NOUN ddj-151 159 4 of of ADP ddj-151 159 5 12 12 NUM ddj-151 159 6 vertices vertex NOUN ddj-151 159 7 of of ADP ddj-151 159 8 hamiltonian hamiltonian ADJ ddj-151 159 9 cycle cycle NOUN ddj-151 159 10 to to ADP ddj-151 159 11 4 4 NUM ddj-151 159 12 disjoint disjoint NOUN ddj-151 159 13 groups group NOUN ddj-151 159 14 such such ADJ ddj-151 159 15 that that SCONJ ddj-151 159 16 within within ADP ddj-151 159 17 given give VERB ddj-151 159 18 group group NOUN ddj-151 159 19 the the DET ddj-151 159 20 distances distance NOUN ddj-151 159 21 between between ADP ddj-151 159 22 the the DET ddj-151 159 23 members member NOUN ddj-151 159 24 of of ADP ddj-151 159 25 group group NOUN ddj-151 159 26 are be AUX ddj-151 159 27 larger large ADJ ddj-151 159 28 than than ADP ddj-151 159 29 one one NUM ddj-151 159 30 unit unit NOUN ddj-151 159 31 so so SCONJ ddj-151 159 32 that that SCONJ ddj-151 159 33 they they PRON ddj-151 159 34 can can AUX ddj-151 159 35 not not PART ddj-151 159 36 belong belong VERB ddj-151 159 37 to to ADP ddj-151 159 38 same same ADJ ddj-151 159 39 triangle triangle NOUN ddj-151 159 40 . . PUNCT ddj-151 160 1 there there PRON ddj-151 160 2 are be VERB ddj-151 160 3 bin(12 bin(12 NOUN ddj-151 160 4 , , PUNCT ddj-151 160 5 4 4 NUM ddj-151 160 6 ) ) PUNCT ddj-151 160 7 × × NOUN ddj-151 160 8 bin(8 bin(8 ADJ ddj-151 160 9 , , PUNCT ddj-151 160 10 4 4 X ddj-151 160 11 ) ) PUNCT ddj-151 160 12 decomposition decomposition NOUN ddj-151 160 13 to to ADP ddj-151 160 14 3 3 NUM ddj-151 160 15 disjoint disjoint NOUN ddj-151 160 16 groups group NOUN ddj-151 160 17 of of ADP ddj-151 160 18 for for ADP ddj-151 160 19 vertices vertex NOUN ddj-151 160 20 , , PUNCT ddj-151 160 21 where where SCONJ ddj-151 160 22 bn(n bn(n NOUN ddj-151 160 23 , , PUNCT ddj-151 160 24 k k NOUN ddj-151 160 25 ) ) PUNCT ddj-151 160 26 = = SYM ddj-151 160 27 n!/(n− n!/(n− PROPN ddj-151 160 28 k)!k k)!k PROPN ddj-151 160 29 ! ! PUNCT ddj-151 160 30 is be AUX ddj-151 160 31 binomial binomial ADJ ddj-151 160 32 coefficient coefficient NOUN ddj-151 160 33 . . PUNCT ddj-151 161 1 2 2 X ddj-151 161 2 . . X ddj-151 162 1 once once SCONJ ddj-151 162 2 the the DET ddj-151 162 3 hamiltonian hamiltonian ADJ ddj-151 162 4 cycle cycle NOUN ddj-151 162 5 has have AUX ddj-151 162 6 been be AUX ddj-151 162 7 fixed fix VERB ddj-151 162 8 and and CCONJ ddj-151 162 9 is be AUX ddj-151 162 10 one one NUM ddj-151 162 11 assumes assume VERB ddj-151 162 12 that that SCONJ ddj-151 162 13 single single ADJ ddj-151 162 14 step step NOUN ddj-151 162 15 along along ADP ddj-151 162 16 cycle cycle NOUN ddj-151 162 17 corresponds correspond NOUN ddj-151 162 18 to to ADP ddj-151 162 19 quint quint NOUN ddj-151 162 20 , , PUNCT ddj-151 162 21 one one PRON ddj-151 162 22 knows know VERB ddj-151 162 23 what what PRON ddj-151 162 24 the the DET ddj-151 162 25 notes note NOUN ddj-151 162 26 associated associate VERB ddj-151 162 27 with with ADP ddj-151 162 28 each each DET ddj-151 162 29 vertex vertex NOUN ddj-151 162 30 is be AUX ddj-151 162 31 and and CCONJ ddj-151 162 32 given give VERB ddj-151 162 33 the the DET ddj-151 162 34 note note NOUN ddj-151 162 35 of of ADP ddj-151 162 36 the the DET ddj-151 162 37 12 12 NUM ddj-151 162 38 - - PUNCT ddj-151 162 39 note note NOUN ddj-151 162 40 scale scale NOUN ddj-151 162 41 one one NUM ddj-151 162 42 knows know VERB ddj-151 162 43 the the DET ddj-151 162 44 number0 number0 NOUN ddj-151 162 45 ≤ ≤ NOUN ddj-151 162 46 n n CCONJ ddj-151 162 47 < < X ddj-151 162 48 12 12 NUM ddj-151 162 49 of of ADP ddj-151 162 50 quint quint NOUN ddj-151 162 51 steps step NOUN ddj-151 162 52 needed need VERB ddj-151 162 53 to to PART ddj-151 162 54 obtain obtain VERB ddj-151 162 55 it it PRON ddj-151 162 56 . . PUNCT ddj-151 163 1 for for ADP ddj-151 163 2 instance instance NOUN ddj-151 163 3 , , PUNCT ddj-151 163 4 for for ADP ddj-151 163 5 the the DET ddj-151 163 6 proposed propose VERB ddj-151 163 7 grouping grouping NOUN ddj-151 163 8 { { PUNCT ddj-151 163 9 c c PROPN ddj-151 163 10 , , PUNCT ddj-151 163 11 c],d c],d X ddj-151 163 12 , , PUNCT ddj-151 163 13 d d X ddj-151 163 14 ] ] X ddj-151 163 15 } } PUNCT ddj-151 163 16 and and CCONJ ddj-151 163 17 its its PRON ddj-151 163 18 two two NUM ddj-151 163 19 transposes transpose NOUN ddj-151 163 20 by by ADP ddj-151 163 21 2 2 NUM ddj-151 163 22 hole hole NOUN ddj-151 163 23 steps step NOUN ddj-151 163 24 one one PRON ddj-151 163 25 can can AUX ddj-151 163 26 assign assign VERB ddj-151 163 27 4 4 NUM ddj-151 163 28 integers integer NOUN ddj-151 163 29 to to ADP ddj-151 163 30 each each DET ddj-151 163 31 group group NOUN ddj-151 163 32 . . PUNCT ddj-151 164 1 the the DET ddj-151 164 2 condition condition NOUN ddj-151 164 3 is be AUX ddj-151 164 4 that that SCONJ ddj-151 164 5 within within ADP ddj-151 164 6 each each DET ddj-151 164 7 group group NOUN ddj-151 164 8 the the DET ddj-151 164 9 notes note NOUN ddj-151 164 10 labelled label VERB ddj-151 164 11 by by ADP ddj-151 164 12 the the DET ddj-151 164 13 integers integer NOUN ddj-151 164 14 have have VERB ddj-151 164 15 minimum minimum ADJ ddj-151 164 16 distance distance NOUN ddj-151 164 17 of of ADP ddj-151 164 18 2 2 NUM ddj-151 164 19 units unit NOUN ddj-151 164 20 between between ADP ddj-151 164 21 themselves themselves PRON ddj-151 164 22 . . PUNCT ddj-151 165 1 3 3 X ddj-151 165 2 . . X ddj-151 165 3 one one PRON ddj-151 165 4 could could AUX ddj-151 165 5 try try VERB ddj-151 165 6 to to PART ddj-151 165 7 understand understand VERB ddj-151 165 8 the the DET ddj-151 165 9 situation situation NOUN ddj-151 165 10 in in ADP ddj-151 165 11 terms term NOUN ddj-151 165 12 of of ADP ddj-151 165 13 the the DET ddj-151 165 14 symmetries symmetry NOUN ddj-151 165 15 of of ADP ddj-151 165 16 the the DET ddj-151 165 17 system system NOUN ddj-151 165 18 . . PUNCT ddj-151 166 1 ( ( PUNCT ddj-151 166 2 a a X ddj-151 166 3 ) ) PUNCT ddj-151 166 4 could could AUX ddj-151 166 5 the the DET ddj-151 166 6 triplet triplet NOUN ddj-151 166 7 { { PUNCT ddj-151 166 8 c c NOUN ddj-151 166 9 , , PUNCT ddj-151 166 10 e e NOUN ddj-151 166 11 , , PUNCT ddj-151 166 12 g g NOUN ddj-151 166 13 ] ] X ddj-151 166 14 } } PUNCT ddj-151 166 15 and and CCONJ ddj-151 166 16 its its PRON ddj-151 166 17 four four NUM ddj-151 166 18 translates translate NOUN ddj-151 166 19 be be AUX ddj-151 166 20 interpreted interpret VERB ddj-151 166 21 as as ADP ddj-151 166 22 z3 z3 PROPN ddj-151 166 23 orbits orbit NOUN ddj-151 166 24 . . PUNCT ddj-151 167 1 could could AUX ddj-151 167 2 suitable suitable VERB ddj-151 167 3 chosen choose VERB ddj-151 167 4 members member NOUN ddj-151 167 5 from from ADP ddj-151 167 6 4 4 NUM ddj-151 167 7 disjoint disjoint NOUN ddj-151 167 8 quartets quartet NOUN ddj-151 167 9 quite quite ADV ddj-151 167 10 general general ADJ ddj-151 167 11 form form NOUN ddj-151 167 12 z3 z3 PROPN ddj-151 167 13 orbits orbit NOUN ddj-151 167 14 . . PUNCT ddj-151 168 1 remark remark PROPN ddj-151 168 2 : : PUNCT ddj-151 168 3 particle particle NOUN ddj-151 168 4 physicists physicist NOUN ddj-151 168 5 notes note VERB ddj-151 168 6 the the DET ddj-151 168 7 analogy analogy NOUN ddj-151 168 8 with with ADP ddj-151 168 9 4 4 NUM ddj-151 168 10 color color NOUN ddj-151 168 11 triplets triplet NOUN ddj-151 168 12 formed form VERB ddj-151 168 13 by by ADP ddj-151 168 14 u u PROPN ddj-151 168 15 and and CCONJ ddj-151 168 16 d d PROPN ddj-151 168 17 quarks quark NOUN ddj-151 168 18 having have VERB ddj-151 168 19 spin spin NOUN ddj-151 168 20 1/2 1/2 NUM ddj-151 168 21 . . PUNCT ddj-151 169 1 z4 z4 PROPN ddj-151 169 2 would would AUX ddj-151 169 3 correspond correspond VERB ddj-151 169 4 to to PART ddj-151 169 5 spin spin VERB ddj-151 169 6 and and CCONJ ddj-151 169 7 color color NOUN ddj-151 169 8 spin spin NOUN ddj-151 169 9 and and CCONJ ddj-151 169 10 z3 z3 VERB ddj-151 169 11 to to PART ddj-151 169 12 color color VERB ddj-151 169 13 . . PUNCT ddj-151 170 1 ( ( PUNCT ddj-151 170 2 b b X ddj-151 170 3 ) ) PUNCT ddj-151 170 4 z4 z4 PROPN ddj-151 170 5 acts act NOUN ddj-151 170 6 as as ADP ddj-151 170 7 symmetries symmetry NOUN ddj-151 170 8 of of ADP ddj-151 170 9 the the DET ddj-151 170 10 tesselation tesselation NOUN ddj-151 170 11 considered consider VERB ddj-151 170 12 and and CCONJ ddj-151 170 13 these these DET ddj-151 170 14 symmetries symmetry NOUN ddj-151 170 15 respect respect VERB ddj-151 170 16 distances distance NOUN ddj-151 170 17 so so SCONJ ddj-151 170 18 that that SCONJ ddj-151 170 19 their their PRON ddj-151 170 20 action action NOUN ddj-151 170 21 on on ADP ddj-151 170 22 a a DET ddj-151 170 23 quartet quartet NOUN ddj-151 170 24 with with ADP ddj-151 170 25 members member NOUN ddj-151 170 26 having have VERB ddj-151 170 27 mutual mutual ADJ ddj-151 170 28 distances distance NOUN ddj-151 170 29 larger large ADJ ddj-151 170 30 than than ADP ddj-151 170 31 unit unit NOUN ddj-151 170 32 creates create VERB ddj-151 170 33 new new ADJ ddj-151 170 34 such such ADJ ddj-151 170 35 quartet quartet NOUN ddj-151 170 36 . . PUNCT ddj-151 171 1 could could AUX ddj-151 171 2 the the DET ddj-151 171 3 triplet triplet NOUN ddj-151 171 4 { { PUNCT ddj-151 171 5 c c NOUN ddj-151 171 6 , , PUNCT ddj-151 171 7 e e NOUN ddj-151 171 8 , , PUNCT ddj-151 171 9 g g NOUN ddj-151 171 10 ] ] X ddj-151 171 11 } } PUNCT ddj-151 171 12 and and CCONJ ddj-151 171 13 its its PRON ddj-151 171 14 four four NUM ddj-151 171 15 translates translate NOUN ddj-151 171 16 by by ADP ddj-151 171 17 an an DET ddj-151 171 18 n−multiple n−multiple PROPN ddj-151 171 19 of of ADP ddj-151 171 20 half half NOUN ddj-151 171 21 note note NOUN ddj-151 171 22 , , PUNCT ddj-151 171 23 n n PROPN ddj-151 171 24 = = SYM ddj-151 171 25 0 0 NUM ddj-151 171 26 , , PUNCT ddj-151 171 27 1 1 NUM ddj-151 171 28 , , PUNCT ddj-151 171 29 2 2 NUM ddj-151 171 30 , , PUNCT ddj-151 171 31 3 3 NUM ddj-151 171 32 correspond correspond NOUN ddj-151 171 33 to to ADP ddj-151 171 34 an an DET ddj-151 171 35 orbit orbit NOUN ddj-151 171 36 z4 z4 PROPN ddj-151 171 37 ? ? PUNCT ddj-151 172 1 could could AUX ddj-151 172 2 the the DET ddj-151 172 3 groups group NOUN ddj-151 172 4 of of ADP ddj-151 172 5 4 4 NUM ddj-151 172 6 notes note NOUN ddj-151 172 7 quite quite ADV ddj-151 172 8 generally generally ADV ddj-151 172 9 correspond correspond VERB ddj-151 172 10 to to ADP ddj-151 172 11 the the DET ddj-151 172 12 orbits orbit NOUN ddj-151 172 13 of of ADP ddj-151 172 14 z4 z4 PROPN ddj-151 172 15 ? ? PUNCT ddj-151 173 1 this this PRON ddj-151 173 2 can can AUX ddj-151 173 3 be be AUX ddj-151 173 4 true true ADJ ddj-151 173 5 only only ADV ddj-151 173 6 if if SCONJ ddj-151 173 7 the the DET ddj-151 173 8 action action NOUN ddj-151 173 9 of of ADP ddj-151 173 10 non non ADJ ddj-151 173 11 - - ADJ ddj-151 173 12 trivial trivial ADJ ddj-151 173 13 z4 z4 PROPN ddj-151 173 14 elements element NOUN ddj-151 173 15 relates relate VERB ddj-151 173 16 only only ADV ddj-151 173 17 vertices vertex NOUN ddj-151 173 18 with with ADP ddj-151 173 19 distance distance NOUN ddj-151 173 20 larger large ADJ ddj-151 173 21 than than ADP ddj-151 173 22 one one NUM ddj-151 173 23 unit unit NOUN ddj-151 173 24 . . PUNCT ddj-151 174 1 4 4 X ddj-151 174 2 . . X ddj-151 174 3 the the DET ddj-151 174 4 group group NOUN ddj-151 174 5 of of ADP ddj-151 174 6 isometries isometry NOUN ddj-151 174 7 of of ADP ddj-151 174 8 the the DET ddj-151 174 9 toric toric ADJ ddj-151 174 10 triangulation triangulation NOUN ddj-151 174 11 acts act VERB ddj-151 174 12 as as ADP ddj-151 174 13 symmetries symmetry NOUN ddj-151 174 14 . . PUNCT ddj-151 175 1 z24 z24 PROPN ddj-151 175 2 = = SYM ddj-151 175 3 z6 z6 PROPN ddj-151 175 4 × × PROPN ddj-151 175 5 z4 z4 PROPN ddj-151 175 6 is be AUX ddj-151 175 7 a a DET ddj-151 175 8 good good ADJ ddj-151 175 9 candidate candidate NOUN ddj-151 175 10 for for ADP ddj-151 175 11 this this DET ddj-151 175 12 group group NOUN ddj-151 175 13 . . PUNCT ddj-151 176 1 z6 z6 PROPN ddj-151 176 2 corresponds correspond VERB ddj-151 176 3 to to ADP ddj-151 176 4 the the DET ddj-151 176 5 rotations rotation NOUN ddj-151 176 6 of of ADP ddj-151 176 7 around around ADV ddj-151 176 8 given give VERB ddj-151 176 9 point point NOUN ddj-151 176 10 of of ADP ddj-151 176 11 triangulation triangulation NOUN ddj-151 176 12 and and CCONJ ddj-151 176 13 should should AUX ddj-151 176 14 leave leave VERB ddj-151 176 15 the the DET ddj-151 176 16 tesselation tesselation NOUN ddj-151 176 17 invariant invariant ADJ ddj-151 176 18 . . PUNCT ddj-151 177 1 the the DET ddj-151 177 2 orbit orbit NOUN ddj-151 177 3 of of ADP ddj-151 177 4 given give VERB ddj-151 177 5 triangle triangle NOUN ddj-151 177 6 defining define VERB ddj-151 177 7 the the DET ddj-151 177 8 set set NOUN ddj-151 177 9 of of ADP ddj-151 177 10 dna dna PROPN ddj-151 177 11 codons codon NOUN ddj-151 177 12 coding code VERB ddj-151 177 13 the the DET ddj-151 177 14 amino amino NOUN ddj-151 177 15 - - PUNCT ddj-151 177 16 acid acid NOUN ddj-151 177 17 represented represent VERB ddj-151 177 18 by by ADP ddj-151 177 19 the the DET ddj-151 177 20 orbit orbit NOUN ddj-151 177 21 would would AUX ddj-151 177 22 correspond correspond VERB ddj-151 177 23 to to ADP ddj-151 177 24 orbit orbit NOUN ddj-151 177 25 of of ADP ddj-151 177 26 subgroups subgroup NOUN ddj-151 177 27 of of ADP ddj-151 177 28 z24 z24 PROPN ddj-151 177 29 . . PUNCT ddj-151 178 1 only only ADV ddj-151 178 2 orbits orbit NOUN ddj-151 178 3 containing contain VERB ddj-151 178 4 orbits orbit NOUN ddj-151 178 5 containing contain VERB ddj-151 178 6 1 1 NUM ddj-151 178 7 , , PUNCT ddj-151 178 8 2 2 NUM ddj-151 178 9 , , PUNCT ddj-151 178 10 3 3 NUM ddj-151 178 11 , , PUNCT ddj-151 178 12 4 4 NUM ddj-151 178 13 or or CCONJ ddj-151 178 14 6 6 NUM ddj-151 178 15 triangles triangle NOUN ddj-151 178 16 are be AUX ddj-151 178 17 allowed allow VERB ddj-151 178 18 by by ADP ddj-151 178 19 the the DET ddj-151 178 20 degeneracies degeneracy NOUN ddj-151 178 21 of of ADP ddj-151 178 22 the the DET ddj-151 178 23 genetic genetic ADJ ddj-151 178 24 code code NOUN ddj-151 178 25 . . PUNCT ddj-151 179 1 these these DET ddj-151 179 2 numbers number NOUN ddj-151 179 3 would would AUX ddj-151 179 4 correspond correspond VERB ddj-151 179 5 to to ADP ddj-151 179 6 degeneracies degeneracy NOUN ddj-151 179 7 that that PRON ddj-151 179 8 is be AUX ddj-151 179 9 the the DET ddj-151 179 10 numbers number NOUN ddj-151 179 11 of of ADP ddj-151 179 12 codons codon NOUN ddj-151 179 13 coding code VERB ddj-151 179 14 for for ADP ddj-151 179 15 given give VERB ddj-151 179 16 amino amino NOUN ddj-151 179 17 - - PUNCT ddj-151 179 18 acid acid NOUN ddj-151 179 19 . . PUNCT ddj-151 180 1 all all DET ddj-151 180 2 these these DET ddj-151 180 3 numbers number NOUN ddj-151 180 4 appear appear VERB ddj-151 180 5 as as ADP ddj-151 180 6 degeneracies degeneracy NOUN ddj-151 180 7 . . PUNCT ddj-151 181 1 issn issn PROPN ddj-151 181 2 : : PUNCT ddj-151 181 3 2159 2159 NUM ddj-151 181 4 - - PUNCT ddj-151 181 5 046x 046x NOUN ddj-151 181 6 dna dna NOUN ddj-151 181 7 decipher decipher PROPN ddj-151 181 8 journal journal PROPN ddj-151 181 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 181 10 published publish VERB ddj-151 181 11 by by ADP ddj-151 181 12 quantumdream quantumdream PROPN ddj-151 181 13 , , PUNCT ddj-151 181 14 inc inc PROPN ddj-151 181 15 . . PROPN ddj-151 181 16 dna dna PROPN ddj-151 181 17 decipher decipher PROPN ddj-151 181 18 journal journal PROPN ddj-151 181 19 | | ADP ddj-151 181 20 november november PROPN ddj-151 181 21 2018 2018 NUM ddj-151 182 1 | | ADV ddj-151 182 2 volume volume NOUN ddj-151 182 3 8 8 NUM ddj-151 182 4 | | NOUN ddj-151 182 5 issue issue VERB ddj-151 182 6 3 3 NUM ddj-151 182 7 | | CCONJ ddj-151 182 8 pp pp ADJ ddj-151 182 9 . . PUNCT ddj-151 183 1 197 197 NUM ddj-151 183 2 - - SYM ddj-151 183 3 205 205 NUM ddj-151 183 4 202 202 NUM ddj-151 183 5 pitkänen pitkänen NOUN ddj-151 183 6 , , PUNCT ddj-151 183 7 m. m. NOUN ddj-151 183 8 , , PUNCT ddj-151 183 9 thoughts thought NOUN ddj-151 183 10 on on ADP ddj-151 183 11 modification modification NOUN ddj-151 183 12 of of ADP ddj-151 183 13 bio bio NOUN ddj-151 183 14 - - NOUN ddj-151 183 15 harmony harmony NOUN ddj-151 183 16 2.3 2.3 NUM ddj-151 183 17 what what PRON ddj-151 183 18 one one PRON ddj-151 183 19 can can AUX ddj-151 183 20 say say VERB ddj-151 183 21 about about ADP ddj-151 183 22 toric toric ADJ ddj-151 183 23 hamiltonian hamiltonian ADJ ddj-151 183 24 cycles cycle NOUN ddj-151 183 25 ? ? PUNCT ddj-151 184 1 first first ADV ddj-151 184 2 some some DET ddj-151 184 3 basic basic ADJ ddj-151 184 4 notions notion NOUN ddj-151 184 5 are be AUX ddj-151 184 6 in in ADP ddj-151 184 7 order order NOUN ddj-151 184 8 . . PUNCT ddj-151 185 1 the the DET ddj-151 185 2 graph graph NOUN ddj-151 185 3 is be AUX ddj-151 185 4 said say VERB ddj-151 185 5 to to PART ddj-151 185 6 be be AUX ddj-151 185 7 equivelar equivelar ADJ ddj-151 185 8 if if SCONJ ddj-151 185 9 it it PRON ddj-151 185 10 is be AUX ddj-151 185 11 a a DET ddj-151 185 12 triangulation triangulation NOUN ddj-151 185 13 of of ADP ddj-151 185 14 a a DET ddj-151 185 15 surface surface NOUN ddj-151 185 16 meaning mean VERB ddj-151 185 17 that that SCONJ ddj-151 185 18 it it PRON ddj-151 185 19 has have VERB ddj-151 185 20 6 6 NUM ddj-151 185 21 edges edge NOUN ddj-151 185 22 emanating emanate VERB ddj-151 185 23 from from ADP ddj-151 185 24 each each DET ddj-151 185 25 vertex vertex NOUN ddj-151 185 26 and and CCONJ ddj-151 185 27 each each DET ddj-151 185 28 face face NOUN ddj-151 185 29 has have VERB ddj-151 185 30 3 3 NUM ddj-151 185 31 vertices vertex NOUN ddj-151 185 32 and and CCONJ ddj-151 185 33 3 3 NUM ddj-151 185 34 edges edge NOUN ddj-151 185 35 [ [ X ddj-151 185 36 1 1 NUM ddj-151 185 37 ] ] PUNCT ddj-151 185 38 . . PUNCT ddj-151 186 1 equivelarity equivelarity NOUN ddj-151 186 2 is be AUX ddj-151 186 3 equivalent equivalent ADJ ddj-151 186 4 with with ADP ddj-151 186 5 the the DET ddj-151 186 6 folllowing folllowe VERB ddj-151 186 7 conditions condition NOUN ddj-151 186 8 ; ; PUNCT ddj-151 187 1 1 1 X ddj-151 187 2 . . X ddj-151 188 1 every every DET ddj-151 188 2 vertex vertex NOUN ddj-151 188 3 is be AUX ddj-151 188 4 6 6 NUM ddj-151 188 5 - - PUNCT ddj-151 188 6 valent valent NOUN ddj-151 188 7 . . PUNCT ddj-151 189 1 2 2 X ddj-151 189 2 . . X ddj-151 189 3 the the DET ddj-151 189 4 edge edge NOUN ddj-151 189 5 graph graph NOUN ddj-151 189 6 is be AUX ddj-151 189 7 6 6 NUM ddj-151 189 8 - - PUNCT ddj-151 189 9 connected connect VERB ddj-151 189 10 . . PUNCT ddj-151 190 1 3 3 X ddj-151 190 2 . . X ddj-151 190 3 the the DET ddj-151 190 4 graph graph NOUN ddj-151 190 5 has have VERB ddj-151 190 6 vertex vertex NOUN ddj-151 190 7 transitive transitive ADJ ddj-151 190 8 automorphism automorphism NOUN ddj-151 190 9 group group NOUN ddj-151 190 10 . . PUNCT ddj-151 191 1 4 4 X ddj-151 191 2 . . X ddj-151 191 3 the the DET ddj-151 191 4 graph graph NOUN ddj-151 191 5 can can AUX ddj-151 191 6 be be AUX ddj-151 191 7 obtained obtain VERB ddj-151 191 8 as as ADP ddj-151 191 9 a a DET ddj-151 191 10 quotient quotient NOUN ddj-151 191 11 of of ADP ddj-151 191 12 the the DET ddj-151 191 13 universal universal ADJ ddj-151 191 14 covering covering NOUN ddj-151 191 15 tesselation tesselation NOUN ddj-151 191 16 ( ( PUNCT ddj-151 191 17 3,6 3,6 NUM ddj-151 191 18 ) ) PUNCT ddj-151 191 19 by by ADP ddj-151 191 20 a a DET ddj-151 191 21 sublattice sublattice NOUN ddj-151 191 22 ( ( PUNCT ddj-151 191 23 subgroup subgroup NOUN ddj-151 191 24 of of ADP ddj-151 191 25 translation translation NOUN ddj-151 191 26 group group NOUN ddj-151 191 27 ) ) PUNCT ddj-151 191 28 . . PUNCT ddj-151 192 1 6 6 NUM ddj-151 192 2 - - PUNCT ddj-151 192 3 connectedness connectedness NOUN ddj-151 192 4 means mean VERB ddj-151 192 5 that that SCONJ ddj-151 192 6 one one PRON ddj-151 192 7 can can AUX ddj-151 192 8 decompose decompose VERB ddj-151 192 9 the the DET ddj-151 192 10 tesselation tesselation NOUN ddj-151 192 11 into into ADP ddj-151 192 12 two two NUM ddj-151 192 13 disconnected disconnected ADJ ddj-151 192 14 pieces piece NOUN ddj-151 192 15 by by ADP ddj-151 192 16 removing remove VERB ddj-151 192 17 6 6 NUM ddj-151 192 18 or or CCONJ ddj-151 192 19 more more ADJ ddj-151 192 20 vertices vertex NOUN ddj-151 192 21 5 5 NUM ddj-151 192 22 . . NOUN ddj-151 192 23 edge edge NOUN ddj-151 192 24 graph graph NOUN ddj-151 192 25 is be AUX ddj-151 192 26 n n ADV ddj-151 192 27 - - PUNCT ddj-151 192 28 connected connect VERB ddj-151 192 29 if if SCONJ ddj-151 192 30 the the DET ddj-151 192 31 elimation elimation NOUN ddj-151 192 32 of of ADP ddj-151 192 33 k k PROPN ddj-151 192 34 < < X ddj-151 192 35 n n PRON ddj-151 192 36 vertices vertice VERB ddj-151 192 37 leaves leave VERB ddj-151 192 38 it it PRON ddj-151 192 39 connected connect VERB ddj-151 192 40 . . PUNCT ddj-151 193 1 it it PRON ddj-151 193 2 is be AUX ddj-151 193 3 known know VERB ddj-151 193 4 that that SCONJ ddj-151 193 5 every every DET ddj-151 193 6 5 5 NUM ddj-151 193 7 - - PUNCT ddj-151 193 8 connected connect VERB ddj-151 193 9 triangulation triangulation NOUN ddj-151 193 10 of of ADP ddj-151 193 11 torus torus NOUN ddj-151 193 12 is be AUX ddj-151 193 13 hamiltonian hamiltonian ADJ ddj-151 194 1 [ [ X ddj-151 194 2 2 2 NUM ddj-151 194 3 ] ] PUNCT ddj-151 194 4 ( ( PUNCT ddj-151 194 5 see see VERB ddj-151 194 6 http://tinyurl.com/y7cartk2 http://tinyurl.com/y7cartk2 NOUN ddj-151 194 7 ) ) PUNCT ddj-151 194 8 . . PUNCT ddj-151 195 1 therefore therefore ADV ddj-151 195 2 also also ADV ddj-151 195 3 6 6 NUM ddj-151 195 4 - - PUNCT ddj-151 195 5 connected connect VERB ddj-151 195 6 ( ( PUNCT ddj-151 195 7 6 6 NUM ddj-151 195 8 , , PUNCT ddj-151 195 9 3)p=2,q=2 3)p=2,q=2 PROPN ddj-151 195 10 tesselation tesselation NOUN ddj-151 195 11 has have VERB ddj-151 195 12 hamiltonian hamiltonian ADJ ddj-151 195 13 cycles cycle NOUN ddj-151 195 14 . . PUNCT ddj-151 196 1 6 6 X ddj-151 196 2 . . PUNCT ddj-151 196 3 the the DET ddj-151 196 4 hamiltonian hamiltonian ADJ ddj-151 196 5 cycles cycle NOUN ddj-151 196 6 for for ADP ddj-151 196 7 the the DET ddj-151 196 8 dual dual ADJ ddj-151 196 9 tesselation tesselation NOUN ddj-151 196 10 are be AUX ddj-151 196 11 not not PART ddj-151 196 12 in in ADP ddj-151 196 13 any any DET ddj-151 196 14 sense sense NOUN ddj-151 196 15 duals dual NOUN ddj-151 196 16 of of ADP ddj-151 196 17 those those PRON ddj-151 196 18 for for ADP ddj-151 196 19 the the DET ddj-151 196 20 tesselation tesselation NOUN ddj-151 196 21 . . PUNCT ddj-151 197 1 for for ADP ddj-151 197 2 instance instance NOUN ddj-151 197 3 , , PUNCT ddj-151 197 4 in in ADP ddj-151 197 5 the the DET ddj-151 197 6 case case NOUN ddj-151 197 7 of of ADP ddj-151 197 8 dodecahedron dodecahedron NOUN ddj-151 197 9 there there PRON ddj-151 197 10 is be VERB ddj-151 197 11 unique unique ADJ ddj-151 197 12 hamiltonian hamiltonian ADJ ddj-151 197 13 cycle cycle NOUN ddj-151 197 14 and and CCONJ ddj-151 197 15 for for ADP ddj-151 197 16 icosahedron icosahedron X ddj-151 197 17 has have VERB ddj-151 197 18 large large ADJ ddj-151 197 19 number number NOUN ddj-151 197 20 of of ADP ddj-151 197 21 cycles cycle NOUN ddj-151 197 22 . . PUNCT ddj-151 198 1 also also ADV ddj-151 198 2 in in ADP ddj-151 198 3 the the DET ddj-151 198 4 case case NOUN ddj-151 198 5 of of ADP ddj-151 198 6 ( ( PUNCT ddj-151 198 7 6 6 NUM ddj-151 198 8 , , PUNCT ddj-151 198 9 3 3 NUM ddj-151 198 10 ) ) PUNCT ddj-151 198 11 tesselations tesselation NOUN ddj-151 198 12 the the DET ddj-151 198 13 duals dual NOUN ddj-151 198 14 have have VERB ddj-151 198 15 different different ADJ ddj-151 198 16 hamilton hamilton PROPN ddj-151 198 17 cycles cycle NOUN ddj-151 198 18 . . PUNCT ddj-151 199 1 in in ADP ddj-151 199 2 fact fact NOUN ddj-151 199 3 , , PUNCT ddj-151 199 4 the the DET ddj-151 199 5 problem problem NOUN ddj-151 199 6 of of ADP ddj-151 199 7 constructing construct VERB ddj-151 199 8 the the DET ddj-151 199 9 hamiltonian hamiltonian ADJ ddj-151 199 10 cycles cycle NOUN ddj-151 199 11 is be AUX ddj-151 199 12 np np INTJ ddj-151 199 13 complete complete ADJ ddj-151 199 14 . . PUNCT ddj-151 200 1 can can AUX ddj-151 200 2 one one PRON ddj-151 200 3 say say VERB ddj-151 200 4 anything anything PRON ddj-151 200 5 about about ADP ddj-151 200 6 the the DET ddj-151 200 7 number number NOUN ddj-151 200 8 of of ADP ddj-151 200 9 hamiltonian hamiltonian ADJ ddj-151 200 10 cycles cycle NOUN ddj-151 200 11 ? ? PUNCT ddj-151 201 1 1 1 X ddj-151 201 2 . . X ddj-151 201 3 for for ADP ddj-151 201 4 dodecahedron dodecahedron NOUN ddj-151 201 5 only only ADV ddj-151 201 6 3 3 NUM ddj-151 201 7 edges edge NOUN ddj-151 201 8 emanates emanate VERB ddj-151 201 9 from from ADP ddj-151 201 10 a a DET ddj-151 201 11 given give VERB ddj-151 201 12 vertex vertex NOUN ddj-151 201 13 and and CCONJ ddj-151 201 14 there there PRON ddj-151 201 15 is be VERB ddj-151 201 16 only only ADV ddj-151 201 17 one one NUM ddj-151 201 18 hamiltonian hamiltonian ADJ ddj-151 201 19 cycle cycle NOUN ddj-151 201 20 . . PUNCT ddj-151 202 1 for for SCONJ ddj-151 202 2 icosahedron icosahedron X ddj-151 202 3 5 5 NUM ddj-151 202 4 edges edge NOUN ddj-151 202 5 emanate emanate VERB ddj-151 202 6 from from ADP ddj-151 202 7 given give VERB ddj-151 202 8 vertex vertex NOUN ddj-151 202 9 and and CCONJ ddj-151 202 10 the the DET ddj-151 202 11 number number NOUN ddj-151 202 12 of of ADP ddj-151 202 13 cycles cycle NOUN ddj-151 202 14 is be AUX ddj-151 202 15 rather rather ADV ddj-151 202 16 large large ADJ ddj-151 202 17 . . PUNCT ddj-151 203 1 hence hence ADV ddj-151 203 2 the the DET ddj-151 203 3 valence valence NOUN ddj-151 203 4 and and CCONJ ddj-151 203 5 also also ADV ddj-151 203 6 closely closely ADV ddj-151 203 7 related relate VERB ddj-151 203 8 notion notion NOUN ddj-151 203 9 of of ADP ddj-151 203 10 n n X ddj-151 203 11 - - PUNCT ddj-151 203 12 connectedness connectedness NOUN ddj-151 203 13 are be AUX ddj-151 203 14 essential essential ADJ ddj-151 203 15 for for ADP ddj-151 203 16 the the DET ddj-151 203 17 existence existence NOUN ddj-151 203 18 of of ADP ddj-151 203 19 hamilton hamilton PROPN ddj-151 203 20 ’s ’s PART ddj-151 203 21 cycles cycle NOUN ddj-151 203 22 . . PUNCT ddj-151 204 1 for for ADP ddj-151 204 2 instance instance NOUN ddj-151 204 3 , , PUNCT ddj-151 204 4 for for ADP ddj-151 204 5 a a DET ddj-151 204 6 graph graph NOUN ddj-151 204 7 consisting consist VERB ddj-151 204 8 of of ADP ddj-151 204 9 two two NUM ddj-151 204 10 connected connected ADJ ddj-151 204 11 graphs graph NOUN ddj-151 204 12 connected connect VERB ddj-151 204 13 by by ADP ddj-151 204 14 single single ADJ ddj-151 204 15 edge edge NOUN ddj-151 204 16 , , PUNCT ddj-151 204 17 there there PRON ddj-151 204 18 exist exist VERB ddj-151 204 19 no no DET ddj-151 204 20 hamilton hamilton PROPN ddj-151 204 21 ’s ’s PART ddj-151 204 22 cycles cycle NOUN ddj-151 204 23 . . PUNCT ddj-151 205 1 for for ADP ddj-151 205 2 toric toric ADJ ddj-151 205 3 triangulations triangulation NOUN ddj-151 205 4 one one PRON ddj-151 205 5 has have VERB ddj-151 205 6 as as ADV ddj-151 205 7 many many ADJ ddj-151 205 8 as as ADP ddj-151 205 9 6 6 NUM ddj-151 205 10 edges edge NOUN ddj-151 205 11 from from ADP ddj-151 205 12 given give VERB ddj-151 205 13 vertex vertex NOUN ddj-151 205 14 and and CCONJ ddj-151 205 15 this this PRON ddj-151 205 16 favors favor VERB ddj-151 205 17 the the DET ddj-151 205 18 formation formation NOUN ddj-151 205 19 of of ADP ddj-151 205 20 a a DET ddj-151 205 21 large large ADJ ddj-151 205 22 number number NOUN ddj-151 205 23 of of ADP ddj-151 205 24 hamiltonian hamiltonian ADJ ddj-151 205 25 cycles cycle NOUN ddj-151 205 26 . . PUNCT ddj-151 206 1 2 2 X ddj-151 206 2 . . X ddj-151 206 3 curves curve NOUN ddj-151 206 4 on on ADP ddj-151 206 5 torus torus NOUN ddj-151 206 6 are be AUX ddj-151 206 7 labelled label VERB ddj-151 206 8 by by ADP ddj-151 206 9 winding wind VERB ddj-151 206 10 numbers number NOUN ddj-151 206 11 ( ( PUNCT ddj-151 206 12 m m NOUN ddj-151 206 13 , , PUNCT ddj-151 206 14 n n CCONJ ddj-151 206 15 ) ) PUNCT ddj-151 206 16 telling tell VERB ddj-151 206 17 the the DET ddj-151 206 18 homology homology NOUN ddj-151 206 19 equivalence equivalence NOUN ddj-151 206 20 class class NOUN ddj-151 206 21 of of ADP ddj-151 206 22 the the DET ddj-151 206 23 cycle cycle NOUN ddj-151 206 24 . . PUNCT ddj-151 207 1 m m PROPN ddj-151 207 2 and and CCONJ ddj-151 207 3 m m VERB ddj-151 207 4 can can AUX ddj-151 207 5 be be AUX ddj-151 207 6 any any DET ddj-151 207 7 integers integer NOUN ddj-151 207 8 . . PUNCT ddj-151 208 1 curve curve VERB ddj-151 208 2 winds wind NOUN ddj-151 208 3 m m VERB ddj-151 208 4 ( ( PUNCT ddj-151 208 5 n n CCONJ ddj-151 208 6 ) ) PUNCT ddj-151 208 7 times time NOUN ddj-151 208 8 around around ADP ddj-151 208 9 the the DET ddj-151 208 10 circle circle NOUN ddj-151 208 11 defining define VERB ddj-151 208 12 the the DET ddj-151 208 13 first first ADJ ddj-151 208 14 ( ( PUNCT ddj-151 208 15 second second ADJ ddj-151 208 16 ) ) PUNCT ddj-151 208 17 equivalence equivalence NOUN ddj-151 208 18 homology homology NOUN ddj-151 208 19 equivalence equivalence NOUN ddj-151 208 20 class class NOUN ddj-151 208 21 . . PUNCT ddj-151 209 1 also also ADV ddj-151 209 2 hamiltonian hamiltonian ADJ ddj-151 209 3 cycles cycle NOUN ddj-151 209 4 are be AUX ddj-151 209 5 characterized characterize VERB ddj-151 209 6 by by ADP ddj-151 209 7 their their PRON ddj-151 209 8 homology homology NOUN ddj-151 209 9 equivalence equivalence NOUN ddj-151 209 10 class class NOUN ddj-151 209 11 , , PUNCT ddj-151 209 12 that that PRON ddj-151 209 13 is be AUX ddj-151 209 14 pair pair NOUN ddj-151 209 15 ( ( PUNCT ddj-151 209 16 m m NOUN ddj-151 209 17 , , PUNCT ddj-151 209 18 n n CCONJ ddj-151 209 19 ) ) PUNCT ddj-151 209 20 of of ADP ddj-151 209 21 integers integer NOUN ddj-151 209 22 . . PUNCT ddj-151 210 1 since since SCONJ ddj-151 210 2 there there PRON ddj-151 210 3 are be VERB ddj-151 210 4 only only ADV ddj-151 210 5 v v ADJ ddj-151 210 6 = = SYM ddj-151 210 7 12 12 NUM ddj-151 210 8 points point NOUN ddj-151 210 9 , , PUNCT ddj-151 210 10 the the DET ddj-151 210 11 numbers number NOUN ddj-151 210 12 ( ( PUNCT ddj-151 210 13 m m NOUN ddj-151 210 14 , , PUNCT ddj-151 210 15 n n CCONJ ddj-151 210 16 ) ) PUNCT ddj-151 210 17 are be AUX ddj-151 210 18 finite finite ADJ ddj-151 210 19 . . PUNCT ddj-151 211 1 by by ADP ddj-151 211 2 periodic periodic ADJ ddj-151 211 3 boundary boundary ADJ ddj-151 211 4 conditions condition NOUN ddj-151 211 5 means mean VERB ddj-151 211 6 that that SCONJ ddj-151 211 7 the the DET ddj-151 211 8 translations translation NOUN ddj-151 211 9 by by ADP ddj-151 211 10 multiples multiple NOUN ddj-151 211 11 of of ADP ddj-151 211 12 2e1 2e1 NUM ddj-151 211 13 + + CCONJ ddj-151 211 14 2e2 2e2 NUM ddj-151 211 15 do do AUX ddj-151 211 16 not not PART ddj-151 211 17 affect affect VERB ddj-151 211 18 the the DET ddj-151 211 19 tesselation tesselation NOUN ddj-151 211 20 ( ( PUNCT ddj-151 211 21 one one PRON ddj-151 211 22 can can AUX ddj-151 211 23 see see VERB ddj-151 211 24 what what PRON ddj-151 211 25 this this PRON ddj-151 211 26 means mean VERB ddj-151 211 27 geometrically geometrically ADV ddj-151 211 28 from from ADP ddj-151 211 29 the the DET ddj-151 211 30 illustration illustration NOUN ddj-151 211 31 at at ADP ddj-151 211 32 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 211 33 ) ) PUNCT ddj-151 211 34 . . PUNCT ddj-151 212 1 does do AUX ddj-151 212 2 this this PRON ddj-151 212 3 mean mean VERB ddj-151 212 4 that that SCONJ ddj-151 212 5 ( ( PUNCT ddj-151 212 6 m m NOUN ddj-151 212 7 , , PUNCT ddj-151 212 8 n n CCONJ ddj-151 212 9 ) ) PUNCT ddj-151 212 10 belongs belong VERB ddj-151 212 11 to to ADP ddj-151 212 12 z2 z2 PROPN ddj-151 212 13 × × PROPN ddj-151 212 14 z2 z2 PROPN ddj-151 212 15 so so SCONJ ddj-151 212 16 that that SCONJ ddj-151 212 17 one one PRON ddj-151 212 18 would would AUX ddj-151 212 19 have have VERB ddj-151 212 20 4 4 NUM ddj-151 212 21 homologically homologically ADV ddj-151 212 22 non non ADJ ddj-151 212 23 - - ADJ ddj-151 212 24 equivalent equivalent ADJ ddj-151 212 25 paths path NOUN ddj-151 212 26 . . PUNCT ddj-151 213 1 are be AUX ddj-151 213 2 all all PRON ddj-151 213 3 four four NUM ddj-151 213 4 homology homology NOUN ddj-151 213 5 classes class NOUN ddj-151 213 6 realized realize VERB ddj-151 213 7 as as ADP ddj-151 213 8 hamiltonian hamiltonian ADJ ddj-151 213 9 cycles cycle NOUN ddj-151 213 10 ? ? PUNCT ddj-151 214 1 does do AUX ddj-151 214 2 given give VERB ddj-151 214 3 homology homology NOUN ddj-151 214 4 class class NOUN ddj-151 214 5 contain contain VERB ddj-151 214 6 several several ADJ ddj-151 214 7 representatives representative NOUN ddj-151 214 8 or or CCONJ ddj-151 214 9 only only ADV ddj-151 214 10 single single ADJ ddj-151 214 11 one one NUM ddj-151 214 12 in in ADP ddj-151 214 13 which which DET ddj-151 214 14 case case NOUN ddj-151 214 15 one one PRON ddj-151 214 16 would would AUX ddj-151 214 17 have have VERB ddj-151 214 18 20 20 NUM ddj-151 214 19 non non ADJ ddj-151 214 20 - - ADJ ddj-151 214 21 equivalent equivalent ADJ ddj-151 214 22 hamiltonian hamiltonian ADJ ddj-151 214 23 cycles cycle NOUN ddj-151 214 24 ? ? PUNCT ddj-151 215 1 it it PRON ddj-151 215 2 turned turn VERB ddj-151 215 3 out out ADP ddj-151 215 4 that that SCONJ ddj-151 215 5 there there PRON ddj-151 215 6 exist exist VERB ddj-151 215 7 programs program NOUN ddj-151 215 8 coding code VERB ddj-151 215 9 for for ADP ddj-151 215 10 an an DET ddj-151 215 11 algorithm algorithm NOUN ddj-151 215 12 for for ADP ddj-151 215 13 finding find VERB ddj-151 215 14 whether whether SCONJ ddj-151 215 15 given give VERB ddj-151 215 16 graph graph NOUN ddj-151 215 17 ( ( PUNCT ddj-151 215 18 much much ADV ddj-151 215 19 more more ADV ddj-151 215 20 general general ADJ ddj-151 215 21 than than ADP ddj-151 215 22 tesselation tesselation NOUN ddj-151 215 23 ) ) PUNCT ddj-151 215 24 has have VERB ddj-151 215 25 hamiltonian hamiltonian ADJ ddj-151 215 26 cycles cycle NOUN ddj-151 215 27 . . PUNCT ddj-151 216 1 having having AUX ddj-151 216 2 told tell VERB ddj-151 216 3 to to PART ddj-151 216 4 jebin jebin VERB ddj-151 216 5 larosh larosh PROPN ddj-151 216 6 about about ADP ddj-151 216 7 the the DET ddj-151 216 8 problem problem NOUN ddj-151 216 9 , , PUNCT ddj-151 216 10 he he PRON ddj-151 216 11 sent send VERB ddj-151 216 12 within within ADP ddj-151 216 13 five five NUM ddj-151 216 14 minutes minute NOUN ddj-151 216 15 a a DET ddj-151 216 16 link link NOUN ddj-151 216 17 to to ADP ddj-151 216 18 a a DET ddj-151 216 19 java java NOUN ddj-151 216 20 algorithm algorithm NOUN ddj-151 216 21 allowing allow VERB ddj-151 216 22 to to PART ddj-151 216 23 show show VERB ddj-151 216 24 whether whether SCONJ ddj-151 216 25 a a DET ddj-151 216 26 given give VERB ddj-151 216 27 graph graph NOUN ddj-151 216 28 is be AUX ddj-151 216 29 hamiltonian hamiltonian ADJ ddj-151 216 30 ( ( PUNCT ddj-151 216 31 see see VERB ddj-151 216 32 http://tinyurl.com/y7y9tr5 http://tinyurl.com/y7y9tr5 NOUN ddj-151 216 33 t t PROPN ddj-151 216 34 ): ): PUNCT ddj-151 216 35 sincere sincere ADJ ddj-151 216 36 thanks thank NOUN ddj-151 216 37 to to ADP ddj-151 216 38 jebin jebin PROPN ddj-151 216 39 ! ! PUNCT ddj-151 217 1 by by ADP ddj-151 217 2 a a DET ddj-151 217 3 suitable suitable ADJ ddj-151 217 4 modification modification NOUN ddj-151 217 5 this this DET ddj-151 217 6 algorithm algorithm NOUN ddj-151 217 7 find find VERB ddj-151 217 8 all all DET ddj-151 217 9 hamiltonian hamiltonian ADJ ddj-151 217 10 cycles cycle NOUN ddj-151 217 11 . . PUNCT ddj-151 218 1 issn issn PROPN ddj-151 218 2 : : PUNCT ddj-151 218 3 2159 2159 NUM ddj-151 218 4 - - PUNCT ddj-151 218 5 046x 046x NOUN ddj-151 218 6 dna dna NOUN ddj-151 218 7 decipher decipher PROPN ddj-151 218 8 journal journal PROPN ddj-151 218 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 218 10 published publish VERB ddj-151 218 11 by by ADP ddj-151 218 12 quantumdream quantumdream PROPN ddj-151 218 13 , , PUNCT ddj-151 218 14 inc inc PROPN ddj-151 218 15 . . PROPN ddj-151 218 16 http://tinyurl.com/y7cartk2 http://tinyurl.com/y7cartk2 PROPN ddj-151 218 17 http://tinyurl.com/y7xfromc http://tinyurl.com/y7xfromc PROPN ddj-151 218 18 http://tinyurl.com/y7y9tr5 http://tinyurl.com/y7y9tr5 PROPN ddj-151 218 19 t t PROPN ddj-151 218 20 dna dna PROPN ddj-151 218 21 decipher decipher PROPN ddj-151 218 22 journal journal PROPN ddj-151 218 23 | | ADP ddj-151 218 24 november november PROPN ddj-151 218 25 2018 2018 NUM ddj-151 218 26 | | ADV ddj-151 218 27 volume volume NOUN ddj-151 218 28 8 8 NUM ddj-151 218 29 | | NOUN ddj-151 218 30 issue issue VERB ddj-151 218 31 3 3 NUM ddj-151 218 32 | | CCONJ ddj-151 218 33 pp pp ADJ ddj-151 218 34 . . PUNCT ddj-151 219 1 197 197 NUM ddj-151 219 2 - - SYM ddj-151 219 3 205 205 NUM ddj-151 219 4 203 203 NUM ddj-151 219 5 pitkänen pitkänen NOUN ddj-151 219 6 , , PUNCT ddj-151 219 7 m. m. NOUN ddj-151 219 8 , , PUNCT ddj-151 219 9 thoughts thought NOUN ddj-151 219 10 on on ADP ddj-151 219 11 modification modification NOUN ddj-151 219 12 of of ADP ddj-151 219 13 bio bio ADJ ddj-151 219 14 - - ADJ ddj-151 219 15 harmony harmony ADJ ddj-151 219 16 figure figure NOUN ddj-151 219 17 1 1 NUM ddj-151 219 18 : : PUNCT ddj-151 219 19 the the DET ddj-151 219 20 number number NOUN ddj-151 219 21 of of ADP ddj-151 219 22 the the DET ddj-151 219 23 vertices vertex NOUN ddj-151 219 24 of of ADP ddj-151 219 25 ( ( PUNCT ddj-151 219 26 v v NOUN ddj-151 219 27 , , PUNCT ddj-151 219 28 f f PROPN ddj-151 219 29 ) ) PUNCT ddj-151 220 1 = = SYM ddj-151 220 2 12 12 NUM ddj-151 220 3 , , PUNCT ddj-151 220 4 24 24 NUM ddj-151 220 5 ) ) PUNCT ddj-151 220 6 torus torus NOUN ddj-151 220 7 tesselation tesselation NOUN ddj-151 220 8 allowing allow VERB ddj-151 220 9 path path NOUN ddj-151 220 10 ( ( PUNCT ddj-151 220 11 0 0 NUM ddj-151 220 12 , , PUNCT ddj-151 220 13 1 1 NUM ddj-151 220 14 , , PUNCT ddj-151 220 15 2 2 NUM ddj-151 220 16 , , PUNCT ddj-151 220 17 3 3 NUM ddj-151 220 18 , , PUNCT ddj-151 220 19 4 4 NUM ddj-151 220 20 , , PUNCT ddj-151 220 21 6 6 NUM ddj-151 220 22 , , PUNCT ddj-151 220 23 5 5 NUM ddj-151 220 24 , , PUNCT ddj-151 220 25 8 8 NUM ddj-151 220 26 , , PUNCT ddj-151 220 27 10 10 NUM ddj-151 220 28 , , PUNCT ddj-151 220 29 7 7 NUM ddj-151 220 30 , , PUNCT ddj-151 220 31 11 11 NUM ddj-151 220 32 , , PUNCT ddj-151 220 33 9 9 NUM ddj-151 220 34 , , PUNCT ddj-151 220 35 0 0 NUM ddj-151 220 36 ) ) PUNCT ddj-151 220 37 as as ADP ddj-151 220 38 one one NUM ddj-151 220 39 particular particular ADJ ddj-151 220 40 hamiltonian hamiltonian ADJ ddj-151 220 41 cycle cycle NOUN ddj-151 220 42 . . PUNCT ddj-151 221 1 1 1 X ddj-151 221 2 . . X ddj-151 221 3 the the PRON ddj-151 221 4 number number NOUN ddj-151 221 5 nh nh PROPN ddj-151 221 6 of of ADP ddj-151 221 7 hamiltonian hamiltonian ADJ ddj-151 221 8 cycles cycle NOUN ddj-151 221 9 is be AUX ddj-151 221 10 expected expect VERB ddj-151 221 11 to to PART ddj-151 221 12 be be AUX ddj-151 221 13 rather rather ADV ddj-151 221 14 large large ADJ ddj-151 221 15 for for ADP ddj-151 221 16 a a DET ddj-151 221 17 torus torus NOUN ddj-151 221 18 triangulation triangulation NOUN ddj-151 221 19 with with ADP ddj-151 221 20 12 12 NUM ddj-151 221 21 vertices vertex NOUN ddj-151 221 22 and and CCONJ ddj-151 221 23 24 24 NUM ddj-151 221 24 triangles triangle NOUN ddj-151 221 25 and and CCONJ ddj-151 221 26 it it PRON ddj-151 221 27 is be AUX ddj-151 221 28 indeed indeed ADV ddj-151 221 29 so so ADV ddj-151 221 30 : : PUNCT ddj-151 221 31 nh nh PROPN ddj-151 221 32 = = PUNCT ddj-151 221 33 27816 27816 NUM ddj-151 221 34 ! ! PUNCT ddj-151 222 1 the the DET ddj-151 222 2 image image NOUN ddj-151 222 3 of of ADP ddj-151 222 4 the the DET ddj-151 222 5 tessellation tessellation NOUN ddj-151 222 6 and and CCONJ ddj-151 222 7 the the DET ddj-151 222 8 numbering numbering NOUN ddj-151 222 9 of of ADP ddj-151 222 10 its its PRON ddj-151 222 11 vertices vertex NOUN ddj-151 222 12 are be AUX ddj-151 222 13 described describe VERB ddj-151 222 14 in in ADP ddj-151 222 15 figure figure NOUN ddj-151 222 16 below below ADV ddj-151 222 17 ( ( PUNCT ddj-151 222 18 see see VERB ddj-151 222 19 fig fig NOUN ddj-151 222 20 . . PUNCT ddj-151 222 21 1 1 NUM ddj-151 222 22 ) ) PUNCT ddj-151 222 23 . . PUNCT ddj-151 223 1 incide incide ADJ ddj-151 223 2 matrix matrix NOUN ddj-151 223 3 a a PRON ddj-151 223 4 characterizes characterize VERB ddj-151 223 5 the the DET ddj-151 223 6 graph graph NOUN ddj-151 223 7 : : PUNCT ddj-151 223 8 if if SCONJ ddj-151 223 9 vertices vertice VERB ddj-151 223 10 i i PRON ddj-151 223 11 and and CCONJ ddj-151 223 12 j j PROPN ddj-151 223 13 are be AUX ddj-151 223 14 connected connect VERB ddj-151 223 15 by by ADP ddj-151 223 16 edge edge NOUN ddj-151 223 17 , , PUNCT ddj-151 223 18 one one PRON ddj-151 223 19 has have VERB ddj-151 223 20 aij aij PROPN ddj-151 223 21 = = SYM ddj-151 223 22 aji aji NOUN ddj-151 223 23 = = SYM ddj-151 223 24 1 1 NUM ddj-151 223 25 , , PUNCT ddj-151 223 26 otherwise otherwise ADV ddj-151 223 27 aij aij PROPN ddj-151 223 28 = = SYM ddj-151 223 29 aji aji NOUN ddj-151 223 30 = = SYM ddj-151 223 31 0 0 PUNCT ddj-151 223 32 and and CCONJ ddj-151 223 33 is be AUX ddj-151 223 34 used use VERB ddj-151 223 35 as as ADP ddj-151 223 36 data datum NOUN ddj-151 223 37 in in ADP ddj-151 223 38 the the DET ddj-151 223 39 algorithm algorithm NOUN ddj-151 223 40 finding find VERB ddj-151 223 41 the the DET ddj-151 223 42 hamiltonian hamiltonian ADJ ddj-151 223 43 cycles cycle NOUN ddj-151 223 44 . . PUNCT ddj-151 224 1 the the DET ddj-151 224 2 cycles cycle NOUN ddj-151 224 3 related relate VERB ddj-151 224 4 by by ADP ddj-151 224 5 the the DET ddj-151 224 6 isometries isometry NOUN ddj-151 224 7 of of ADP ddj-151 224 8 torus torus NOUN ddj-151 224 9 tessellation tessellation NOUN ddj-151 224 10 are be AUX ddj-151 224 11 however however ADV ddj-151 224 12 equivalent equivalent ADJ ddj-151 224 13 . . PUNCT ddj-151 225 1 the the DET ddj-151 225 2 guess guess NOUN ddj-151 225 3 is be AUX ddj-151 225 4 that that SCONJ ddj-151 225 5 the the DET ddj-151 225 6 group group NOUN ddj-151 225 7 of of ADP ddj-151 225 8 isometries isometry NOUN ddj-151 225 9 is be AUX ddj-151 225 10 g g PROPN ddj-151 225 11 = = SYM ddj-151 225 12 z2,reflo z2,reflo PROPN ddj-151 225 13 ( ( PUNCT ddj-151 225 14 z4,trozn z4,trozn NUM ddj-151 225 15 , , PUNCT ddj-151 225 16 rot rot NOUN ddj-151 225 17 ) ) PUNCT ddj-151 225 18 . . PUNCT ddj-151 226 1 zn zn X ddj-151 226 2 , , PUNCT ddj-151 226 3 rot rot VERB ddj-151 226 4 is be AUX ddj-151 226 5 a a DET ddj-151 226 6 subgroup subgroup NOUN ddj-151 226 7 of of ADP ddj-151 226 8 local local ADJ ddj-151 226 9 z6,rot z6,rot PROPN ddj-151 226 10 . . PUNCT ddj-151 227 1 a a DET ddj-151 227 2 priori priori ADJ ddj-151 227 3 n n X ddj-151 227 4 ∈ ∈ PROPN ddj-151 227 5 { { PUNCT ddj-151 227 6 1 1 NUM ddj-151 227 7 , , PUNCT ddj-151 227 8 2 2 NUM ddj-151 227 9 , , PUNCT ddj-151 227 10 3 3 NUM ddj-151 227 11 , , PUNCT ddj-151 227 12 6 6 NUM ddj-151 227 13 } } PUNCT ddj-151 227 14 is be AUX ddj-151 227 15 allowed allow VERB ddj-151 227 16 . . PUNCT ddj-151 228 1 on on ADP ddj-151 228 2 basis basis NOUN ddj-151 228 3 of of ADP ddj-151 228 4 [ [ X ddj-151 228 5 1 1 X ddj-151 228 6 ] ] X ddj-151 228 7 i i PRON ddj-151 228 8 have have AUX ddj-151 228 9 understood understand VERB ddj-151 228 10 that that SCONJ ddj-151 228 11 one one PRON ddj-151 228 12 has have VERB ddj-151 228 13 n n NOUN ddj-151 228 14 = = SYM ddj-151 228 15 3 3 NUM ddj-151 228 16 but but CCONJ ddj-151 228 17 that that DET ddj-151 228 18 one one PRON ddj-151 228 19 can can AUX ddj-151 228 20 express express VERB ddj-151 228 21 the the DET ddj-151 228 22 local local ADJ ddj-151 228 23 action action NOUN ddj-151 228 24 of of ADP ddj-151 228 25 z6,rot z6,rot PROPN ddj-151 228 26 as as ADP ddj-151 228 27 the the DET ddj-151 228 28 action action NOUN ddj-151 228 29 of of ADP ddj-151 228 30 the the DET ddj-151 228 31 semidirect semidirect NOUN ddj-151 228 32 product product NOUN ddj-151 228 33 z2,refl z2,refl NUM ddj-151 228 34 × × NOUN ddj-151 228 35 z3,rot z3,rot X ddj-151 228 36 at at ADP ddj-151 228 37 a a DET ddj-151 228 38 point point NOUN ddj-151 228 39 of of ADP ddj-151 228 40 tesselation tesselation NOUN ddj-151 228 41 ( ( PUNCT ddj-151 228 42 see see VERB ddj-151 228 43 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 228 44 ) ) PUNCT ddj-151 228 45 . . PUNCT ddj-151 229 1 the the DET ddj-151 229 2 identity identity NOUN ddj-151 229 3 of of ADP ddj-151 229 4 the the DET ddj-151 229 5 global global ADJ ddj-151 229 6 actions action NOUN ddj-151 229 7 z2,refl×z3,rot z2,refl×z3,rot PROPN ddj-151 229 8 and and CCONJ ddj-151 229 9 z6,rot z6,rot PROPN ddj-151 229 10 does do AUX ddj-151 229 11 not not PART ddj-151 229 12 look look VERB ddj-151 229 13 feasible feasible ADJ ddj-151 229 14 to to ADP ddj-151 229 15 me i PRON ddj-151 229 16 . . PUNCT ddj-151 230 1 therefore therefore ADV ddj-151 230 2 g g PROPN ddj-151 230 3 = = PUNCT ddj-151 230 4 z2,refl z2,refl NUM ddj-151 230 5 o o X ddj-151 230 6 ( ( PUNCT ddj-151 230 7 z4,tr z4,tr NOUN ddj-151 230 8 o o PROPN ddj-151 230 9 z3,rot z3,rot PROPN ddj-151 230 10 ) ) PUNCT ddj-151 230 11 with with ADP ddj-151 230 12 order order NOUN ddj-151 230 13 ord(g ord(g NUM ddj-151 230 14 ) ) PUNCT ddj-151 230 15 = = SYM ddj-151 230 16 24 24 NUM ddj-151 230 17 will will AUX ddj-151 230 18 be be AUX ddj-151 230 19 assumed assume VERB ddj-151 230 20 in in ADP ddj-151 230 21 the the DET ddj-151 230 22 following follow VERB ddj-151 230 23 ( ( PUNCT ddj-151 230 24 note note VERB ddj-151 230 25 that that SCONJ ddj-151 230 26 for for ADP ddj-151 230 27 icosahedral icosahedral ADJ ddj-151 230 28 tesselation tesselation NOUN ddj-151 230 29 one one NOUN ddj-151 230 30 has have VERB ddj-151 230 31 ord(g ord(g PROPN ddj-151 230 32 ) ) PUNCT ddj-151 230 33 = = SYM ddj-151 231 1 120 120 NUM ddj-151 231 2 so so SCONJ ddj-151 231 3 that that SCONJ ddj-151 231 4 there there PRON ddj-151 231 5 is be VERB ddj-151 231 6 symmetry symmetry NOUN ddj-151 231 7 breaking breaking NOUN ddj-151 231 8 ) ) PUNCT ddj-151 231 9 . . PUNCT ddj-151 232 1 z4 z4 PROPN ddj-151 232 2 would would AUX ddj-151 232 3 have have VERB ddj-151 232 4 as as ADP ddj-151 232 5 generators generator NOUN ddj-151 232 6 the the DET ddj-151 232 7 translations translation NOUN ddj-151 232 8 e1 e1 PROPN ddj-151 232 9 and and CCONJ ddj-151 232 10 e2 e2 NOUN ddj-151 232 11 defining define VERB ddj-151 232 12 the the DET ddj-151 232 13 conformal conformal ADJ ddj-151 232 14 equivalence equivalence NOUN ddj-151 232 15 class class NOUN ddj-151 232 16 of of ADP ddj-151 232 17 torus torus NOUN ddj-151 232 18 . . PUNCT ddj-151 233 1 the the DET ddj-151 233 2 multiples multiple NOUN ddj-151 233 3 of of ADP ddj-151 233 4 2(e1 2(e1 NUM ddj-151 233 5 + + CCONJ ddj-151 233 6 e2 e2 PROPN ddj-151 233 7 ) ) PUNCT ddj-151 233 8 would would AUX ddj-151 233 9 leave leave VERB ddj-151 233 10 the the DET ddj-151 233 11 tesselation tesselation NOUN ddj-151 233 12 invariant invariant ADJ ddj-151 233 13 . . PUNCT ddj-151 234 1 if if SCONJ ddj-151 234 2 these these DET ddj-151 234 3 arguments argument NOUN ddj-151 234 4 are be AUX ddj-151 234 5 correct correct ADJ ddj-151 234 6 , , PUNCT ddj-151 234 7 the the DET ddj-151 234 8 number number NOUN ddj-151 234 9 of of ADP ddj-151 234 10 isometry isometry NOUN ddj-151 234 11 equivalence equivalence NOUN ddj-151 234 12 classes class NOUN ddj-151 234 13 of of ADP ddj-151 234 14 cycles cycle NOUN ddj-151 234 15 would would AUX ddj-151 234 16 satisfy satisfy VERB ddj-151 234 17 nh nh PROPN ddj-151 234 18 , , PUNCT ddj-151 234 19 i i PRON ddj-151 234 20 ≥ ≥ VERB ddj-151 234 21 nh/24 nh/24 PROPN ddj-151 234 22 = = SYM ddj-151 234 23 1159 1159 NUM ddj-151 234 24 . . PUNCT ddj-151 235 1 2 2 X ddj-151 235 2 . . X ddj-151 235 3 the the DET ddj-151 235 4 actual actual ADJ ddj-151 235 5 number number NOUN ddj-151 235 6 is be AUX ddj-151 235 7 obtained obtain VERB ddj-151 235 8 as as ADP ddj-151 235 9 sum sum NOUN ddj-151 235 10 of of ADP ddj-151 235 11 cycles cycle NOUN ddj-151 235 12 characterized characterize VERB ddj-151 235 13 by by ADP ddj-151 235 14 groups group NOUN ddj-151 235 15 h h PROPN ddj-151 235 16 ⊂ ⊂ PROPN ddj-151 235 17 z12 z12 PROPN ddj-151 235 18 leaving leave VERB ddj-151 235 19 the the DET ddj-151 235 20 cycle cycle NOUN ddj-151 235 21 invariant invariant ADJ ddj-151 235 22 and and CCONJ ddj-151 235 23 one one PRON ddj-151 235 24 can can AUX ddj-151 235 25 write write VERB ddj-151 235 26 nh nh PROPN ddj-151 235 27 , , PUNCT ddj-151 235 28 i i PRON ddj-151 235 29 = = PUNCT ddj-151 235 30 ∑ ∑ PUNCT ddj-151 235 31 h(ord(h)/ord(g))n0(h h(ord(h)/ord(g))n0(h NUM ddj-151 235 32 ) ) PUNCT ddj-151 235 33 , , PUNCT ddj-151 235 34 where where SCONJ ddj-151 235 35 n0(h n0(h X ddj-151 235 36 ) ) PUNCT ddj-151 235 37 is be AUX ddj-151 235 38 the the DET ddj-151 235 39 number number NOUN ddj-151 235 40 of of ADP ddj-151 235 41 cycles cycle NOUN ddj-151 235 42 invariant invariant ADJ ddj-151 235 43 under under ADP ddj-151 235 44 h. h. PROPN ddj-151 235 45 what what PRON ddj-151 235 46 can can AUX ddj-151 235 47 one one PRON ddj-151 235 48 say say VERB ddj-151 235 49 about about ADP ddj-151 235 50 the the DET ddj-151 235 51 symmetry symmetry NOUN ddj-151 235 52 group group NOUN ddj-151 235 53 h h NOUN ddj-151 235 54 for for ADP ddj-151 235 55 the the DET ddj-151 235 56 cycle cycle NOUN ddj-151 235 57 ? ? PUNCT ddj-151 236 1 1 1 X ddj-151 236 2 . . X ddj-151 236 3 suppose suppose VERB ddj-151 236 4 that that SCONJ ddj-151 236 5 the the DET ddj-151 236 6 isometry isometry PROPN ddj-151 236 7 group group NOUN ddj-151 236 8 g g NOUN ddj-151 236 9 leaving leave VERB ddj-151 236 10 the the DET ddj-151 236 11 tesselation tesselation NOUN ddj-151 236 12 invariant invariant ADJ ddj-151 236 13 decomposes decompose NOUN ddj-151 236 14 into into ADP ddj-151 236 15 semi semi ADJ ddj-151 236 16 - - ADJ ddj-151 236 17 direct direct ADJ ddj-151 236 18 product product NOUN ddj-151 236 19 g g NOUN ddj-151 236 20 = = SYM ddj-151 236 21 z2,refl z2,refl NUM ddj-151 236 22 o o X ddj-151 236 23 ( ( PUNCT ddj-151 236 24 z4,tr z4,tr NOUN ddj-151 236 25 o o PROPN ddj-151 236 26 z3,rot z3,rot PROPN ddj-151 236 27 ) ) PUNCT ddj-151 236 28 , , PUNCT ddj-151 236 29 where where SCONJ ddj-151 236 30 z3,rot z3,rot PROPN ddj-151 236 31 leaves leave VERB ddj-151 236 32 invariant invariant VERB ddj-151 236 33 the the DET ddj-151 236 34 starting starting NOUN ddj-151 236 35 point point NOUN ddj-151 236 36 of of ADP ddj-151 236 37 the the DET ddj-151 236 38 cycle cycle NOUN ddj-151 236 39 . . PUNCT ddj-151 237 1 the the DET ddj-151 237 2 group group NOUN ddj-151 237 3 h h NOUN ddj-151 237 4 decomposes decompose VERB ddj-151 237 5 into into ADP ddj-151 237 6 a a DET ddj-151 237 7 semi semi ADJ ddj-151 237 8 - - ADJ ddj-151 237 9 direct direct ADJ ddj-151 237 10 product product NOUN ddj-151 237 11 h h NOUN ddj-151 237 12 = = SYM ddj-151 237 13 z2,refl z2,refl NUM ddj-151 237 14 o o NOUN ddj-151 237 15 ( ( PUNCT ddj-151 237 16 zm zm PROPN ddj-151 237 17 , , PUNCT ddj-151 237 18 tr tr VERB ddj-151 237 19 × × NOUN ddj-151 237 20 z3,rot z3,rot NUM ddj-151 237 21 ) ) PUNCT ddj-151 237 22 as as ADP ddj-151 237 23 subgroup subgroup NOUN ddj-151 237 24 of of ADP ddj-151 237 25 g g PROPN ddj-151 237 26 = = SYM ddj-151 237 27 z2,refl z2,refl NUM ddj-151 237 28 o o X ddj-151 237 29 ( ( PUNCT ddj-151 237 30 z4,tr z4,tr NOUN ddj-151 237 31 × × PROPN ddj-151 237 32 z3,rot z3,rot NUM ddj-151 237 33 ) ) PUNCT ddj-151 237 34 . . PUNCT ddj-151 238 1 2 2 X ddj-151 238 2 . . X ddj-151 238 3 zn zn NOUN ddj-151 238 4 , , PUNCT ddj-151 238 5 rot rot AUX ddj-151 238 6 associated associate VERB ddj-151 238 7 with with ADP ddj-151 238 8 the the DET ddj-151 238 9 starting starting NOUN ddj-151 238 10 point point NOUN ddj-151 238 11 of of ADP ddj-151 238 12 cycle cycle NOUN ddj-151 238 13 must must AUX ddj-151 238 14 leave leave VERB ddj-151 238 15 the the DET ddj-151 238 16 cycle cycle NOUN ddj-151 238 17 invariant invariant ADJ ddj-151 238 18 at at ADP ddj-151 238 19 each each DET ddj-151 238 20 point point NOUN ddj-151 238 21 . . PUNCT ddj-151 239 1 applied apply VERB ddj-151 239 2 to to ADP ddj-151 239 3 the the DET ddj-151 239 4 starting starting NOUN ddj-151 239 5 point point NOUN ddj-151 239 6 , , PUNCT ddj-151 239 7 the the DET ddj-151 239 8 action action NOUN ddj-151 239 9 of of ADP ddj-151 239 10 h h NOUN ddj-151 239 11 , , PUNCT ddj-151 239 12 if if SCONJ ddj-151 239 13 non non ADJ ddj-151 239 14 - - ADJ ddj-151 239 15 trivial trivial ADJ ddj-151 239 16 that that PRON ddj-151 239 17 is be AUX ddj-151 239 18 z3,rot z3,rot PROPN ddj-151 239 19 , , PUNCT ddj-151 239 20 must must AUX ddj-151 239 21 transform transform VERB ddj-151 239 22 the the DET ddj-151 239 23 outgoing outgoing ADJ ddj-151 239 24 edge edge NOUN ddj-151 239 25 to to ADP ddj-151 239 26 incoming incoming ADJ ddj-151 239 27 edge edge NOUN ddj-151 239 28 . . PUNCT ddj-151 240 1 this this PRON ddj-151 240 2 is be AUX ddj-151 240 3 not not PART ddj-151 240 4 possible possible ADJ ddj-151 240 5 since since SCONJ ddj-151 240 6 z3 z3 PROPN ddj-151 240 7 has have VERB ddj-151 240 8 no no DET ddj-151 240 9 idempotent idempotent ADJ ddj-151 240 10 elements element NOUN ddj-151 240 11 so so SCONJ ddj-151 240 12 that that SCONJ ddj-151 240 13 one one PRON ddj-151 240 14 can can AUX ddj-151 240 15 have have VERB ddj-151 240 16 only only ADV ddj-151 240 17 n n ADJ ddj-151 240 18 = = SYM ddj-151 240 19 1 1 X ddj-151 240 20 . . PUNCT ddj-151 241 1 this this PRON ddj-151 241 2 gives give VERB ddj-151 241 3 h h NOUN ddj-151 241 4 = = PUNCT ddj-151 241 5 z2,refl z2,refl NUM ddj-151 241 6 o o NOUN ddj-151 241 7 ( ( PUNCT ddj-151 241 8 zm zm PROPN ddj-151 241 9 , , PUNCT ddj-151 241 10 tr tr VERB ddj-151 241 11 . . X ddj-151 242 1 m m VERB ddj-151 243 1 = = NOUN ddj-151 243 2 1 1 NUM ddj-151 243 3 , , PUNCT ddj-151 243 4 2 2 NUM ddj-151 243 5 and and CCONJ ddj-151 243 6 m m VERB ddj-151 243 7 = = NOUN ddj-151 243 8 4 4 NUM ddj-151 243 9 are be AUX ddj-151 243 10 possible possible ADJ ddj-151 243 11 . . PUNCT ddj-151 244 1 3 3 X ddj-151 244 2 . . X ddj-151 244 3 should should AUX ddj-151 244 4 one one NUM ddj-151 244 5 require require VERB ddj-151 244 6 that that SCONJ ddj-151 244 7 the the DET ddj-151 244 8 action action NOUN ddj-151 244 9 of of ADP ddj-151 244 10 h h NOUN ddj-151 244 11 leaves leave NOUN ddj-151 244 12 invariant invariant VERB ddj-151 244 13 the the DET ddj-151 244 14 starting starting NOUN ddj-151 244 15 point point NOUN ddj-151 244 16 defining define VERB ddj-151 244 17 the the DET ddj-151 244 18 scale scale NOUN ddj-151 244 19 associated associate VERB ddj-151 244 20 with with ADP ddj-151 244 21 the the DET ddj-151 244 22 harmony harmony NOUN ddj-151 244 23 ? ? PUNCT ddj-151 245 1 if if SCONJ ddj-151 245 2 this this PRON ddj-151 245 3 is be AUX ddj-151 245 4 the the DET ddj-151 245 5 case case NOUN ddj-151 245 6 , , PUNCT ddj-151 245 7 then then ADV ddj-151 245 8 only only ADV ddj-151 245 9 the the DET ddj-151 245 10 group group NOUN ddj-151 245 11 h h NOUN ddj-151 245 12 = = SYM ddj-151 245 13 z2,refl z2,refl PROPN ddj-151 245 14 would would AUX ddj-151 245 15 remain remain VERB ddj-151 245 16 and and CCONJ ddj-151 245 17 invariance invariance NOUN ddj-151 245 18 under under ADP ddj-151 245 19 zrefl zrefl PROPN ddj-151 245 20 would would AUX ddj-151 245 21 mean mean VERB ddj-151 245 22 invariance invariance NOUN ddj-151 245 23 under under ADP ddj-151 245 24 reflection reflection NOUN ddj-151 245 25 with with ADP ddj-151 245 26 respect respect NOUN ddj-151 245 27 to to ADP ddj-151 245 28 the the DET ddj-151 245 29 axis axis NOUN ddj-151 245 30 defined define VERB ddj-151 245 31 by by ADP ddj-151 245 32 e1 e1 PROPN ddj-151 245 33 issn issn PROPN ddj-151 245 34 : : PUNCT ddj-151 245 35 2159 2159 NUM ddj-151 245 36 - - PUNCT ddj-151 245 37 046x 046x NOUN ddj-151 245 38 dna dna NOUN ddj-151 245 39 decipher decipher PROPN ddj-151 245 40 journal journal PROPN ddj-151 245 41 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 245 42 published publish VERB ddj-151 245 43 by by ADP ddj-151 245 44 quantumdream quantumdream PROPN ddj-151 245 45 , , PUNCT ddj-151 245 46 inc inc PROPN ddj-151 245 47 . . PUNCT ddj-151 246 1 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe ADJ ddj-151 246 2 dna dna ADJ ddj-151 246 3 decipher decipher ADJ ddj-151 246 4 journal journal NOUN ddj-151 246 5 | | ADP ddj-151 246 6 november november PROPN ddj-151 246 7 2018 2018 NUM ddj-151 246 8 | | ADV ddj-151 246 9 volume volume NOUN ddj-151 246 10 8 8 NUM ddj-151 246 11 | | NOUN ddj-151 246 12 issue issue VERB ddj-151 246 13 3 3 NUM ddj-151 246 14 | | CCONJ ddj-151 246 15 pp pp ADJ ddj-151 246 16 . . PUNCT ddj-151 247 1 197 197 NUM ddj-151 247 2 - - SYM ddj-151 247 3 205 205 NUM ddj-151 247 4 204 204 NUM ddj-151 247 5 pitkänen pitkänen NOUN ddj-151 247 6 , , PUNCT ddj-151 247 7 m. m. NOUN ddj-151 247 8 , , PUNCT ddj-151 247 9 thoughts thought NOUN ddj-151 247 10 on on ADP ddj-151 247 11 modification modification NOUN ddj-151 247 12 of of ADP ddj-151 247 13 bio bio NOUN ddj-151 247 14 - - NOUN ddj-151 247 15 harmony harmony NOUN ddj-151 247 16 or or CCONJ ddj-151 247 17 e2 e2 PROPN ddj-151 247 18 . . PUNCT ddj-151 248 1 the the DET ddj-151 248 2 orbit orbit NOUN ddj-151 248 3 of of ADP ddj-151 248 4 triangle triangle NOUN ddj-151 248 5 under under ADP ddj-151 248 6 z2,refl z2,refl NUM ddj-151 248 7 would would AUX ddj-151 248 8 consist consist VERB ddj-151 248 9 of of ADP ddj-151 248 10 2 2 NUM ddj-151 248 11 triangles triangle NOUN ddj-151 248 12 always always ADV ddj-151 248 13 and and CCONJ ddj-151 248 14 one one PRON ddj-151 248 15 would would AUX ddj-151 248 16 obtain obtain VERB ddj-151 248 17 12 12 NUM ddj-151 248 18 codon codon NOUN ddj-151 248 19 doublets doublet NOUN ddj-151 248 20 instead instead ADV ddj-151 248 21 of of ADP ddj-151 248 22 10 10 NUM ddj-151 248 23 as as ADP ddj-151 248 24 in in ADP ddj-151 248 25 the the DET ddj-151 248 26 case case NOUN ddj-151 248 27 of of ADP ddj-151 248 28 icosahedral icosahedral ADJ ddj-151 248 29 code code NOUN ddj-151 248 30 . . PUNCT ddj-151 249 1 if if SCONJ ddj-151 249 2 this this DET ddj-151 249 3 argument argument NOUN ddj-151 249 4 is be AUX ddj-151 249 5 correct correct ADJ ddj-151 249 6 , , PUNCT ddj-151 249 7 the the DET ddj-151 249 8 possible possible ADJ ddj-151 249 9 symmetry symmetry NOUN ddj-151 249 10 groups group NOUN ddj-151 249 11 h h NOUN ddj-151 249 12 would would AUX ddj-151 249 13 be be AUX ddj-151 249 14 z0 z0 PROPN ddj-151 249 15 and and CCONJ ddj-151 249 16 z2,refl z2,refl PROPN ddj-151 249 17 . . PUNCT ddj-151 250 1 for for ADP ddj-151 250 2 icosahedral icosahedral ADJ ddj-151 250 3 code code NOUN ddj-151 250 4 both both CCONJ ddj-151 250 5 zrot zrot VERB ddj-151 250 6 and and CCONJ ddj-151 250 7 z2refl z2refl NUM ddj-151 250 8 occur occur VERB ddj-151 250 9 but but CCONJ ddj-151 250 10 z2,refl z2,refl NOUN ddj-151 250 11 does do AUX ddj-151 250 12 not not PART ddj-151 250 13 occur occur VERB ddj-151 250 14 as as ADP ddj-151 250 15 a a DET ddj-151 250 16 non non ADJ ddj-151 250 17 - - ADJ ddj-151 250 18 trivial trivial ADJ ddj-151 250 19 factor factor NOUN ddj-151 250 20 of of ADP ddj-151 250 21 h h NOUN ddj-151 250 22 in in ADP ddj-151 250 23 this this DET ddj-151 250 24 case case NOUN ddj-151 250 25 . . PUNCT ddj-151 251 1 the the DET ddj-151 251 2 almost almost ADV ddj-151 251 3 exact exact ADJ ddj-151 251 4 u u NOUN ddj-151 251 5 ↔ ↔ PROPN ddj-151 251 6 c c NOUN ddj-151 251 7 and and CCONJ ddj-151 251 8 a↔ a↔ PROPN ddj-151 251 9 g g NOUN ddj-151 251 10 symmetry symmetry NOUN ddj-151 251 11 of of ADP ddj-151 251 12 the the DET ddj-151 251 13 genetic genetic ADJ ddj-151 251 14 code code NOUN ddj-151 251 15 would would AUX ddj-151 251 16 naturally naturally ADV ddj-151 251 17 correspond correspond VERB ddj-151 251 18 to to ADP ddj-151 251 19 z2,refl z2,refl PROPN ddj-151 251 20 symmetry symmetry NOUN ddj-151 251 21 . . PUNCT ddj-151 252 1 therefore therefore ADV ddj-151 252 2 the the DET ddj-151 252 3 predictions prediction NOUN ddj-151 252 4 need need AUX ddj-151 252 5 not not PART ddj-151 252 6 change change VERB ddj-151 252 7 from from ADP ddj-151 252 8 those those PRON ddj-151 252 9 of of ADP ddj-151 252 10 the the DET ddj-151 252 11 icosahedral icosahedral ADJ ddj-151 252 12 model model NOUN ddj-151 252 13 except except SCONJ ddj-151 252 14 that that SCONJ ddj-151 252 15 the the DET ddj-151 252 16 4 4 NUM ddj-151 252 17 additional additional ADJ ddj-151 252 18 codons codon NOUN ddj-151 252 19 emerge emerge VERB ddj-151 252 20 more more ADV ddj-151 252 21 naturally naturally ADV ddj-151 252 22 . . PUNCT ddj-151 253 1 the the DET ddj-151 253 2 predictions prediction NOUN ddj-151 253 3 would would AUX ddj-151 253 4 be be AUX ddj-151 253 5 also also ADV ddj-151 253 6 essentially essentially ADV ddj-151 253 7 unique unique ADJ ddj-151 253 8 . . PUNCT ddj-151 254 1 4 4 X ddj-151 254 2 . . X ddj-151 254 3 if if SCONJ ddj-151 254 4 h h NOUN ddj-151 254 5 is be AUX ddj-151 254 6 trivial trivial ADJ ddj-151 254 7 z1 z1 NOUN ddj-151 254 8 , , PUNCT ddj-151 254 9 the the DET ddj-151 254 10 cycle cycle NOUN ddj-151 254 11 would would AUX ddj-151 254 12 have have VERB ddj-151 254 13 no no DET ddj-151 254 14 symmetries symmetry NOUN ddj-151 254 15 and and CCONJ ddj-151 254 16 the the DET ddj-151 254 17 orbits orbit NOUN ddj-151 254 18 of of ADP ddj-151 254 19 triangles triangle NOUN ddj-151 254 20 would would AUX ddj-151 254 21 contain contain VERB ddj-151 254 22 only only ADV ddj-151 254 23 one one NUM ddj-151 254 24 triangle triangle NOUN ddj-151 254 25 and and CCONJ ddj-151 254 26 the the DET ddj-151 254 27 correspondence correspondence NOUN ddj-151 254 28 between between ADP ddj-151 254 29 dna dna PROPN ddj-151 254 30 codons codon NOUN ddj-151 254 31 and and CCONJ ddj-151 254 32 amino amino NOUN ddj-151 254 33 - - PUNCT ddj-151 254 34 acids acid NOUN ddj-151 254 35 would would AUX ddj-151 254 36 be be AUX ddj-151 254 37 one one NUM ddj-151 254 38 - - PUNCT ddj-151 254 39 to to ADP ddj-151 254 40 - - PUNCT ddj-151 254 41 one one NUM ddj-151 254 42 . . PUNCT ddj-151 255 1 one one PRON ddj-151 255 2 would would AUX ddj-151 255 3 speak speak VERB ddj-151 255 4 of of ADP ddj-151 255 5 disharmony disharmony NOUN ddj-151 255 6 . . PUNCT ddj-151 256 1 icosahedral icosahedral ADJ ddj-151 256 2 hamiltonian hamiltonian ADJ ddj-151 256 3 cycles cycle NOUN ddj-151 256 4 can can AUX ddj-151 256 5 also also ADV ddj-151 256 6 be be AUX ddj-151 256 7 of of ADP ddj-151 256 8 this this DET ddj-151 256 9 kind kind NOUN ddj-151 256 10 . . PUNCT ddj-151 257 1 if if SCONJ ddj-151 257 2 they they PRON ddj-151 257 3 are be AUX ddj-151 257 4 realized realize VERB ddj-151 257 5 in in ADP ddj-151 257 6 the the DET ddj-151 257 7 genetic genetic ADJ ddj-151 257 8 code code NOUN ddj-151 257 9 , , PUNCT ddj-151 257 10 the the DET ddj-151 257 11 almost almost ADV ddj-151 257 12 exact exact ADJ ddj-151 257 13 u u PRON ddj-151 257 14 ↔ ↔ PROPN ddj-151 257 15 c c NOUN ddj-151 257 16 and and CCONJ ddj-151 257 17 a a DET ddj-151 257 18 ↔ ↔ PROPN ddj-151 257 19 g g NOUN ddj-151 257 20 symmetry symmetry NOUN ddj-151 257 21 is be AUX ddj-151 257 22 lost lose VERB ddj-151 257 23 and and CCONJ ddj-151 257 24 the the DET ddj-151 257 25 degeneracies degeneracy NOUN ddj-151 257 26 of of ADP ddj-151 257 27 codons codon NOUN ddj-151 257 28 assignable assignable ADJ ddj-151 257 29 to to ADP ddj-151 257 30 20 20 NUM ddj-151 258 1 + + SYM ddj-151 258 2 20 20 NUM ddj-151 258 3 icosahedral icosahedral ADJ ddj-151 258 4 codons codon NOUN ddj-151 258 5 increase increase NOUN ddj-151 258 6 by by ADP ddj-151 258 7 one one NUM ddj-151 258 8 unit unit NOUN ddj-151 258 9 so so SCONJ ddj-151 258 10 that that SCONJ ddj-151 258 11 one one PRON ddj-151 258 12 obtains obtain VERB ddj-151 258 13 for for ADP ddj-151 258 14 instance instance NOUN ddj-151 258 15 degeneracy degeneracy NOUN ddj-151 258 16 7 7 NUM ddj-151 258 17 instead instead ADV ddj-151 258 18 of of ADP ddj-151 258 19 6 6 NUM ddj-151 258 20 not not PART ddj-151 258 21 realized realize VERB ddj-151 258 22 in in ADP ddj-151 258 23 nature nature NOUN ddj-151 258 24 . . PUNCT ddj-151 259 1 what what PRON ddj-151 259 2 can can AUX ddj-151 259 3 one one PRON ddj-151 259 4 say say VERB ddj-151 259 5 about about ADP ddj-151 259 6 the the DET ddj-151 259 7 character character NOUN ddj-151 259 8 of of ADP ddj-151 259 9 toric toric ADJ ddj-151 259 10 harmonies harmony NOUN ddj-151 259 11 on on ADP ddj-151 259 12 basis basis NOUN ddj-151 259 13 of of ADP ddj-151 259 14 this this DET ddj-151 259 15 picture picture NOUN ddj-151 259 16 . . PUNCT ddj-151 260 1 1 1 X ddj-151 260 2 . . X ddj-151 260 3 it it PRON ddj-151 260 4 has have AUX ddj-151 260 5 been be AUX ddj-151 260 6 already already ADV ddj-151 260 7 found find VERB ddj-151 260 8 that that SCONJ ddj-151 260 9 the the DET ddj-151 260 10 proposal proposal NOUN ddj-151 260 11 involving involve VERB ddj-151 260 12 three three NUM ddj-151 260 13 disjoint disjoint ADJ ddj-151 260 14 quartets quartet NOUN ddj-151 260 15 of of ADP ddj-151 260 16 subsequent subsequent ADJ ddj-151 260 17 notes note NOUN ddj-151 260 18 can can AUX ddj-151 260 19 reproduce reproduce VERB ddj-151 260 20 the the DET ddj-151 260 21 basic basic ADJ ddj-151 260 22 chords chord NOUN ddj-151 260 23 of of ADP ddj-151 260 24 basic basic ADJ ddj-151 260 25 major major ADJ ddj-151 260 26 and and CCONJ ddj-151 260 27 minor minor ADJ ddj-151 260 28 harmonies harmony NOUN ddj-151 260 29 . . PUNCT ddj-151 261 1 the the DET ddj-151 261 2 challenge challenge NOUN ddj-151 261 3 is be AUX ddj-151 261 4 to to PART ddj-151 261 5 prove prove VERB ddj-151 261 6 that that SCONJ ddj-151 261 7 it it PRON ddj-151 261 8 can can AUX ddj-151 261 9 be be AUX ddj-151 261 10 assigned assign VERB ddj-151 261 11 to to ADP ddj-151 261 12 some some DET ddj-151 261 13 hamiltonian hamiltonian ADJ ddj-151 261 14 cycle(s cycle(s PROPN ddj-151 261 15 ) ) PUNCT ddj-151 261 16 . . PUNCT ddj-151 262 1 the the DET ddj-151 262 2 proposal proposal NOUN ddj-151 262 3 is be AUX ddj-151 262 4 that that SCONJ ddj-151 262 5 the the DET ddj-151 262 6 quartets quartet NOUN ddj-151 262 7 are be AUX ddj-151 262 8 obtained obtain VERB ddj-151 262 9 by by ADP ddj-151 262 10 z3 z3 PROPN ddj-151 262 11 rot rot PROPN ddj-151 262 12 symmetry symmetry NOUN ddj-151 262 13 from from ADP ddj-151 262 14 each each DET ddj-151 262 15 other other ADJ ddj-151 262 16 and and CCONJ ddj-151 262 17 that that SCONJ ddj-151 262 18 the the DET ddj-151 262 19 notes note NOUN ddj-151 262 20 of of ADP ddj-151 262 21 each each DET ddj-151 262 22 quartet quartet NOUN ddj-151 262 23 are be AUX ddj-151 262 24 obtained obtain VERB ddj-151 262 25 by by ADP ddj-151 262 26 z4,tr z4,tr NOUN ddj-151 262 27 symmetry symmetry NOUN ddj-151 262 28 . . PUNCT ddj-151 263 1 2 2 X ddj-151 263 2 . . X ddj-151 263 3 a a DET ddj-151 263 4 key key ADJ ddj-151 263 5 observation observation NOUN ddj-151 263 6 is be AUX ddj-151 263 7 that that SCONJ ddj-151 263 8 classical classical ADJ ddj-151 263 9 harmonies harmony NOUN ddj-151 263 10 involve involve VERB ddj-151 263 11 chords chord NOUN ddj-151 263 12 containing contain VERB ddj-151 263 13 1 1 NUM ddj-151 263 14 quint quint NOUN ddj-151 263 15 but but CCONJ ddj-151 263 16 not not PART ddj-151 263 17 2 2 NUM ddj-151 263 18 or or CCONJ ddj-151 263 19 no no DET ddj-151 263 20 quints quint NOUN ddj-151 263 21 at at ADV ddj-151 263 22 all all ADV ddj-151 263 23 . . PUNCT ddj-151 264 1 the the DET ddj-151 264 2 number number NOUN ddj-151 264 3 of of ADP ddj-151 264 4 chords chord NOUN ddj-151 264 5 in in ADP ddj-151 264 6 torus torus NOUN ddj-151 264 7 harmonies harmony NOUN ddj-151 264 8 is be AUX ddj-151 264 9 24 24 NUM ddj-151 264 10 = = SYM ddj-151 264 11 2× 2× NUM ddj-151 264 12 12 12 NUM ddj-151 264 13 and and CCONJ ddj-151 264 14 twice twice DET ddj-151 264 15 the the DET ddj-151 264 16 number number NOUN ddj-151 264 17 of of ADP ddj-151 264 18 notes note NOUN ddj-151 264 19 . . PUNCT ddj-151 265 1 the the DET ddj-151 265 2 number number NOUN ddj-151 265 3 of of ADP ddj-151 265 4 intervals interval NOUN ddj-151 265 5 in in ADP ddj-151 265 6 turn turn NOUN ddj-151 265 7 is be AUX ddj-151 265 8 36 36 NUM ddj-151 265 9 , , PUNCT ddj-151 265 10 3 3 NUM ddj-151 265 11 times time NOUN ddj-151 265 12 the the DET ddj-151 265 13 number number NOUN ddj-151 265 14 of of ADP ddj-151 265 15 the the DET ddj-151 265 16 notes note NOUN ddj-151 265 17 . . PUNCT ddj-151 266 1 this this PRON ddj-151 266 2 allows allow VERB ddj-151 266 3 a a DET ddj-151 266 4 situation situation NOUN ddj-151 266 5 in in ADP ddj-151 266 6 which which PRON ddj-151 266 7 each each DET ddj-151 266 8 triangle triangle NOUN ddj-151 266 9 contains contain VERB ddj-151 266 10 one one NUM ddj-151 266 11 edge edge NOUN ddj-151 266 12 of of ADP ddj-151 266 13 the the DET ddj-151 266 14 hamiltonian hamiltonian ADJ ddj-151 266 15 cycle cycle NOUN ddj-151 266 16 so so SCONJ ddj-151 266 17 that that SCONJ ddj-151 266 18 all all DET ddj-151 266 19 3 3 NUM ddj-151 266 20 - - NOUN ddj-151 266 21 chords chord NOUN ddj-151 266 22 indeed indeed ADV ddj-151 266 23 have have VERB ddj-151 266 24 exactly exactly ADV ddj-151 266 25 one one NUM ddj-151 266 26 quint quint NOUN ddj-151 266 27 . . PUNCT ddj-151 267 1 3 3 X ddj-151 267 2 . . X ddj-151 267 3 by by ADP ddj-151 267 4 the the DET ddj-151 267 5 above above ADJ ddj-151 267 6 argument argument NOUN ddj-151 267 7 harmony harmony NOUN ddj-151 267 8 possesses possesse NOUN ddj-151 267 9 z2 z2 PROPN ddj-151 267 10 symmetry symmetry PROPN ddj-151 267 11 or or CCONJ ddj-151 267 12 no no DET ddj-151 267 13 symmetry symmetry NOUN ddj-151 267 14 at at ADV ddj-151 267 15 all all ADV ddj-151 267 16 and and CCONJ ddj-151 267 17 one one NUM ddj-151 267 18 has have VERB ddj-151 267 19 12 12 NUM ddj-151 267 20 codon codon NOUN ddj-151 267 21 doublets doublet NOUN ddj-151 267 22 . . PUNCT ddj-151 268 1 for for ADP ddj-151 268 2 these these DET ddj-151 268 3 harmonies harmony NOUN ddj-151 268 4 each each DET ddj-151 268 5 edge edge NOUN ddj-151 268 6 of of ADP ddj-151 268 7 cycle cycle NOUN ddj-151 268 8 is be AUX ddj-151 268 9 shared share VERB ddj-151 268 10 by by ADP ddj-151 268 11 two two NUM ddj-151 268 12 neighboring neighboring NOUN ddj-151 268 13 triangles triangle NOUN ddj-151 268 14 containing contain VERB ddj-151 268 15 the the DET ddj-151 268 16 same same ADJ ddj-151 268 17 quint quint NOUN ddj-151 268 18 . . PUNCT ddj-151 269 1 a a DET ddj-151 269 2 possible possible ADJ ddj-151 269 3 identification identification NOUN ddj-151 269 4 is be AUX ddj-151 269 5 as as ADV ddj-151 269 6 major major ADJ ddj-151 269 7 and and CCONJ ddj-151 269 8 minor minor ADJ ddj-151 269 9 chords chord NOUN ddj-151 269 10 with with ADP ddj-151 269 11 same same ADJ ddj-151 269 12 quint quint NOUN ddj-151 269 13 . . PUNCT ddj-151 270 1 the the DET ddj-151 270 2 changing changing NOUN ddj-151 270 3 of of ADP ddj-151 270 4 the the DET ddj-151 270 5 direction direction NOUN ddj-151 270 6 of of ADP ddj-151 270 7 the the DET ddj-151 270 8 scale scale NOUN ddj-151 270 9 and and CCONJ ddj-151 270 10 the the DET ddj-151 270 11 reflection reflection NOUN ddj-151 270 12 with with ADP ddj-151 270 13 respect respect NOUN ddj-151 270 14 to to ADP ddj-151 270 15 the the DET ddj-151 270 16 edges edge NOUN ddj-151 270 17 the the DET ddj-151 270 18 hamiltonian hamiltonian ADJ ddj-151 270 19 cycle cycle NOUN ddj-151 270 20 would would AUX ddj-151 270 21 transforms transform VERB ddj-151 270 22 major major ADJ ddj-151 270 23 chords chord NOUN ddj-151 270 24 and and CCONJ ddj-151 270 25 minor minor ADJ ddj-151 270 26 chords chord NOUN ddj-151 270 27 along along ADP ddj-151 270 28 it it PRON ddj-151 270 29 to to ADP ddj-151 270 30 each each DET ddj-151 270 31 other other ADJ ddj-151 270 32 and and CCONJ ddj-151 270 33 change change VERB ddj-151 270 34 the the DET ddj-151 270 35 mood mood NOUN ddj-151 270 36 from from ADP ddj-151 270 37 glad glad ADJ ddj-151 270 38 to to PART ddj-151 270 39 sad sad ADJ ddj-151 270 40 and and CCONJ ddj-151 270 41 vice vice ADV ddj-151 270 42 versa versa ADV ddj-151 270 43 . . PUNCT ddj-151 271 1 the the DET ddj-151 271 2 proposed propose VERB ddj-151 271 3 harmony harmony NOUN ddj-151 271 4 indeed indeed ADV ddj-151 271 5 contains contain VERB ddj-151 271 6 classical classical ADJ ddj-151 271 7 chords chord NOUN ddj-151 271 8 with with ADP ddj-151 271 9 one one NUM ddj-151 271 10 quint quint NOUN ddj-151 271 11 per per ADP ddj-151 271 12 chord chord NOUN ddj-151 271 13 and and CCONJ ddj-151 271 14 for for ADP ddj-151 271 15 f f PROPN ddj-151 271 16 , , PUNCT ddj-151 271 17 a a PRON ddj-151 271 18 , , PUNCT ddj-151 271 19 c c NOUN ddj-151 271 20 ] ] X ddj-151 271 21 both both CCONJ ddj-151 271 22 minor minor ADJ ddj-151 271 23 and and CCONJ ddj-151 271 24 major major ADJ ddj-151 271 25 chords chord NOUN ddj-151 271 26 are be AUX ddj-151 271 27 possible possible ADJ ddj-151 271 28 . . PUNCT ddj-151 272 1 there there PRON ddj-151 272 2 are be VERB ddj-151 272 3 4 4 NUM ddj-151 272 4 transposes transpose NOUN ddj-151 272 5 of of ADP ddj-151 272 6 this this DET ddj-151 272 7 harmony harmony NOUN ddj-151 272 8 . . PUNCT ddj-151 273 1 4 4 X ddj-151 273 2 . . X ddj-151 273 3 also also ADV ddj-151 273 4 hamiltonian hamiltonian ADJ ddj-151 273 5 cycles cycle NOUN ddj-151 273 6 for for ADP ddj-151 273 7 which which PRON ddj-151 273 8 n n PRON ddj-151 273 9 triangles triangle NOUN ddj-151 273 10 contain contain VERB ddj-151 273 11 two two NUM ddj-151 273 12 edges edge NOUN ddj-151 273 13 of of ADP ddj-151 273 14 hamiltonian hamiltonian ADJ ddj-151 273 15 path path NOUN ddj-151 273 16 ( ( PUNCT ddj-151 273 17 cgd cgd PROPN ddj-151 273 18 type type NOUN ddj-151 273 19 chords chord NOUN ddj-151 273 20 ) ) PUNCT ddj-151 273 21 and and CCONJ ddj-151 273 22 n n PROPN ddj-151 273 23 triangles triangle NOUN ddj-151 273 24 contain contain VERB ddj-151 273 25 no no DET ddj-151 273 26 edges edge NOUN ddj-151 273 27 . . PUNCT ddj-151 274 1 this this DET ddj-151 274 2 situation situation NOUN ddj-151 274 3 is be AUX ddj-151 274 4 less less ADV ddj-151 274 5 symmetric symmetric ADJ ddj-151 274 6 and and CCONJ ddj-151 274 7 could could AUX ddj-151 274 8 correspond correspond VERB ddj-151 274 9 to to ADP ddj-151 274 10 a a DET ddj-151 274 11 situtation situtation NOUN ddj-151 274 12 without without ADP ddj-151 274 13 any any DET ddj-151 274 14 symmetry symmetry NOUN ddj-151 274 15 at at ADV ddj-151 274 16 all all ADV ddj-151 274 17 . . PUNCT ddj-151 275 1 5 5 X ddj-151 275 2 . . X ddj-151 275 3 one one PRON ddj-151 275 4 can can AUX ddj-151 275 5 ask ask VERB ddj-151 275 6 whether whether SCONJ ddj-151 275 7 the the DET ddj-151 275 8 classical classical ADJ ddj-151 275 9 harmonies harmony NOUN ddj-151 275 10 corresponds correspond NOUN ddj-151 275 11 to to ADP ddj-151 275 12 24 24 NUM ddj-151 275 13 codons codon NOUN ddj-151 275 14 assignable assignable ADJ ddj-151 275 15 to to ADP ddj-151 275 16 the the DET ddj-151 275 17 toric toric ADJ ddj-151 275 18 harmony harmony NOUN ddj-151 275 19 and and CCONJ ddj-151 275 20 to to ADP ddj-151 275 21 the the DET ddj-151 275 22 24 24 NUM ddj-151 275 23 amino amino NOUN ddj-151 275 24 - - PUNCT ddj-151 275 25 acids acid NOUN ddj-151 275 26 being be AUX ddj-151 275 27 thus thus ADV ddj-151 275 28 realizable realizable ADJ ddj-151 275 29 using use VERB ddj-151 275 30 only only ADV ddj-151 275 31 amino amino NOUN ddj-151 275 32 - - PUNCT ddj-151 275 33 acids acid NOUN ddj-151 275 34 . . PUNCT ddj-151 276 1 if if SCONJ ddj-151 276 2 so so ADV ddj-151 276 3 , , PUNCT ddj-151 276 4 the the DET ddj-151 276 5 two two NUM ddj-151 276 6 icosahedral icosahedral ADJ ddj-151 276 7 harmonies harmony NOUN ddj-151 276 8 would would AUX ddj-151 276 9 represent represent VERB ddj-151 276 10 kind kind ADV ddj-151 276 11 of of ADP ddj-151 276 12 non non ADJ ddj-151 276 13 - - ADJ ddj-151 276 14 classical classical ADJ ddj-151 276 15 exotics exotic NOUN ddj-151 276 16 . . PUNCT ddj-151 277 1 3 3 NUM ddj-151 277 2 appendix appendix NOUN ddj-151 277 3 : : PUNCT ddj-151 277 4 some some DET ddj-151 277 5 facts fact NOUN ddj-151 277 6 about about ADP ddj-151 277 7 toric toric ADJ ddj-151 277 8 tesselations tesselation NOUN ddj-151 277 9 genus genus VERB ddj-151 277 10 g g PROPN ddj-151 277 11 = = SYM ddj-151 277 12 1 1 NUM ddj-151 277 13 ( ( PUNCT ddj-151 277 14 torus torus PROPN ddj-151 277 15 ) ) PUNCT ddj-151 277 16 is be AUX ddj-151 277 17 unique unique ADJ ddj-151 277 18 in in ADP ddj-151 277 19 that that SCONJ ddj-151 277 20 it it PRON ddj-151 277 21 allows allow VERB ddj-151 277 22 infinite infinite ADJ ddj-151 277 23 number number NOUN ddj-151 277 24 of of ADP ddj-151 277 25 tesselations tesselation NOUN ddj-151 277 26 as as ADP ddj-151 277 27 analogs analog NOUN ddj-151 277 28 of of ADP ddj-151 277 29 planar planar ADJ ddj-151 277 30 lattices lattice NOUN ddj-151 277 31 with with ADP ddj-151 277 32 periodicic periodicic PROPN ddj-151 277 33 boundary boundary ADJ ddj-151 277 34 conditions condition NOUN ddj-151 277 35 . . PUNCT ddj-151 278 1 g g NOUN ddj-151 278 2 = = SYM ddj-151 278 3 0 0 NUM ddj-151 278 4 allows allow VERB ddj-151 278 5 only only ADV ddj-151 278 6 platonic platonic ADJ ddj-151 278 7 solids solid NOUN ddj-151 278 8 as as ADP ddj-151 278 9 tesselations tesselation NOUN ddj-151 278 10 and and CCONJ ddj-151 278 11 g g PROPN ddj-151 278 12 > > SYM ddj-151 278 13 1 1 NUM ddj-151 278 14 allows allow VERB ddj-151 278 15 very very ADV ddj-151 278 16 few few ADJ ddj-151 278 17 tesselations tesselation NOUN ddj-151 278 18 . . PUNCT ddj-151 279 1 the the DET ddj-151 279 2 article article NOUN ddj-151 279 3 [ [ X ddj-151 279 4 1 1 X ddj-151 279 5 ] ] PUNCT ddj-151 279 6 gives give VERB ddj-151 279 7 a a DET ddj-151 279 8 nice nice ADJ ddj-151 279 9 review review NOUN ddj-151 279 10 about about ADP ddj-151 279 11 toric toric ADJ ddj-151 279 12 tesselations tesselation NOUN ddj-151 279 13 . . PUNCT ddj-151 280 1 issn issn PROPN ddj-151 280 2 : : PUNCT ddj-151 280 3 2159 2159 NUM ddj-151 280 4 - - PUNCT ddj-151 280 5 046x 046x NOUN ddj-151 280 6 dna dna NOUN ddj-151 280 7 decipher decipher PROPN ddj-151 280 8 journal journal PROPN ddj-151 280 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 280 10 published publish VERB ddj-151 280 11 by by ADP ddj-151 280 12 quantumdream quantumdream PROPN ddj-151 280 13 , , PUNCT ddj-151 280 14 inc inc PROPN ddj-151 280 15 . . PROPN ddj-151 280 16 dna dna PROPN ddj-151 280 17 decipher decipher PROPN ddj-151 280 18 journal journal PROPN ddj-151 280 19 | | ADP ddj-151 280 20 november november PROPN ddj-151 280 21 2018 2018 NUM ddj-151 281 1 | | ADV ddj-151 281 2 volume volume NOUN ddj-151 281 3 8 8 NUM ddj-151 281 4 | | NOUN ddj-151 281 5 issue issue VERB ddj-151 281 6 3 3 NUM ddj-151 281 7 | | CCONJ ddj-151 281 8 pp pp ADJ ddj-151 281 9 . . PUNCT ddj-151 282 1 197 197 NUM ddj-151 282 2 - - SYM ddj-151 282 3 205 205 NUM ddj-151 282 4 205 205 NUM ddj-151 282 5 pitkänen pitkänen NOUN ddj-151 282 6 , , PUNCT ddj-151 282 7 m. m. NOUN ddj-151 282 8 , , PUNCT ddj-151 282 9 thoughts thought NOUN ddj-151 282 10 on on ADP ddj-151 282 11 modification modification NOUN ddj-151 282 12 of of ADP ddj-151 282 13 bio bio NOUN ddj-151 282 14 - - NOUN ddj-151 282 15 harmony harmony NOUN ddj-151 282 16 1 1 NUM ddj-151 282 17 . . X ddj-151 282 18 toric toric ADJ ddj-151 282 19 tesselations tesselation NOUN ddj-151 282 20 correspond correspond VERB ddj-151 282 21 to to ADP ddj-151 282 22 tesselations tesselation NOUN ddj-151 282 23 of of ADP ddj-151 282 24 plane plane NOUN ddj-151 282 25 by by ADP ddj-151 282 26 periodic periodic ADJ ddj-151 282 27 boundary boundary ADJ ddj-151 282 28 conditions condition NOUN ddj-151 282 29 . . PUNCT ddj-151 283 1 torus torus PROPN ddj-151 283 2 tesselation tesselation NOUN ddj-151 283 3 allows allow VERB ddj-151 283 4 a a DET ddj-151 283 5 universal universal ADJ ddj-151 283 6 covering covering NOUN ddj-151 283 7 identifiable identifiable ADJ ddj-151 283 8 as as ADP ddj-151 283 9 counterpart counterpart NOUN ddj-151 283 10 of of ADP ddj-151 283 11 infinite infinite ADJ ddj-151 283 12 lattice lattice NOUN ddj-151 283 13 in in ADP ddj-151 283 14 plane plane NOUN ddj-151 283 15 . . PUNCT ddj-151 284 1 there there PRON ddj-151 284 2 are be VERB ddj-151 284 3 infinite infinite ADJ ddj-151 284 4 number number NOUN ddj-151 284 5 of of ADP ddj-151 284 6 coverings covering NOUN ddj-151 284 7 of of ADP ddj-151 284 8 given give VERB ddj-151 284 9 tesselation tesselation NOUN ddj-151 284 10 labelled label VERB ddj-151 284 11 by by ADP ddj-151 284 12 two two NUM ddj-151 284 13 integers integer NOUN ddj-151 284 14 ( ( PUNCT ddj-151 284 15 m m NOUN ddj-151 284 16 , , PUNCT ddj-151 284 17 n n CCONJ ddj-151 284 18 ) ) PUNCT ddj-151 284 19 since since SCONJ ddj-151 284 20 the the DET ddj-151 284 21 homology homology NOUN ddj-151 284 22 group group NOUN ddj-151 284 23 of of ADP ddj-151 284 24 torus torus PROPN ddj-151 284 25 is be AUX ddj-151 284 26 z×z z×z NUM ddj-151 284 27 . . PUNCT ddj-151 285 1 the the DET ddj-151 285 2 tesselation tesselation NOUN ddj-151 285 3 is be AUX ddj-151 285 4 obtained obtain VERB ddj-151 285 5 by by ADP ddj-151 285 6 dividing divide VERB ddj-151 285 7 z×z z×z PROPN ddj-151 285 8 by by ADP ddj-151 285 9 its its PRON ddj-151 285 10 normal normal ADJ ddj-151 285 11 subgroup subgroup NOUN ddj-151 285 12 . . PUNCT ddj-151 286 1 also also ADV ddj-151 286 2 the the DET ddj-151 286 3 rotation rotation NOUN ddj-151 286 4 group group NOUN ddj-151 286 5 z6 z6 PROPN ddj-151 286 6 acts act VERB ddj-151 286 7 as as ADP ddj-151 286 8 group group NOUN ddj-151 286 9 leaving leave VERB ddj-151 286 10 the the DET ddj-151 286 11 tesselation tesselation NOUN ddj-151 286 12 invariant invariant ADJ ddj-151 286 13 and and CCONJ ddj-151 286 14 correspond correspond VERB ddj-151 286 15 to to ADP ddj-151 286 16 the the DET ddj-151 286 17 rotation rotation NOUN ddj-151 286 18 leaving leave VERB ddj-151 286 19 invariant invariant ADJ ddj-151 286 20 the the DET ddj-151 286 21 lattice lattice NOUN ddj-151 286 22 cell cell NOUN ddj-151 286 23 consisting consist VERB ddj-151 286 24 of of ADP ddj-151 286 25 6 6 NUM ddj-151 286 26 vertices vertex NOUN ddj-151 286 27 around around ADV ddj-151 286 28 given give VERB ddj-151 286 29 vertex vertex NOUN ddj-151 286 30 . . PUNCT ddj-151 287 1 2 2 X ddj-151 287 2 . . X ddj-151 287 3 the the DET ddj-151 287 4 tesselation tesselation NOUN ddj-151 287 5 is be AUX ddj-151 287 6 called call VERB ddj-151 287 7 decomposable decomposable ADJ ddj-151 287 8 if if SCONJ ddj-151 287 9 there there PRON ddj-151 287 10 is be VERB ddj-151 287 11 a a DET ddj-151 287 12 k k ADJ ddj-151 287 13 - - ADJ ddj-151 287 14 sheeted sheeted ADJ ddj-151 287 15 covering covering NOUN ddj-151 287 16 map map NOUN ddj-151 287 17 ( ( PUNCT ddj-151 287 18 map map NOUN ddj-151 287 19 corresponds correspond VERB ddj-151 287 20 to to ADP ddj-151 287 21 a a DET ddj-151 287 22 collection collection NOUN ddj-151 287 23 of of ADP ddj-151 287 24 charts chart NOUN ddj-151 287 25 ) ) PUNCT ddj-151 287 26 characterized characterize VERB ddj-151 287 27 by by ADP ddj-151 287 28 the the DET ddj-151 287 29 subgroup subgroup NOUN ddj-151 287 30 of of ADP ddj-151 287 31 the the DET ddj-151 287 32 isometries isometry NOUN ddj-151 287 33 of of ADP ddj-151 287 34 the the DET ddj-151 287 35 covering covering NOUN ddj-151 287 36 of of ADP ddj-151 287 37 the the DET ddj-151 287 38 tesselation tesselation NOUN ddj-151 287 39 which which PRON ddj-151 287 40 corresponds correspond VERB ddj-151 287 41 to to ADP ddj-151 287 42 a a DET ddj-151 287 43 sub sub NOUN ddj-151 287 44 - - NOUN ddj-151 287 45 tesselation tesselation NOUN ddj-151 287 46 . . PUNCT ddj-151 288 1 this this DET ddj-151 288 2 subgroup subgroup NOUN ddj-151 288 3 is be AUX ddj-151 288 4 charactrized charactrize VERB ddj-151 288 5 by by ADP ddj-151 288 6 a a DET ddj-151 288 7 pair pair NOUN ddj-151 288 8 ( ( PUNCT ddj-151 288 9 p p X ddj-151 288 10 , , PUNCT ddj-151 288 11 q q NOUN ddj-151 288 12 ) ) PUNCT ddj-151 288 13 of of ADP ddj-151 288 14 integers integer NOUN ddj-151 288 15 being be AUX ddj-151 288 16 generated generate VERB ddj-151 288 17 by by ADP ddj-151 288 18 the the DET ddj-151 288 19 translation translation NOUN ddj-151 288 20 pe1 pe1 PROPN ddj-151 288 21 + + PROPN ddj-151 288 22 qe2 qe2 PROPN ddj-151 288 23 and and CCONJ ddj-151 288 24 2π/6 2π/6 NUM ddj-151 288 25 rotation rotation NOUN ddj-151 288 26 . . PUNCT ddj-151 289 1 the the DET ddj-151 289 2 unit unit NOUN ddj-151 289 3 vectors vector NOUN ddj-151 289 4 can can AUX ddj-151 289 5 be be AUX ddj-151 289 6 chosen choose VERB ddj-151 289 7 to to PART ddj-151 289 8 be be AUX ddj-151 289 9 e1 e1 NOUN ddj-151 289 10 = = SYM ddj-151 289 11 ( ( PUNCT ddj-151 289 12 1 1 NUM ddj-151 289 13 , , PUNCT ddj-151 289 14 0 0 NUM ddj-151 289 15 ) ) PUNCT ddj-151 289 16 and and CCONJ ddj-151 289 17 e2 e2 PROPN ddj-151 289 18 = = SYM ddj-151 289 19 ( ( PUNCT ddj-151 289 20 1 1 NUM ddj-151 289 21 , , PUNCT ddj-151 289 22 √ √ PROPN ddj-151 289 23 3)/2 3)/2 NUM ddj-151 289 24 for for ADP ddj-151 289 25 triangular triangular NOUN ddj-151 289 26 tesselation tesselation NOUN ddj-151 289 27 ( ( PUNCT ddj-151 289 28 presumably presumably ADV ddj-151 289 29 this this DET ddj-151 289 30 tesselation tesselation NOUN ddj-151 289 31 is be AUX ddj-151 289 32 regular regular ADJ ddj-151 289 33 tesselation tesselation NOUN ddj-151 289 34 with with ADP ddj-151 289 35 the the DET ddj-151 289 36 conformal conformal ADJ ddj-151 289 37 equivalence equivalence NOUN ddj-151 289 38 class class NOUN ddj-151 289 39 of of ADP ddj-151 289 40 torus torus NOUN ddj-151 289 41 fixed fix VERB ddj-151 289 42 by by ADP ddj-151 289 43 the the DET ddj-151 289 44 angle angle NOUN ddj-151 289 45 between between ADP ddj-151 289 46 e1 e1 PROPN ddj-151 289 47 and and CCONJ ddj-151 289 48 e2 e2 PROPN ddj-151 289 49 ) ) PUNCT ddj-151 289 50 . . PUNCT ddj-151 290 1 line line NOUN ddj-151 290 2 reflection reflection NOUN ddj-151 290 3 transforms transform VERB ddj-151 290 4 ( ( PUNCT ddj-151 290 5 3 3 NUM ddj-151 290 6 , , PUNCT ddj-151 290 7 6)p 6)p NUM ddj-151 290 8 , , PUNCT ddj-151 290 9 q q X ddj-151 290 10 to to PART ddj-151 290 11 ( ( PUNCT ddj-151 290 12 3 3 NUM ddj-151 290 13 , , PUNCT ddj-151 290 14 6)q 6)q NUM ddj-151 290 15 , , PUNCT ddj-151 290 16 p p X ddj-151 290 17 ( ( PUNCT ddj-151 290 18 see see VERB ddj-151 290 19 fig fig NOUN ddj-151 290 20 1 1 NUM ddj-151 290 21 of of ADP ddj-151 290 22 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe PROPN ddj-151 290 23 ) ) PUNCT ddj-151 290 24 . . PUNCT ddj-151 291 1 the the DET ddj-151 291 2 tesselation tesselation NOUN ddj-151 291 3 is be AUX ddj-151 291 4 invariant invariant ADJ ddj-151 291 5 under under ADP ddj-151 291 6 reflections reflection NOUN ddj-151 291 7 regular regular ADJ ddj-151 291 8 -if -if PRON ddj-151 291 9 pq(p− pq(p− NOUN ddj-151 291 10 q q NOUN ddj-151 291 11 ) ) PUNCT ddj-151 291 12 = = SYM ddj-151 292 1 0 0 X ddj-151 292 2 . . PUNCT ddj-151 293 1 the the DET ddj-151 293 2 peculiar peculiar ADJ ddj-151 293 3 looking look VERB ddj-151 293 4 form form NOUN ddj-151 293 5 of of ADP ddj-151 293 6 the the DET ddj-151 293 7 conditions condition NOUN ddj-151 293 8 follows follow VERB ddj-151 293 9 from from ADP ddj-151 293 10 the the DET ddj-151 293 11 identitity identitity NOUN ddj-151 293 12 ( ( PUNCT ddj-151 293 13 3 3 NUM ddj-151 293 14 , , PUNCT ddj-151 293 15 6)q 6)q NUM ddj-151 293 16 , , PUNCT ddj-151 293 17 p p NOUN ddj-151 293 18 = = X ddj-151 293 19 ( ( PUNCT ddj-151 293 20 3 3 NUM ddj-151 293 21 , , PUNCT ddj-151 293 22 6)p+q,−q 6)p+q,−q NUM ddj-151 293 23 ( ( PUNCT ddj-151 293 24 also also ADV ddj-151 293 25 p p X ddj-151 293 26 = = NOUN ddj-151 293 27 0 0 NUM ddj-151 293 28 or or CCONJ ddj-151 293 29 q q NOUN ddj-151 293 30 = = SYM ddj-151 293 31 0 0 NUM ddj-151 293 32 is be AUX ddj-151 293 33 possble possble ADJ ddj-151 293 34 ) ) PUNCT ddj-151 293 35 note note NOUN ddj-151 293 36 that that SCONJ ddj-151 293 37 the the DET ddj-151 293 38 tesselation tesselation NOUN ddj-151 293 39 ( ( PUNCT ddj-151 293 40 3 3 NUM ddj-151 293 41 , , PUNCT ddj-151 293 42 6)2,2 6)2,2 NUM ddj-151 293 43 is be AUX ddj-151 293 44 invariant invariant ADJ ddj-151 293 45 under under ADP ddj-151 293 46 reflection reflection NOUN ddj-151 293 47 and and CCONJ ddj-151 293 48 thus thus ADV ddj-151 293 49 non non ADJ ddj-151 293 50 - - ADJ ddj-151 293 51 chiral chiral ADJ ddj-151 293 52 . . PUNCT ddj-151 294 1 3 3 X ddj-151 294 2 . . X ddj-151 294 3 the the DET ddj-151 294 4 number number NOUN ddj-151 294 5 v v NOUN ddj-151 294 6 of of ADP ddj-151 294 7 vertices vertex NOUN ddj-151 294 8 of of ADP ddj-151 294 9 the the DET ddj-151 294 10 triangular triangular NOUN ddj-151 294 11 itesselation itesselation NOUN ddj-151 294 12 is be AUX ddj-151 294 13 given give VERB ddj-151 294 14 by by ADP ddj-151 294 15 v v NOUN ddj-151 294 16 = = SYM ddj-151 294 17 p2 p2 PROPN ddj-151 294 18 + + CCONJ ddj-151 294 19 q2 q2 NOUN ddj-151 294 20 + + CCONJ ddj-151 294 21 pq pq NOUN ddj-151 294 22 . . PUNCT ddj-151 295 1 the the DET ddj-151 295 2 regular regular ADJ ddj-151 295 3 tesselation tesselation NOUN ddj-151 295 4 ( ( PUNCT ddj-151 295 5 p p X ddj-151 295 6 , , PUNCT ddj-151 295 7 q q NOUN ddj-151 295 8 ) ) PUNCT ddj-151 295 9 = = SYM ddj-151 295 10 ( ( PUNCT ddj-151 295 11 2 2 NUM ddj-151 295 12 , , PUNCT ddj-151 295 13 2 2 NUM ddj-151 295 14 ) ) PUNCT ddj-151 295 15 has have VERB ddj-151 295 16 12 12 NUM ddj-151 295 17 vertices vertex NOUN ddj-151 295 18 and and CCONJ ddj-151 295 19 is be AUX ddj-151 295 20 the the DET ddj-151 295 21 interesting interesting ADJ ddj-151 295 22 one one NUM ddj-151 295 23 in in ADP ddj-151 295 24 the the DET ddj-151 295 25 recent recent ADJ ddj-151 295 26 case case NOUN ddj-151 295 27 . . PUNCT ddj-151 296 1 it it PRON ddj-151 296 2 is be AUX ddj-151 296 3 the the DET ddj-151 296 4 smallest small ADJ ddj-151 296 5 regular regular ADJ ddj-151 296 6 tesselation tesselation NOUN ddj-151 296 7 . . PUNCT ddj-151 297 1 for for ADP ddj-151 297 2 given give VERB ddj-151 297 3 ( ( PUNCT ddj-151 297 4 p p X ddj-151 297 5 , , PUNCT ddj-151 297 6 q q NOUN ddj-151 297 7 ) ) PUNCT ddj-151 297 8 one one PRON ddj-151 297 9 can can AUX ddj-151 297 10 have have VERB ddj-151 297 11 several several ADJ ddj-151 297 12 non non ADJ ddj-151 297 13 - - ADJ ddj-151 297 14 equivalent equivalent ADJ ddj-151 297 15 pairs pair NOUN ddj-151 297 16 ( ( PUNCT ddj-151 297 17 p p X ddj-151 297 18 , , PUNCT ddj-151 297 19 q q NOUN ddj-151 297 20 ) ) PUNCT ddj-151 297 21 defining define VERB ddj-151 297 22 combinatorially combinatorially ADV ddj-151 297 23 non non ADJ ddj-151 297 24 - - ADJ ddj-151 297 25 equivalent equivalent ADJ ddj-151 297 26 tesselations tesselation NOUN ddj-151 297 27 . . PUNCT ddj-151 298 1 my my PRON ddj-151 298 2 interpretation interpretation NOUN ddj-151 298 3 is be AUX ddj-151 298 4 that that SCONJ ddj-151 298 5 they they PRON ddj-151 298 6 correspond correspond VERB ddj-151 298 7 to to ADP ddj-151 298 8 different different ADJ ddj-151 298 9 conformal conformal ADJ ddj-151 298 10 equivalence equivalence NOUN ddj-151 298 11 classes class NOUN ddj-151 298 12 for for ADP ddj-151 298 13 torus torus NOUN ddj-151 298 14 : : PUNCT ddj-151 298 15 the the DET ddj-151 298 16 intuitive intuitive ADJ ddj-151 298 17 expectation expectation NOUN ddj-151 298 18 is be AUX ddj-151 298 19 that that SCONJ ddj-151 298 20 this this PRON ddj-151 298 21 should should AUX ddj-151 298 22 not not PART ddj-151 298 23 affect affect VERB ddj-151 298 24 the the DET ddj-151 298 25 topology topology NOUN ddj-151 298 26 of of ADP ddj-151 298 27 tesselation tesselation NOUN ddj-151 298 28 nor nor CCONJ ddj-151 298 29 hamiltonian hamiltonian ADJ ddj-151 298 30 cycles cycle NOUN ddj-151 298 31 . . PUNCT ddj-151 299 1 for for ADP ddj-151 299 2 ( ( PUNCT ddj-151 299 3 6 6 NUM ddj-151 299 4 , , PUNCT ddj-151 299 5 3)p 3)p NUM ddj-151 299 6 , , PUNCT ddj-151 299 7 q q NOUN ddj-151 299 8 = = SYM ddj-151 299 9 ( ( PUNCT ddj-151 299 10 6 6 NUM ddj-151 299 11 , , PUNCT ddj-151 299 12 3)2,2 3)2,2 NUM ddj-151 299 13 with with ADP ddj-151 299 14 s s PROPN ddj-151 299 15 ( ( PUNCT ddj-151 299 16 v v NOUN ddj-151 299 17 = = SYM ddj-151 299 18 12 12 NUM ddj-151 299 19 , , PUNCT ddj-151 299 20 f f PROPN ddj-151 299 21 = = SYM ddj-151 299 22 24 24 NUM ddj-151 299 23 ) ) PUNCT ddj-151 299 24 there there PRON ddj-151 299 25 are be VERB ddj-151 299 26 1 1 NUM ddj-151 299 27 + + SYM ddj-151 299 28 6 6 NUM ddj-151 299 29 = = SYM ddj-151 299 30 7 7 NUM ddj-151 299 31 combinatorially combinatorially ADV ddj-151 299 32 non non ADJ ddj-151 299 33 - - ADJ ddj-151 299 34 equivalent equivalent ADJ ddj-151 299 35 tesselations tesselation NOUN ddj-151 299 36 : : PUNCT ddj-151 299 37 one one NUM ddj-151 299 38 non non ADJ ddj-151 299 39 - - ADJ ddj-151 299 40 chiral chiral ADJ ddj-151 299 41 and and CCONJ ddj-151 299 42 6 6 NUM ddj-151 299 43 chiral chiral ADJ ddj-151 299 44 ones one NOUN ddj-151 299 45 . . PUNCT ddj-151 300 1 quite quite ADV ddj-151 300 2 generally generally ADV ddj-151 300 3 , , PUNCT ddj-151 300 4 the the DET ddj-151 300 5 tesselations tesselation NOUN ddj-151 300 6 with with ADP ddj-151 300 7 v v NOUN ddj-151 300 8 vertices vertex NOUN ddj-151 300 9 with with ADP ddj-151 300 10 v v NUM ddj-151 300 11 mod mod NOUN ddj-151 300 12 4 4 NUM ddj-151 300 13 = = SYM ddj-151 300 14 0 0 PUNCT ddj-151 300 15 ( ( PUNCT ddj-151 300 16 as as ADP ddj-151 300 17 in in ADP ddj-151 300 18 the the DET ddj-151 300 19 case case NOUN ddj-151 300 20 of of ADP ddj-151 300 21 v v NOUN ddj-151 300 22 = = SYM ddj-151 300 23 12 12 NUM ddj-151 300 24 ) ) PUNCT ddj-151 300 25 allow allow VERB ddj-151 300 26 one one NUM ddj-151 300 27 map map NOUN ddj-151 300 28 ( ( PUNCT ddj-151 300 29 chart chart NOUN ddj-151 300 30 consisting consist VERB ddj-151 300 31 of of ADP ddj-151 300 32 faces face NOUN ddj-151 300 33 ) ) PUNCT ddj-151 300 34 with with ADP ddj-151 300 35 isotropy isotropy ADJ ddj-151 300 36 group group NOUN ddj-151 300 37 of of ADP ddj-151 300 38 order order NOUN ddj-151 300 39 2 2 NUM ddj-151 300 40 and and CCONJ ddj-151 300 41 6 6 NUM ddj-151 300 42 maps map NOUN ddj-151 300 43 with with ADP ddj-151 300 44 isotropy isotropy ADJ ddj-151 300 45 group group NOUN ddj-151 300 46 of of ADP ddj-151 300 47 order order NOUN ddj-151 300 48 4 4 NUM ddj-151 300 49 . . PUNCT ddj-151 301 1 these these DET ddj-151 301 2 variants variant NOUN ddj-151 301 3 are be AUX ddj-151 301 4 labelled label VERB ddj-151 301 5 by by ADP ddj-151 301 6 an an DET ddj-151 301 7 sl(2,z sl(2,z NOUN ddj-151 301 8 ) ) PUNCT ddj-151 301 9 matrix matrix NOUN ddj-151 301 10 ( ( PUNCT ddj-151 301 11 a a DET ddj-151 301 12 , , PUNCT ddj-151 301 13 b b NOUN ddj-151 301 14 ; ; PUNCT ddj-151 301 15 0 0 NUM ddj-151 301 16 , , PUNCT ddj-151 301 17 c c NOUN ddj-151 301 18 ) ) PUNCT ddj-151 301 19 with with ADP ddj-151 301 20 determinant determinant ADJ ddj-151 301 21 equal equal ADJ ddj-151 301 22 to to ADP ddj-151 301 23 v v NOUN ddj-151 301 24 = = SYM ddj-151 301 25 ac ac PROPN ddj-151 301 26 . . PUNCT ddj-151 302 1 for for ADP ddj-151 302 2 v v NOUN ddj-151 302 3 = = SYM ddj-151 302 4 12 12 NUM ddj-151 302 5 one one NOUN ddj-151 302 6 has have VERB ddj-151 302 7 decompositions decomposition NOUN ddj-151 302 8 12 12 NUM ddj-151 302 9 = = SYM ddj-151 302 10 1 1 NUM ddj-151 302 11 × × NOUN ddj-151 302 12 12 12 NUM ddj-151 302 13 , , PUNCT ddj-151 302 14 12 12 NUM ddj-151 302 15 = = SYM ddj-151 302 16 2 2 NUM ddj-151 302 17 × × NOUN ddj-151 302 18 6 6 NUM ddj-151 302 19 , , PUNCT ddj-151 302 20 12 12 NUM ddj-151 302 21 = = SYM ddj-151 302 22 3 3 NUM ddj-151 302 23 × × NOUN ddj-151 302 24 4 4 NUM ddj-151 302 25 . . PUNCT ddj-151 303 1 −c −c VERB ddj-151 303 2 < < X ddj-151 303 3 b b X ddj-151 303 4 < < X ddj-151 303 5 a a DET ddj-151 303 6 − − PROPN ddj-151 303 7 c c NOUN ddj-151 303 8 is be AUX ddj-151 303 9 unique unique ADJ ddj-151 303 10 modulo modulo NOUN ddj-151 303 11 a. a. NOUN ddj-151 303 12 in in ADP ddj-151 303 13 the the DET ddj-151 303 14 recent recent ADJ ddj-151 303 15 case case NOUN ddj-151 303 16 one one NUM ddj-151 303 17 as as ADP ddj-151 303 18 ac ac PROPN ddj-151 303 19 = = PROPN ddj-151 303 20 12 12 NUM ddj-151 303 21 allowing allow VERB ddj-151 303 22 ( ( PUNCT ddj-151 303 23 a a PRON ddj-151 303 24 , , PUNCT ddj-151 303 25 c c NOUN ddj-151 303 26 ) ) PUNCT ddj-151 303 27 ∈ ∈ NOUN ddj-151 303 28 { { PUNCT ddj-151 303 29 ( ( PUNCT ddj-151 303 30 1 1 NUM ddj-151 303 31 , , PUNCT ddj-151 303 32 12 12 NUM ddj-151 303 33 ) ) PUNCT ddj-151 303 34 , , PUNCT ddj-151 303 35 ( ( PUNCT ddj-151 303 36 2 2 NUM ddj-151 303 37 , , PUNCT ddj-151 303 38 6 6 NUM ddj-151 303 39 ) ) PUNCT ddj-151 303 40 , , PUNCT ddj-151 303 41 ( ( PUNCT ddj-151 303 42 3 3 NUM ddj-151 303 43 , , PUNCT ddj-151 303 44 4 4 NUM ddj-151 303 45 ) ) PUNCT ddj-151 303 46 } } PUNCT ddj-151 303 47 and and CCONJ ddj-151 303 48 pairs pair NOUN ddj-151 303 49 obtained obtain VERB ddj-151 303 50 by by ADP ddj-151 303 51 permuting permute VERB ddj-151 303 52 a a PRON ddj-151 303 53 and and CCONJ ddj-151 303 54 c. c. NOUN ddj-151 303 55 these these DET ddj-151 303 56 matrices matrix NOUN ddj-151 303 57 need need AUX ddj-151 303 58 not not PART ddj-151 303 59 define define VERB ddj-151 303 60 combinatorially combinatorially ADV ddj-151 303 61 different different ADJ ddj-151 303 62 tesselations tesselation NOUN ddj-151 303 63 since since SCONJ ddj-151 303 64 modular modular ADJ ddj-151 303 65 transformations transformation NOUN ddj-151 303 66 generate generate VERB ddj-151 303 67 equivalent equivalent ADJ ddj-151 303 68 matrices matrix NOUN ddj-151 303 69 . . PUNCT ddj-151 304 1 acknowledgements acknowledgement NOUN ddj-151 304 2 : : PUNCT ddj-151 305 1 i i PRON ddj-151 305 2 am be AUX ddj-151 305 3 grateful grateful ADJ ddj-151 305 4 to to PART ddj-151 305 5 jebin jebin VERB ddj-151 305 6 larosh larosh PROPN ddj-151 305 7 for for ADP ddj-151 305 8 finding find VERB ddj-151 305 9 the the DET ddj-151 305 10 java java NOUN ddj-151 305 11 algorith algorith NOUN ddj-151 305 12 allowing allow VERB ddj-151 305 13 to to PART ddj-151 305 14 find find VERB ddj-151 305 15 hamilton hamilton PROPN ddj-151 305 16 cycles cycle NOUN ddj-151 305 17 for for ADP ddj-151 305 18 given give VERB ddj-151 305 19 graph graph NOUN ddj-151 305 20 . . PUNCT ddj-151 306 1 references reference NOUN ddj-151 306 2 [ [ X ddj-151 306 3 1 1 X ddj-151 306 4 ] ] PUNCT ddj-151 306 5 ulrich ulrich PROPN ddj-151 306 6 brehm brehm PROPN ddj-151 306 7 and and CCONJ ddj-151 306 8 wolfgang wolfgang PROPN ddj-151 306 9 kühnel kühnel PROPN ddj-151 306 10 . . PUNCT ddj-151 307 1 equivelar equivelar ADJ ddj-151 307 2 maps map NOUN ddj-151 307 3 on on ADP ddj-151 307 4 the the DET ddj-151 307 5 torus torus NOUN ddj-151 307 6 . . PUNCT ddj-151 308 1 european european PROPN ddj-151 308 2 journal journal PROPN ddj-151 308 3 of of ADP ddj-151 308 4 combinatorics combinatorics PROPN ddj-151 308 5 . . PUNCT ddj-151 309 1 https https PROPN ddj-151 309 2 : : PUNCT ddj-151 309 3 // // NUM ddj-151 309 4 doi doi X ddj-151 309 5 . . PUNCT ddj-151 310 1 org/ org/ PRON ddj-151 310 2 10 10 NUM ddj-151 310 3 . . PUNCT ddj-151 311 1 1016/ 1016/ NUM ddj-151 311 2 j. j. PROPN ddj-151 311 3 ejc ejc PROPN ddj-151 311 4 . . PUNCT ddj-151 311 5 2008 2008 NUM ddj-151 311 6 . . PUNCT ddj-151 312 1 01 01 NUM ddj-151 312 2 . . NUM ddj-151 312 3 010 010 NUM ddj-151 312 4 , , PUNCT ddj-151 313 1 29(8):1843–1861 29(8):1843–1861 NUM ddj-151 313 2 , , PUNCT ddj-151 313 3 2008 2008 NUM ddj-151 313 4 . . PUNCT ddj-151 314 1 [ [ X ddj-151 314 2 2 2 X ddj-151 314 3 ] ] PUNCT ddj-151 314 4 thomas thomas PROPN ddj-151 314 5 r r PROPN ddj-151 314 6 and and CCONJ ddj-151 314 7 yu yu PROPN ddj-151 314 8 x. x. NOUN ddj-151 314 9 five five NUM ddj-151 314 10 - - PUNCT ddj-151 314 11 connected connect VERB ddj-151 314 12 toroidal toroidal ADJ ddj-151 314 13 graphs graph NOUN ddj-151 314 14 are be AUX ddj-151 314 15 hamiltonian hamiltonian ADJ ddj-151 314 16 . . PUNCT ddj-151 315 1 journal journal NOUN ddj-151 315 2 of of ADP ddj-151 315 3 combinatorial combinatorial ADJ ddj-151 315 4 theory theory NOUN ddj-151 315 5 , , PUNCT ddj-151 315 6 series series PROPN ddj-151 315 7 b b PROPN ddj-151 315 8 , , PUNCT ddj-151 315 9 69(tb961713):79–96 69(tb961713):79–96 NUM ddj-151 315 10 , , PUNCT ddj-151 315 11 1997 1997 NUM ddj-151 315 12 . . PUNCT ddj-151 316 1 [ [ X ddj-151 316 2 3 3 NUM ddj-151 316 3 ] ] X ddj-151 316 4 pitkänen pitkänen NOUN ddj-151 316 5 m. m. NOUN ddj-151 316 6 geometric geometric ADJ ddj-151 316 7 theory theory NOUN ddj-151 316 8 of of ADP ddj-151 316 9 harmony harmony PROPN ddj-151 316 10 . . PUNCT ddj-151 317 1 available available ADJ ddj-151 317 2 at at ADP ddj-151 317 3 : : PUNCT ddj-151 317 4 http://tgdtheory.fi/public_html/ http://tgdtheory.fi/public_html/ NOUN ddj-151 317 5 articles article NOUN ddj-151 317 6 / / SYM ddj-151 317 7 harmonytheory.pdf harmonytheory.pdf X ddj-151 317 8 , , PUNCT ddj-151 317 9 2014 2014 NUM ddj-151 317 10 . . PUNCT ddj-151 318 1 [ [ X ddj-151 318 2 4 4 NUM ddj-151 318 3 ] ] X ddj-151 318 4 pitkänen pitkänen NOUN ddj-151 318 5 m. m. NOUN ddj-151 318 6 new new ADJ ddj-151 318 7 results result NOUN ddj-151 318 8 in in ADP ddj-151 318 9 the the DET ddj-151 318 10 model model NOUN ddj-151 318 11 of of ADP ddj-151 318 12 bio bio NOUN ddj-151 318 13 - - NOUN ddj-151 318 14 harmony harmony NOUN ddj-151 318 15 . . PUNCT ddj-151 319 1 available available ADJ ddj-151 319 2 at at ADP ddj-151 319 3 : : PUNCT ddj-151 319 4 http://tgdtheory.fi/public http://tgdtheory.fi/public NUM ddj-151 319 5 _ _ PROPN ddj-151 319 6 html html NOUN ddj-151 319 7 / / SYM ddj-151 319 8 articles article NOUN ddj-151 319 9 / / SYM ddj-151 319 10 harmonynew.pdf harmonynew.pdf PROPN ddj-151 319 11 , , PUNCT ddj-151 319 12 2018 2018 NUM ddj-151 319 13 . . PUNCT ddj-151 320 1 issn issn PROPN ddj-151 320 2 : : PUNCT ddj-151 320 3 2159 2159 NUM ddj-151 320 4 - - PUNCT ddj-151 320 5 046x 046x NOUN ddj-151 320 6 dna dna NOUN ddj-151 320 7 decipher decipher PROPN ddj-151 320 8 journal journal PROPN ddj-151 320 9 www.www.dnadecipher.com www.www.dnadecipher.com PROPN ddj-151 320 10 published publish VERB ddj-151 320 11 by by ADP ddj-151 320 12 quantumdream quantumdream PROPN ddj-151 320 13 , , PUNCT ddj-151 320 14 inc inc PROPN ddj-151 320 15 . . PUNCT ddj-151 321 1 http://tinyurl.com/ya6g9kwe http://tinyurl.com/ya6g9kwe ADJ ddj-151 321 2 https://doi.org/10.1016/j.ejc.2008.01.010 https://doi.org/10.1016/j.ejc.2008.01.010 ADJ ddj-151 321 3 http://tgdtheory.fi/public_html/articles/harmonytheory.pdf http://tgdtheory.fi/public_html/articles/harmonytheory.pdf PROPN ddj-151 321 4 http://tgdtheory.fi/public_html/articles/harmonytheory.pdf http://tgdtheory.fi/public_html/articles/harmonytheory.pdf PROPN ddj-151 321 5 http://tgdtheory.fi/public_html/articles/harmonynew.pdf http://tgdtheory.fi/public_html/articles/harmonynew.pdf PROPN ddj-151 321 6 http://tgdtheory.fi/public_html/articles/harmonynew.pdf http://tgdtheory.fi/public_html/articles/harmonynew.pdf PROPN ddj-151 321 7 introduction introduction NOUN ddj-151 321 8 icosahedral icosahedral ADJ ddj-151 321 9 bio bio ADJ ddj-151 321 10 - - PUNCT ddj-151 321 11 harmonies harmonies ADJ ddj-151 321 12 explanation explanation NOUN ddj-151 321 13 for for ADP ddj-151 321 14 the the DET ddj-151 321 15 number number NOUN ddj-151 321 16 12 12 NUM ddj-151 321 17 of of ADP ddj-151 321 18 notes note NOUN ddj-151 321 19 of of ADP ddj-151 321 20 12 12 NUM ddj-151 321 21 - - PUNCT ddj-151 321 22 note note NOUN ddj-151 321 23 scale scale NOUN ddj-151 321 24 is be AUX ddj-151 321 25 it it PRON ddj-151 321 26 possible possible ADJ ddj-151 321 27 to to PART ddj-151 321 28 have have VERB ddj-151 321 29 toric toric ADJ ddj-151 321 30 harmonies harmony NOUN ddj-151 321 31 ? ? PUNCT ddj-151 322 1 the the DET ddj-151 322 2 number number NOUN ddj-151 322 3 of of ADP ddj-151 322 4 triangles triangle NOUN ddj-151 322 5 in in ADP ddj-151 322 6 the the DET ddj-151 322 7 12 12 NUM ddj-151 322 8 - - PUNCT ddj-151 322 9 vertex vertex NOUN ddj-151 322 10 tesselation tesselation NOUN ddj-151 322 11 is be AUX ddj-151 322 12 24 24 NUM ddj-151 322 13 : : PUNCT ddj-151 322 14 curse curse NOUN ddj-151 322 15 or or CCONJ ddj-151 322 16 blessing blessing NOUN ddj-151 322 17 ? ? PUNCT ddj-151 323 1 a a DET ddj-151 323 2 more more ADV ddj-151 323 3 detailed detailed ADJ ddj-151 323 4 model model NOUN ddj-151 323 5 for for ADP ddj-151 323 6 toric toric ADJ ddj-151 323 7 harmonies harmony NOUN ddj-151 323 8 what what PRON ddj-151 323 9 one one PRON ddj-151 323 10 can can AUX ddj-151 323 11 say say VERB ddj-151 323 12 about about ADP ddj-151 323 13 toric toric ADJ ddj-151 323 14 hamiltonian hamiltonian ADJ ddj-151 323 15 cycles cycle NOUN ddj-151 323 16 ? ? PUNCT ddj-151 324 1 appendix appendix NOUN ddj-151 324 2 : : PUNCT ddj-151 324 3 some some DET ddj-151 324 4 facts fact NOUN ddj-151 324 5 about about ADP ddj-151 324 6 toric toric ADJ ddj-151 324 7 tesselations tesselation NOUN