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DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

1 

Article 
 

Algebraic and Geometric Representations of the Genetic Code 
 

Richard L Amoroso
,1

, Peter Rowlands
2
 

 
1Noetic Advanced Studies Institute, USA 

2University of Liverpool, UK 

 

Abstract 
Algebraic and geometric representations of the genetic code are used to show their functions for 

coding amino acids. The algebra is a 64-part vector quaternion combination, and the related 

geometry is based on the structure of the regular icosidodecahedron. An almost perfect pattern 

suggesting that this is a biologically significant way of representing the genetic code that may 

lead to a deeper understanding of a relationship between geometry and teleological life principles 

of complex self-organization. 
 

Keywords: Genetic code, triplet codons, amino acids, vector-quaternion algebra, teleology, 

Icosidodecahedron. 

 

 

1. Introduction 
 

We explore an interesting way to represent the genetic code using a correspondence between 

algebra and geometry. The algebraic component is based on an Icosian calculus with a non-

commutative algebraic structure discovered by William Rowan Hamilton in 1856, which he 

called quaternions. In modern terms, Hamilton produced a group presentation of the icosahedral 

rotation group by generators and relations. 

 

Hamilton’s discovery was derived from his attempts to find an algebra of ‘triplets’ that he 

believed would reflect the three Cartesian axes in a manner extending the complex numbers, 

which took the form, 
2 2 2 1i j k ijk     . The symbols of the icosian calculus can be equated 

to moves between vertices on a dodecahedron. 

 

 

2. The Algebraic Representation 
 

In previous work [1-3] we have used various mathematical structures to represent the genetic 

code, including a 64-part vector quaternion algebra, which is isomorphic to the algebra of the 

quantum mechanical Dirac equation, and a combination of the faces and vertices of a regular 

icosidodecahedron. Here, we aim to show that it is possible to represent the codon structures 

both algebraically and geometrically in a way that relates to their function in coding for amino 

acids. 

                                                        
Correspondence: Prof. Richard L. Amoroso, Director of Physics Lab., Noetic Advanced Studies Institute, Utah, USA. 

https://orcid.org/0000-0003-2405-9034; http://www.noeticadvancedstudies.us E-mail: amoroso@noeticadvancestudies.us 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

2 

It is based on a vector-quaternion algebra whose units can be represented as follows [4-6]: 

  

        i j k   vector 

        i     pseudoscalar 

        i j k   quaternion 

        1     scalar 

 

They can be considered as the units of two spaces: (‘real’, constructed from i, j, k) and 

(‘vacuum’, constructed from i, i, j, k, 1). In principle, any self-organizing system, whether 

physical, chemical or biological, forms a space, which has a kind of distorted mirror image in 

another ‘space’ representing the rest of the universe. The double space creates the entire 

combination of system and ‘vacuum’ as a zero totality. 

 

In principle, any self-organizing system, whether physical, chemical or biological, forms a space, 

which has a kind of distorted mirror image in another ‘space’ representing the rest of the 

universe. The double space creates the entire combination of system and ‘vacuum’ as a zero 

totality. The algebraic structure has an exact parallel with a geometric one which can be 

represented using Platonic or Archimedean solids in which each structure has a dual which could 

be imagined as constructed in another space. 

 

The units of the vector-quaternion algebra constructing the double space can be represented as 

follows: 

 

  i  j  k ii ij ik i 1   –i  –j  –k –ii –ij –ik –i –1 

  i  j  k ii ii ik     –i  –j  –k –ii –ii –ik 

  ii  ij  ik iii iii iik     –ii  –ij  –ik –iii –iii –iik 

  ji  jj  jk iji iji ijk     –ji  –jj  –jk –iji –iji –ijk 

  ki  kj  kk iki iki ikk     –ki  –kj  –kk –iki –iki –ikk 
 

An alternative ordering would separate the four complex numbers from 12 nilpotent structures, 

each formed from 5 units. Here, we create a subset of 60 units, which has significance in the 

dodecahedral and icosahedral representations and in Hamilton’s Icosian calculus: 

 

       1              –1           

     ii ij ik ik j  –ii –ij –ik –ik –j 

     ji jj jk ii k  –ji –jj –jk –ii –k 

     ki kj kk ij i  –ki –kj –kk –ij –i 

 

     i                    –i 

     iii iij iik ik j  –iii –iij –iik –ik –j 

     iji ijj ijk ii k  –iji –ijj –ijk –ii –k 

     iki ikj ikk ij i  –iki –ikj –ikk –ij –i 
 

One way of generating the 64 units is by taking the product of 4 options × 4 options × 4 options, 

as is done in the case of the genetic code, where each of three bases may be U (or T), G, A or C. 

To represent this algebraically, we may use the vector units i, j, k and 1 for the options U, G, A 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-xx 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

3 

and C for the first base. Then, for the second base, we may represent U, G, A and C by the 

quaternion units i, j, k and 1. Then U, G, A and C on the third base may be represented by the 

units of complex algebra 1, i, –1, –i. Using three different algebras (vectors, quaternions and 

complex numbers) allows us to track the three bases individually according to their positions in 

the codon. 

 

We have previously grouped the amino acids produced by the genetic code mechanism 

according to the second base in the codon which produced it. The second base seems, in this 

respect, the most important, and the third base the least, becoming in some sense almost 

redundant. Using this division, the 64 codons fall naturally into 4 groups of 16: 

 
            amino codon  first   second   third 

               acid     base   base   base 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The 

Group II 

 

 Cys UGU i j 1 

  UGC i j i 

 Trp UGG i j –i 

 STOP UGA i j –1 

 Gly GGU  j j 1  

  GGC  j j i 

  GGA  j j –1 
  GGG j j –i 

 Ser  AGU  k j 1 

  AGC k j i 

 Arg CGU 1 j 1 

  CGC  1 j i 

  CGA  1 j –1 

  CGG  1 j –i 

  AGA k j –1 

  AGG k j –i 

Group I 

 

 Phe UUU    i i 1 

  UUC i i i 

 Leu UUA  i i –1  
  UUG*  i i –i 

  CUU  1 i 1 

  CUC  1 i i 

  CUA  1 i –1 

  CUG* 1 i –i 

 Val GUU  j i 1 

  GUC  j i i 

  GUA  j i –1 

  GUG  j i –i 

 Ile AUU  k i 1 

  AUC  k i i 
  AUA* k i –1 

 Met AUG* k i –i 

Group III 

 

 STOP  UAA  i k –1 
  UAG i k –i 

 Tyr UAU i k 1 

  UAC i k i 

 Asp GAU  j k 1 

  GAC j k i 

 Glu GAA  j k –1 

  GAG  j k –i 

 Lys AAA   k k –1 

  AAG k k –i 

 Asn AAU  k k 1 

  AAC k k –i 

 His CAU  1 k 1 
  CAC 1 k i 

 Gln CAA  1 k –1 

  CAG 1  k –i 

Group IV 

 

 Ser  UCU  i 1 1 
  UCC  i 1 i 

  UCA  i 1 –1 

  UCG i 1 –i 

 Ala GCU  j 1 1 

  GCC  j 1 i 

  GCA  j 1 –1 

  GCG j 1 –i 

 Thr ACU  k 1 1 

  ACC  k 1 i 

  ACA  k 1 –1 

  ACG k 1 –i 

 Pro CCU  1 1 1 
  CCC  1 1 i 

  CCA  1 1 –1 

  CCG 1 1 –i 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

4 

asterisks represent codons that can act as a START. Notably they are all in the same group. 

Conveniently also (though this is mainly an artefact of our representation) all the START and 

STOP codons are represented by negative units. The way that the various structures are relevant 

to the formation of amino acids will become clearer in the following table: 

 

Group I 

Phenylalanine 

 UUU UUC 

   ii          iii 

Leucine  

 UUA UUG* CUU CUC CUA CUG* 

  –ii        –iii           i           ii          –i    –ii 

Valine  

 GUU GUC GUA GUG 

    ij          iij          –ij        –iij 

Isoleucine 

 AUU AUC AUA* 

   ik          iik        –ik 

Methionine  

 AUG* 

 –iik 

 

 

Group II 

Cysteine  

 UGU UGC 

 ji     iji 

Tryptophan 

 UGG 

 –iji  

STOP 

 UGA 

 –ji 

Glycine 

 GGU GGC GGA GGG 

 –jj        –ijj          jj        ijj 

Serine 

 AGU AGC 

 jk        ijk 

Arginine 

 CGU CGC CGA CGG AGA AGG 

 j         ij          –j        –ij        –jk       –ijk 

 

 

 

 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-xx 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

5 

Group III 

STOP 

 UAA UAG 

 –ki       –iki 

Tyrosine 

 UAU UAC  

 ki        iki 

Aspartate 

 GAU GAC  

 kj         ikj 

Glutamate 

 GAA GAU  

 –kj        –ikj 

Lysine 

 AAA AAG 

 –kk       –ikk 

Asparagine 

 AAU AAC 

 kk        ikk 

Histidine 

 CAU CAC 

 k         ik 

Glutamine 

 CAA CAG 

 –k        –ik 

 

Group IV 

Serine 

 UCU UCC UCA UCG 

 i          ii         –i          ii 

Alanine 

 GCU GCC GCA GCG 

 j          ij          –j        –ij 

Threonine 

 ACU ACC ACA ACG 

 k         ik         –k        –ik 

Proline 

 CCU CCC CCA CCG 

 1          i         –1         –i 

 

We can see the pattern is nearly perfect; illustrated, particularly by the complete regularity of 

Groups II and IV. Only the two serine codons in Group II are anomalous (not in their biological 

group), and they will be in any arrangement. Arginine and serine in this group seem to each have 

two codons that could, originally, have coded a different amino acid. Almost certainly, the 

codons for arginine and serine have become mixed at some stage in biological evolution. (We 

may note also that arginine seems to be an exception to the general tendency for the more 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

6 

complicated amino acid molecules to be coded using fewer codon pathways.) The table we have 

given is only for one species, and it may be that evolutionary drift may be determined to some 

extent by the variations in the patterns from the assumed perfect norm. The codon for 

tryptophan, notably, can become the STOP codon in some species, and vice versa. 
 

 

3. The Geometrical Representation 
 

Algebraic and geometrical structures are fundamentally dual. Where there is an algebra, there is 

also a geometry, and vice versa. It is easy to show that this is the case here. The four groups of 

codons can now be represented on the faces of a regular icosidodecahedron, divided into four 

equal sections. (We could use the combined faces plus vertices of a dodecahedron or 

icosahedron.) The negative units are not shown in the figures, but can be assumed either to be 

represented on the corresponding vertices of the dual rhombic triacontahedron, or on the inner, 

rather than outer, surface of the icosidodecahedron. 

 
 

 
 

Figure 1. Algebraic geometry for representing codons on the icosidodecahedron. 
 

 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-xx 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

7 

The codons can be represented on these diagrams in the form: 

 

 

 
 

   
 

Figure 2. codons representations 
 

 

The amino acids coded can be represented as follows: 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

8 

 

 
 

Figure 3. Amino acids on the icosidodecahedron. 

 

Here, the position of serine in two separate groups gives an idea of how the four sections might 

be connected. One of the significant aspects of the first three sections in alternative arrangement 

(below) is that the algebraic units of the three pentagons in each, combined with those of the two 

outer triangles on the lowest pentagon form the basis of a nilpotent structure, such as we find in 

the amplitude term in the nilpotent Dirac equation (ikE + iipx + ijpy + ikpz + jm). At the same 

time, the five triangles taken together form the basis of another nilpotent structure. So, the three 

inner triangles and the three pentagons display a duality in that either group can be used with the 

two outer triangles to generate a set of nilpotent units (though with their roles switched in the 

two cases). This provides another way of generating 12 nilpotent structures from the algebra. 

Even using the first version of the icosidodecahedral sections, we can connect these triangles 

making up the nilpotent units with the upper pentagons, and so maintain the nilpotent structure. 

Nilpotency is one of the assumed bases of the overall pattern that we have described as Nature’s 

code, and it appears to be the means by which a self-organizing system connects with its external 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-xx 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

9 

environment. Its presence in these geometric structures indicates the real importance of geometry 

in the genetic code as a route towards self-organization. 

                                    

 
 

Figure 4. Quaternion algebraic representation. 
 

 

 

Appendix: Note on the Regular Icosidodecahedron 
 

Geometrically, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and 

twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two 

triangles and two pentagons meeting at each vertex. It also has 60 identical edges, each 

separating a triangle from a pentagon. Because of this, it is one of the Archimedean solids 
 



DNA Decipher Journal | October 2021 | Volume 11 | Issue 1 | pp 01-10 

Amoroso, R. L. & Rowlands, P., Algebraic and Geometric Representations of the Genetic Code 

 

 

ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 

www.dnadecipher.com 

 

10 

   
Figure 5. Two views of the icosidodecahedron. 

 

All Archimedean solids can be produced from Platonic solids, by ‘cutting the edges’ of the 

platonic solid.  Likewise, Platonic solids can be turned into Archimedean solids by following a 

series of rules for their construction. 
 

Interestingly, in Cartesian coordinates, the vertices of an icosidodecahedron with unit edges are 

given by the even permutations of 

 

 

 
2

0,0

1
, ,

2 2 2



 



 
  
 

 , 

 

where   is the golden ratio, 
1 5

2


  [7].  

 
Received June 8, 2021; Accepted June 26, 2021 

 

References 
 

[1] Hill, V.J. and Rowlands, P. (2008) Nature’s code, AIP Conference Proceedings, 1051, 117-126. 

[2] Hill, V.J. and Rowlands, P. (2010) Nature’s fundamental symmetry breaking, International Journal  

      of Computing Anticipatory Systems, 25, 144-159. 

[3] Hill, V.J. and Rowlands, P. (2010) The numbers of nature’s code, International Journal of  
      Computing Anticipatory Systems, 25, 160-175. 

[4] Rowlands, P. (2007) Zero to Infinity: The Foundations of Physics, Singapore and Hackensack,  

      N.J., World Scientific. 
[5] Rowlands, P. (2010) Dual vector spaces and physical singularities, AIP Conference Proceedings,  

      1316, 102-111. 

[6] Amoroso, R.L., Rowlands P., Kauffman, L.H. (2013) Exploring novel cyclic extensions of  
      Hamilton’s dual-quaternion algebra, in R.L. Amoroso, L.H. Kauffman, P. Rowlands (eds.) The  

      Physics of Reality Space, Time, Matter, Cosmos, Proceedings of the 8th Symposium Honoring  

      Mathematical Physicist Jean-Pierre Vigier, pp. 81-92, Singapore: World Scientific;  

      https://vixra.org/pdf/1711.0468v1.pdf. 
[7] Weisstein, E. W. (2010) Icosahedral Group, MathWorld, A Wolfram Web  

      Resource; https://mathworld.wolfram.com/IcosahedralGroup.html. 


