

































je G2ii2`b �M/ LQi2b

1p2M i?Qm;? � ?mK�M +�M i`BpB�HHv /2+B/2 �M �`#Bi`�`BHv HQM;
#Bibi`BM; Q7 Rb Bb MQi `�M/QK- h?2Q`2K j b?Qrb Bb �M BKTQbbB#H2
i�bF 7Q` � ;2M2`�HBx2/ �H;Q`Bi?KX PMHv � bT2+B}+ �H;Q`Bi?K-
bm+? �b 2t2KTHB}2/ BM h?2Q`2K k- +�M /Q bQX

h?Bb +QM+HmbBQM Bb � #Bi +QmMi2`@BMimBiBp2- bBM+2 Bi K2�Mb i?�i
rBi?Qmi /QK�BM FMQrH2/;2- �M �H;Q`Bi?K ;Bp2M �M 2ti`2K2Hv
HQM; b2[m2M+2 Q7 Rb rQmH/ #2 mMbm`2 r?2i?2` i?2 b2[m2M+2 Bb
+QKTH2i2Hv `�M/QKX q?2M �bF2/ iQ T`2/B+i i?2 M2ti /B;Bi- i?2
�H;Q`Bi?K +�M QMHv ;Bp2 �M 2[m�H r2B;?iBM; iQ y �M/ RX

S`QpBM; i?2 .2`Bp�iBp2 Q7 sin(!) lbBM;
i?2 Svi?�;Q`2�M h?2Q`2K �M/ i?2
lMBi *B`+H2
CQM�i?�M "�`iH2ii
.PA, RyXjjyR9fBbbMXke9y@8e8kXjXRX#�`iH2iiXR

h?2 /2`Bp�iBp2 Q7 sin(!) Ur?2`2 ! Bb K2�bm`2/ BM `�/B�MbV Bb
;Bp2M BM bi�M/�`/ +�H+mHmb �b cos(!)X h?2 T`QQ7 7Q` i?Bb Bb
mbm�HHv #�b2/ QM � HBKBi, lim

!!0
sin(!)

! = 1X h?2 T`QQ7- Tmi
bBKTHv- Bb,

" = sin(!) URV
" + d" = sin(! + d!) UkV

d" = sin(! + d!) " sin(!) UjV
d" = sin(!) cos(d!) + cos(!) sin(d!) " sin(!) U9V
d" = sin(!) + cos(!) sin(d!) " sin(!) U8V
d" = cos(!) sin(d!) UeV
d"
d! = cos(!) sin(d!)

d! UdV
d"
d! = cos(!) U3V

q?BH2 i?2`2 Bb MQi?BM; r`QM; rBi? i?2 T`QQ7 T2` b2- A ?�p2
�Hr�vb 7QmM/ Bi mMb�iBb7vBM;- miBHBxBM; i`B;QMQK2i`v B/2MiBiB2b
72r bim/2Mib `2K2K#2`X �//BiBQM�HHv- Bi Bb mbm�HHv �++QKT�@
MB2/ rBi? �M 2tTH�M�iBQM Q7 i?2 HBKBi Q7 sin "

" i?�i Bb ?�`/ 7Q`
bim/2Mib iQ /2+BT?2`X h?2`27Q`2- i?Bb T�T2` 2M/2�pQ`b iQ T`Q@
pB/2 � KQ`2 bi`�B;?i7Q`r�`/ T`QQ7 #�b2/ QM KQ`2 #�bB+ K�i?@
2K�iB+�H �bb2`iBQMb- 7QmM/2/ QM i?2 Svi?�;Q`2�M i?2Q`2K �M/
i?2 mMBi +B`+H2X Ai /Q2bMǶi `2KQp2 i?2 ;Bp2M HBKBi BM Bib 2M@
iB`2iv- #mi `�i?2` ;Bp2b KQ`2 bi`�B;?i7Q`r�`/- +�H+mHmb@Q`B2Mi2/

`2�bQMBM; 7Q` /QBM; � bBKBH�` QT2`�iBQMX Ai Bb /2#�i�#H2 ?Qr
Km+? /Bz2`2Mi Bi Bb BM FBM/ 7`QK i?2 bi�M/�`/ T`QQ7- #mi BM
�Mv +�b2 A i?BMF Bi Bb � KQ`2 bi`�B;?i7Q`r�`/- BMi2`2biBM;- �M/
BMbi`m+iBp2 r�v Q7 HQQFBM; �i Bi 7Q` bim/2MibX Ai b?Qrb U�V i?2
TQr2` Q7 +�H+mHmb- U#V i?2 TQr2` Q7 /Bz2`2MiB�H i?BMFBM;- �M/
U+V ?Qr /Bb+Qp2`B2b +�M #2 K�/2 7`QK #�bB+ T`BM+BTH2bX

"�bB+ �bbmKTiBQMb

h?Bb T`QQ7 rBHH #2 �M�HvxBM; i`B�M;H2b /`�rM QM i?2 mMBi +B`+H2X
PM � mMBi +B`+H2- i?2 ?vTQi2Mmb2 rBHH �Hr�vb #2 1X 6B;m`2 R
b?Qrb i?2 ;2M2`�H b2imTX ! rBHH #2 i?2 �M;H2 K2�bm`2/ BM
`�/B�Mb- # rBHH #2 i?2 �/D�+2Mi- �M/ $ rBHH #2 i?2 QTTQbBi2X

6B;m`2 R, � h`B�M;H2 AMb+`B#2/ PMiQ � lMBi *B`+H2

h?2 Svi?�;Q`2�M i?2Q`2K ;Bp2b i?2 7QHHQrBM;,

#2 + $2 = 1 UNV
$2 = 1 " #2 URyV
#2 = 1 " $2 URRV

URkV

aBM+2 i?2 ?vTQi2Mmb2 Bb 1- sin(!) = $ �M/ cos(!) = #X h?2
/2`Bp�iBp2 Q7 sin(!) rBi? `2bT2+i iQ !- i?2`27Q`2- rBHH #2 d#

d" X
h?2`27Q`2- i?2 T`QQ7 rBHH #2 bm++2bb7mH B7 Bi +�M /2KQMbi`�i2

https://dx.doi.org/10.33014/issn.2640-5652.3.1.bartlett.1


oQHmK2 j- Abbm2 R

S`QpBM; i?2 .2`Bp�iBp2 Q7 sin(!) lbBM; i?2 Svi?�;Q`2�M h?2Q`2K �M/ i?2 lMBi *B`+H2 jd

i?2 7QHHQrBM; 2[mBp�H2M+v,

d$
d! = # URjV

.Bz2`2MiB�H �M�HvbBb

h�FBM; 6B;m`2 R �M/ #m/;BM; i?2 �M;H2 #v d! vB2H/b i?2 TB+im`2
b?QrM BM 6B;m`2 kX

6B;m`2 k, *?�M;2b BM i?2 h`B�M;H2 "�b2/ QM d!

� 72r BKTQ`i�Mi MQi2b QM 6B;m`2 k,

RX �HH +?�M;2b �`2 #2BM; 2tT`2bb2/ �b �//BM; /Bz2`2MiB�Hb-
2p2M B7 i?2 /Bz2`2MiB�H Bib2H7 Bb M2;�iBp2X h?Bb Bb r?v #+d#
BM i?2 ;`�T? Bb b?Q`i2` i?�M # QM Bib QrMX

kX aBM+2 i?Bb Bb i?2 mMBi +B`+H2- i?2 �M;H2 +?�M;2 Bb B/2MiB+�H
iQ i?2 +B`+mK72`2M+2 +?�M;2 UbBM+2 i?2 `�/Bmb Bb R- i?2M
i?2 +B`+mK72`2M+2 Bb 2%- i?2 MmK#2` Q7 `�/B�Mb BM � +B`+H2VX

jX aBM+2 i?2 +?�M;2 Bb BM}MBi2bBK�H- �M/ i?Bb Bb � bKQQi? �M/
+QMiBMmQmb };m`2- i?2M i?2 +?�M;2 QM i?2 /Bz2`2MiB�H Bb
HBM2�`X AM Qi?2` rQ`/b- i?2 TB+im`2 Bb xQQK2/ BM 2MQm;?
i?�i i?2 �`+ DQBMBM; i?2 irQ i`B�M;H2b +�M #2 i`2�i2/ �b B7
Bi r2`2 � bi`�B;?i HBM2XR

RLQi2 i?�i i?Bb Bb #�bB+�HHv 2[mBp�H2Mi iQ i?2 HBKBi lim
!!0

sin(!)
! - #mi

"2+�mb2 Q7 i?Bb H�bi TQBMi- i?2 H2M;i? Q7 d! +�M #2 /2i2`KBM2/
mbBM; i?2 /Bbi�M+2 7Q`KmH�- r?2`2 i?2 ?Q`BxQMi�H �M/ p2`iB+�H
+?�M;2b �`2 bBKTHv ;Bp2M #v d# �M/ d$,

d! =
!

d$2 + d#2 UR9V

6BM�HHv- i?2 /Bz2`2MiB�H Q7 UNV +�M #2 i�F2M iQ +QK2 mT rBi?,

#2 + $2 = 1
2# d# + 2$ d$ = 0 UR8V
# d# + $ d$ = 0 UReV

# d# = "$ d$ URdV
d# = " $

#
d$ UR3V

J�FBM; i?2 S`QQ7

ai�`iBM; rBi? UR9V- bm#biBimiBQMb �M/ bBKTHB}+�iBQMb +�M #2
K�/2 �b 7QHHQrb,

d! =
!

d$2 + d#2 URNV

=

"
d$2 +

#
" $
#

d$
$2

UkyV

=

"
d$2 + $2

#2 d$2 UkRV

=

"
d$2 + 1 " #2

#2 d$2 UkkV

=

"
d$2 + d$2

#2 " d$2 UkjV

=

"
d$2

#2 Uk9V

d! = d$
#

Uk8V

LQi2 i?�i Uk8V +QmH/ �HbQ ?�p2 #22M M2;�iBp2X AMbT2+iBQM Q7
6B;m`2 k b?Qrb i?�i d$ rBHH �Hr�vb ?�p2 i?2 b�K2 bB;M �b #
UBM+`2�bBM; mMiBH # Bb x2`Q- i?2M /2+`2�bBM; r?BH2 # Bb M2;�iBp2VX
h?2`27Q`2- +?QQbBM; i?2 TQbBiBp2 b[m�`2 `QQi Bb i?2 p�HB/ +?QB+2X

�b bi�i2/ �i i?2 #2;BMMBM;- i?2 ;Q�H Bb iQ };m`2 Qmi �M �Hi2`@
M�iBp2 `2�/BM; Q7 d#

d" X lbBM; Uk8V- i?Bb +�M #2 bBKTHB}2/ �b
7QHHQrb,

d$
d! =

d$
d#
$

=
d$
1

#

d$ = # UkeV

�b b?QrM BM URjV- i?Bb T`Qp2b i?�i i?2 /2`Bp�iBp2 Q7 sin(!) Bb
BM/22/ cos(!)X
bi�i2/ BM � KQ`2 bi`�B;?i7Q`r�`/ r�v i?�i Bb `2T2�i2/Hv BM +�H+mHmb i?BMF@
BM;X



j3 G2ii2`b �M/ LQi2b

q?�i Bb bB;MB}+�Mi �#Qmi i?Bb T`QQ7 Bb i?�i Bi `2HB2b 2MiB`2Hv QM
i?2 #�bB+běi?2 Svi?�;Q`2�M i?2Q`2K- i?2 mMBi +B`+H2- i?2 /27@
BMBiBQM Q7 bBM2 �M/ +QbBM2- i?2 /2}MBiBQM Q7 i?2 `�/B�M K2�bm`2
Q7 �M �M;H2- i?2 /Bbi�M+2 7Q`KmH�- �M/ i?2 TQr2` `mH2X

� _2bTQMb2 iQ *HmMMǶb �tBQKb Q7
JQ`�HBiv
C _ JBHH2`
.PA, RyXjjyR9fBbbMXke9y@8e8kXjXRXKBHH2`XR

h?Bb �`iB+H2 Qz2`b � #`B27 +`BiB[m2 Q7 *HmMMǶb 7QmM/�iBQM�HBbK
r?B+? ;`QmM/b KQ`�H /2+BbBQM K�FBM; BM r?�i ?2 +�HHb i?2
i?`22 7mM/�K2Mi�H �tBQKb Q7 2tBbi2M+2- +QMb+BQmbM2bb- �M/
B/2MiBiv U*HmMM- kyRNVX Ai b?Qrb ?Qr ?Bb +QKKBiK2Mi iQ
M2Q@SH�iQMBbK- Q` TQbbB#Hv T�Mi?2BbK- +`2�i2b �i H2�bi i?`22
BM+Q?2`2M+B2b r?2`2BM � T`BQ`B Bb � TQbi2`BQ`B- BM/BpB/m�HBiv Bb
�M BHHmbBQM- �M/ Q#D2+iBp2 KQ`�HBiv Bb bm#D2+iBp2X 6Q` *HmMMǶb
KQ`�H T?BHQbQT?v iQ Qz2` T`�+iB+�H p�Hm2- i?2b2 BMi2`M�H +QM@
~B+ib Kmbi #2 `2bQHp2/X

AMi`Q/m+iBQM

AM ?Bb �`iB+H2- �tBQKb Q7 JQ`�HBiv- *HmMM �`;m2b i?�i KQ`�HBiv
Bb �M � T`BQ`B i`mi? i?�i Bb Q#D2+iBp2Hv FMQrM iQ 2p2`v T2`bQMX
>2 #2HB2p2b i?�i i?2 7mM/�K2Mi�H �tBQKb Q7 2tBbi2M+2- +QM@
b+BQmbM2bb �M/ B/2MiBiv K�F2 HB72 Bib2H7 i?2 mHiBK�i2 Q#D2+iBp2
bi�M/�`/ 7Q` 2�+? T2`bQMǶb bm#D2+iBp2 KQ`�H +?QB+2bX h?2`2@
7Q`2- �b � ;2M2`�H `mH2- �Mv KQ`�H +?QB+2 r?B+? #2M2}ib HB72 BM
;2M2`�H Bb � KQ`�H ;QQ/X �Mv KQ`�H +?QB+2 r?B+? ?m`ib HB72 BM
;2M2`�H Bb � KQ`�H 2pBHX *HmMM `2D2+ib b2H}b?M2bb �M/ miBHBi�`B�M@
BbK �b pB�#H2 K2i?Q/b 7Q` +?QQbBM; r?�i Bb ;QQ/X AMbi2�/- ?2
�`;m2b i?�i Qm` BM/BpB/m�H +?QB+2b Kmbi #2 ;mB/2/ #v r?�i ?2
+QMbB/2`b i?2 7Qm` +�`/BM�H pB`im2b /2}M2/ #v ?BbiQ`v, DmbiB+2-
T`m/2M+2- i2KT2`�M+2 �M/ +Qm`�;2X

6BM/BM; �M Q#D2+iBp2 ;`QmM/ 7Q` KQ`�H ;QQ/ Bb � /�mMiBM; i�bF
7Q` �Mv T?BHQbQT?2`X �M/ r?BH2 *HmMMǶb i?`22 �tBQKb �`2 BK@
TQ`i�Mi- i?2 ;`QmM/BM; 7Q` ?Bb Qp2`�`+?BM; KQ`�H T?BHQbQT?v Bb
T`Q#H2K�iB+X 6Q` *HmMM- 2p2`v T2`bQM b?�`2b BM i?2 b�K2 � T`B@
Q`B mMBp2`b�H +QMb+BQmbM2bb r?B+? Bb � MQM`2/m+iBp2 2K2`;2Mi
T`QT2`iv Q7 i?2 #BQHQ;B+�H bi`m+im`2b i?�i /2}M2 ?mK�MM2bbX
A7 Bi Bb i`m2 i?�i 2tBbi2M+2- +QMb+BQmbM2bb �M/ B/2MiBiv 2tBbi
� T`BQ`B iQ ?mK�M HB72 BM bQK2 7Q`K Q7 M2Q@SH�iQMB+ `2�HKě

Q` TQbbB#Hv BM � T�Mi?2BbiB+ mMBp2`b2ě�i H2�bi i?`22 BMi2`M�H
+QM~B+ib �`Bb2X

*QM~B+i OR, � S`BQ`B Bb � SQbi2`BQ`B

*HmMM H2�Mb ?2�pBHv QM �vM _�M/ 7Q` /2}MBM; ?Bb �tBQKb #mi
/2T�`ib 7`QK _�M/ r?Q i�m;?i i?�i +QMb+BQmbM2bb �M/ KQ`�HBiv
�`2 � TQbi2`BQ`BX h?Bb /BbiBM+iBQM Bb +`BiB+�H 7Q` *HmMM �b ?2
?QT2b iQ bmbi�BM ?Bb +QKKBiK2Mi iQ #Qi? Q#D2+iBp2 KQ`�HBiv
�M/ 7`22 rBHHX >2 r`Bi2b- ǳ�i i?2 2M/ Q7 i?2 /�v- KQ`�HBiv Bb
�#Qmi 7`22 rBHH- +?QB+2b- �M/ /2+BbBQMbX h?2b2 i?BM;b �HH 2tBbi
rBi?BM Qm` +QMb+BQmbM2bb U98VXǴ >2`2 Bb r?2`2 i?2 BM+Q?2`2M+2
}`bi K�MB72bibX "v /2}MBiBQM- � T`BQ`B K2�Mb i?�i KQ`�HBiv
Kmbi 2tBbi BM/2T2M/2Mi Q7 �Mv T2`bQMǶb 2tT2`B2M+2X u2i- *HmMM
�HbQ T`2bmK2b i?�i KQ`�HBiv 2tBbib i?`Qm;? i?2 2t2`+Bb2 Q7 QM2Ƕb
7`22 rBHHX :Bp2M i?2b2 +H�BKb- KQ`�HBiv Kmbi �HbQ #2 � TQbi2`BQ`B
#2+�mb2 Bi /2T2M/b mTQM ?Qr 2�+? BM/BpB/m�H T2`bQM 2t2`+Bb2b
i?2B` 7`22 rBHHX *HmMMǶb T`2bmTTQbBiBQM Q7 � T`BQ`B KQ`�HBiv K�v
#2 T`2b2`p2/ B7 ?2 �bbmK2b 7`22 rBHH Bb �HbQ �M 2tT`2bbBQM Q7 i?2
� T`BQ`B mMBp2`b�H +QMb+BQmbM2bbX >Qr2p2`- i?Bb �bbmKTiBQM
H2�/b iQ � b2+QM/ +QM~B+i 7Q` ?Qr *HmMM /2}M2b B/2MiBivX

*QM~B+i Ok, AM/BpB/m�HBiv Bb �M AHHmbBQM

6Q` *HmMM- +QMb+BQmbM2bb Bb MQi � T`QT2`iv Q7 T2`bQM?QQ/- #mi
�M 2K2`;2Mi T`QT2`iv Q7 i?2 T?vbB+�H `2�HK i?�i 2tBbi2/ #2@
7Q`2 �Mv BM/BpB/m�HX h?�i Bb iQ b�v- �HH ?mK�Mb b?�`2 BM i?2
QM2 MQM`2/m+iBp2 � T`BQ`B mMBp2`b�H +QMb+BQmbM2bbX �i i?2 b�K2
iBK2- *HmMM �`;m2b i?�i i?2 i2`K ǳAǴ Bb �M 2tT`2bbBQM Q7 `�iBQ@
M�H i?Qm;?i r?B+? 2bi�#HBb?2b QM2Ƕb bT2+B}+ B/2MiBiv rBi?BM i?2
mMBp2`b�H +QMb+BQmbM2bb2bX "mi 2p2M B7 ǳAǴ 2bi�#HBb?2b Kv T2`@
bQM�H 2tBbi2M+2- Bi `2K�BMb �M 2tBbi2M+2 QMHv rBi?BM i?2 H�`;2`
�tBQK Q7 2tBbi2M+2X Ai b22Kb iQ 7QHHQr 7`QK *HmMMǶb QrM /2}MB@
iBQMb i?�i i?2 T2`+2TiBQM Q7 BM/BpB/m�HBiv- �M/ #v 2ti2MbBQM 7`22
+?QB+2- Bb QMHv �M BHHmbBQMX h?Bb +`2�i2b �i H2�bi QM2 bB;MB}+�Mi
BMi2`M�H +QM~B+ib 7Q` *HmMMǶb �tBQKbX

*HmMM �`;m2b i?�i i?2 +QMb2[m2M+2b Q7 Qm` /2+BbBQMb �`2 2tT2`B@
2M+2/ QMHv rBi?BM i?2 `2�HK Q7 T2`bQM�H +QMb+BQmbM2bb- r?B+?
MQ Qi?2` T2`bQM +�M Q#b2`p2X AM +QMi`�bi- *HmMM b�vb i?�i
r2 +�M Q#b2`p2 2tBbi2M+2X u2i- 7Q` *HmMM- KQ`�HBiv /Q2b MQi
K�MB72bi BM i?2 �tBQK Q7 2tBbi2M+2X >Qr2p2`- B7 2�+? T2`bQMǶb
+QMb+BQmbM2bb Bb � b?�`2/ � T`BQ`B `2�HBiv- ?Qr Bb Bi #2vQM/ Kv
TQr2`b Q7 Q#b2`p�iBQM\ A7 A +�M ?�p2 �r�`2M2bb Q7 Kv QrM
+QMb+BQmbM2bb- �M/ i?�i +QMb+BQmbM2bb Bb iB2/ iQ i?2 mMBp2`b�H-
i?2M #v /2}MBiBQM A Kmbi �HbQ ?�p2 �++2bb iQ mM/2`bi�M/BM;
i?2 +QMb+BQmbM2bb Q7 Qi?2`b #2+�mb2 i?2v iQQ �`2 iB2/ iQ i?2
b�K2 mMBp2`b�H �tBQKX 1p2M KQ`2- B7 +QMb+BQmbM2bb Bb �M 2K2`@
;2Mi T`QT2`iv Q7 2tBbi2M+2- ?Qr /Q2b Bi `2K�BM BM/2T2M/2Mi Q7
2tBbi2M+2 �b Bi `2H�i2b iQ KQ`�HBiv\ h?Bb BM+Q?2`2M+2 H2�/b iQ

https://dx.doi.org/10.33014/issn.2640-5652.3.1.miller.1

	About This Journal
	The Purpose of the Journal
	Paper Submission Policies
	Other Journal Content

	From the Editors
	Sam S Rakover and Baruch CahlonSam S Rakover and Baruch CahlonWhen is Explanation Transitive? A Methodological Note
	Introduction
	Explanatory-transitivity
	Discussion
	Acknowledgments

	Robert J. Marks IIRobert J. Marks IITiling Efflorescence of Expanding Kernels in a Fixed Periodic Array: Generalizing the Flower-Of-Life
	Introduction
	Other Expanding Kernels 
	Properties of Expanding Kernels of Varying Periodicity
	 Analysis
	Conclusions
	Appendices

	Letters and Notes
	Eric HollowayDeciding a Bitstring of 1s is Non-Random is Impossible in General
	Jonathan BartlettProving the Derivative of sin(x) Using the Pythagorean Theorem and the Unit Circle
	J R MillerA Response to Clunn's Axioms of Morality
	Jorge FernandezIs Information Content a Single, Static Quantity?

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