oQHmK2 j- Abbm2 R

G2ii2`b �M/ LQi2b

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LQM@_�M/QK Bb AKTQbbB#H2 BM :2M2`�H
1`B+ >QHHQr�v
.PA, RyXjjyR9fBbbMXke9y@8e8kXjXRX?QHHQr�vXR

h?2Q`2K R U*?�BiBMǶb BM+QKTH2i2M2bb i?2Q`2KVX 6Q` 2p2`v
�tBQK�iB+ T`QQ7 bvbi2K ! i?�i +�M #2 2M+Q/2/ BM �i H2�bi "!

#Bib- i?2`2 Bb � +QMbi�Mi # bm+? i?�i i?2 EQHKQ;Q`Qp +QKTH2tBiv
Q7 � #Bibi`BM; $ +�MMQi #2 T`Qp2M iQ #2 H�`;2` i?�M #X

q2 +�M T?`�b2 i?Bb BM i2`Kb Q7 �M �H;Q`Bi?K %! ($, &, ') i?�i
+?2+Fb 2p2`v TQbbB#H2 T`QQ7 BM ! X q?2M �TTHB2/ iQ � #Bibi`BM; $
�M/ � p�Hm2 &- %! ($, &, ') QmiTmib � T`QQ7 BM ! i?�i ( ($) > &-
Q` 7�BHb �7i2` ' bi2Tb �M/ QmiTmib � MmHH- {}-

!# : "$, & # #, ', %! ($, &, ') = {}.

S`QQ7X h?Bb Bb �TT�`2Mi #v �bbmKBM; i?2 QTTQbBi2-

"&, !$, ', %! ($, &, ') ! {}. URV

q2 +QMbi`m+i �M �H;Q`Bi?K )"
! r?B+?- 7Q` � ;Bp2M &- T2`7Q`Kb �

#`2�/i? }`bi b2�`+? i?`Qm;? �HH #Bibi`BM;b $ �M/ bi2T �KQmMib
' rBi? %! ($, &, ') mMiBH Bi }M/b � T`QQ7 i?�i ( ($) > &X )"

!
i?2M QmiTmib $X "v �bbmKTiBQM BM 1[m�iBQM R- )"

! rBHH �Hr�vb
?�HiX

h?2 EQHKQ;Q`Qp +QKTH2tBiv Q7 )"
! Bb ( ()"

! ) $ ( (%! )+log &+
*- r?2`2 * Bb � +QMbi�Mi 7Q` i?2 Qp2`?2�/ BM )"

! X h?2 p�Hm2 Q7
* Bb BM/2T2M/2Mi Q7 &- bQ rBHH MQi p�`v �b & +?�M;2bX

q2 i?2M b2i & = # bm+? i?�i ( ()#
! ) $ ( (%! ) + log & + * < #X

aBM+2 )#
! rBHH ?�Hi- Bi rBHH QmiTmi � $ bm+? i?�i ( ($) > #X

>Qr2p2`- ( ()#
! ) < #- `2bmHiBM; BM � +QMi`�/B+iBQMX h?mb- 7Q`

�HH & # # Qm` �bbmKTiBQM BM 1[m�iBQM R Bb 7�Hb2X

h?2`27Q`2- h?2Q`2K R Bb i`m2X !

h?2Q`2K k U*�M T`Qp2 MQM@`�M/QKM2bbVX q2 /2}M2 � 7mM+@
iBQM +! ($, &, ') r?B+? QmiTmib T`QQ7b BM ! Q7 i?2 7Q`K ( ($) <
,($)- r?B+? �`2 T`QQ7b Q7 MQM@`�M/QKM2bbX Ai Bb T�`�K2i2`Bx2/
BM i?2 b�K2 r�v �b %! ($, &, ')X

6Q` �HH H2M;i?b # Q7 #Bibi`BM;b $- i?2`2 Bb � ! i?�i +�M T`Qp2 �i
H2�bi QM2 #Bibi`BM; Q7 H2M;i? ,($) = # Bb MQM@`�M/QK-

!$"#, &, ',+! ($, &, ') ! {} % ,($) = #.

S`QQ7X q2 #2;BM #v �bbmKBM; i?2 +QMi`�`v

!#&"# > #&, &, ',+! ($, &, ') = {} % ,($) = #. UkV

h?2 7QHHQrBM; �tBQK�iB+ bvbi2K !1$ Bb � 7�HbB}+�iBQM Q7 1[m�@
iBQM kX

�tBQKb Q7 !1$ �`2,

RX ( ({1}20) < 20X

kX A7 ( ($) < ,($)- i?2M ( ($1) < ,($1)X

jX A7 ( ($) < ,($)- i?2M $ Bb MQM@`�M/QKX

6Q` �Mv #Bibi`BM; Q7 Rb $1$ r?2`2 ,($1$) # 20- !1$ +�M T`Qp2
i?2 #Bibi`BM; Bb MQM@`�M/QKX Ai /Q2b i?Bb #v BM+`2K2Mi�HHv
#mBH/BM; � #Bibi`BM; Q7 Rb mMiBH i?2 BMTmi Bb K�i+?2/X h?2
�tBQKb Q7 !1$ �`2 i`m2 �M/ �HH T`QQ7b �`2 #v BM/m+iBQM- bQ �HH
T`QQ7b �`2 i`m2X 1[m�iBQM k Bb +QMi`�/B+i2/X !

h?2Q`2K j U*�MMQi ;2M2`�HHv T`Qp2 MQM@`�M/QKM2bbVX h?2`2
Bb MQ �tBQK�iB+ bvbi2K i?�i +�M /2+B/2 i?2 MQM@`�M/QKM2bb Q7
2p2`v MQM@`�M/QK #Bibi`BM;X

S`QQ7X q?BH2 � /Qp2i�BHBM; �H;Q`Bi?K +�M QmiTmi T`QQ7b Q7
MQM@`�M/QKM2bb 7Q` 2p2`v MQM@`�M/QK #Bibi`BM;- i?2`2 Bb MQ
/2+BbBQM T`Q+2/m`2 i?�i +�M /2+B/2 r?2i?2` i?2 /Qp2i�BHBM; �H@
;Q`Bi?K rBHH ?�HiX A7 i?2`2 r2`2- i?2M i?Bb /2+BbBQM T`Q+2/m`2
+�M 2MmK2`�i2 �HH `�M/QK #Bibi`BM;b- +QMi`�/B+iBM; h?2Q`2K
RX !

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


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/ 0 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 + 02 = 1 UNV
02 = 1 ( /2 URyV
/2 = 1 ( 02 URRV

URkV

aBM+2 i?2 ?vTQi2Mmb2 Bb 1- sin(-) = 0 �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

	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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