Cellular automata are widely used in undergraduate physics courses to educate students in elementary programming and for project work. Cellular automata are coded with simple rules yet provide a rich if well-trodden landscape for exploring aspects of physics such as diffusion and magnetism. Mathematical games, such as the minority game or the prisoner's dilemma, are also amenable to project work with the added dimension of applications in finance, econophysics, and social physics. Conway's classical game of life is both a mathematical game and a cellular automaton. We exploit adaptations of Conway's game of life as an opportunity for undergraduate students to explore new territory within the safe haven of an easy-to-implement cellular automaton. Students may discover new “lifeforms” comprising collections of live, dead, and part-live cells, and explore the escalation of floating-point errors leading to chaos-like behavior, amongst many phenomena not observed in Conway's classical counterpart.

1.
J.
Von Neumann
, “
On the theory of parlor games
,”
Math. Ann.
100
,
295
(
1928
).
https://doi.org/10.1007/BF01448847
Google Scholar
Crossref
Search ADS
 
2.
J.
Von Neumann
 and
O.
Morgenstern
,
Game Theory and Economic Behavior
(
Princeton U. P
.,
Princeton, NJ
,
1944
).
Google Scholar
 
3.
M.
Gardner
, “
The fantastic combinations of John Conway's new solitaire game ‘life’
,”
Sci. Am.
223
,
120
–
123
(
1970
).
https://doi.org/10.1038/scientificamerican1070-120
Google Scholar
Crossref
Search ADS
 
4.
C.
Hauert
 and
G.
Szabó
, “
Game theory and physics
,”
Am. J. Phys.
73
,
405–414
(
2005
).
https://doi.org/10.1119/1.1848514
Google Scholar
Crossref
Search ADS
 
5.
D.
Challet
 and
Y.-C.
Zhang
, “
On the minority game: Analytical and numerical studies
,”
Phys. A
256
,
514–532
(
1998
).
https://doi.org/10.1016/S0378-4371(98)00260-X
Google Scholar
Crossref
Search ADS
 
6.
G.
Faraco
,
P.
Pantano
, and
R.
Servidio
, “
The use of cellular automata in the learning of emergence
,”
Comput. Educ.
47
,
280–297
(
2006
).
https://doi.org/10.1016/j.compedu.2004.10.005
Google Scholar
Crossref
Search ADS
 
7.
A.
Dhatsuwan
 and
M.
Precharattana
, “
BLOCKYLAND: A cellular automata-based game to enhance logical thinking
,”
Simul. Gaming
47
,
445–464
(
2016
).
https://doi.org/10.1177/1046878116643468
Google Scholar
Crossref
Search ADS
 
8.
J.
Ogborn
, “
A future for modelling in science education
,”
J. Comput. Assist. Learn.
6
,
103–112
(
1990
).
https://doi.org/10.1111/j.1365-2729.1990.tb00355.x
Google Scholar
Crossref
Search ADS
 
9.
S.
Wolfram
, “
Universality and complexity in cellular automata
,”
Phys. D: Nonlinear Phenom.
10
,
1
–
35
(
1984
).
https://doi.org/10.1016/0167-2789(84)90245-8
Google Scholar
Crossref
Search ADS
 
10.
Wikipedia,
“
Conway's game of life
,” (
2019
), last accessed on February 21, 2019.
11.
R. P.
Feynman
, “
Simulating physics with computers
, ”
Int. J. Theor. Phys.
21
,
467–488
(
1982
).
https://doi.org/10.1007/BF02650179
Google Scholar
Crossref
Search ADS
 
12.
J.
Eisert
,
M.
Wilkens
, and
M.
Lewenstein
, “
Quantum games and quantum strategies
,”
Phys. Rev. Lett.
83
,
3077–3080
(
1999
).
https://doi.org/10.1103/PhysRevLett.83.3077
Google Scholar
Crossref
Search ADS
 
13.
A. P.
Flitney
 and
D.
Abbott
, “
A semi-quantum version of the game of life
,” e-print arXiv:quant-ph/0208149 (
2002
).
Google Scholar
 
14.
A.
López-Incera
 and
W.
Dür
, “
Entangle me! A game to demonstrate the principles of quantum mechanics
,”
Am. J. Phys.
87
,
95–101
(
2019
).
https://doi.org/10.1119/1.5086275
Google Scholar
Crossref
Search ADS
 
15.
A. P.
Flitney
 and
D.
Abbott
,
Advances in Dynamic Games
(
Springer
,
New York
,
2005
), pp.
667
–
679
.
Google Scholar
Crossref
Search ADS
 
16.
A. P.
Flitney
 and
D.
Abbott
,
Quantum Aspects of Life
, edited by
D.
Abbott
,
P. C.
Davies
, and
A. K.
Pati
 (
World Scientific
,
Singapore
,
2008
), pp.
233
–
250
.
Google Scholar
Crossref
Search ADS
 
17.
G. S.
Engel
,
T. R.
Calhoun
,
E. L.
Read
,
T.-K.
Ahn
,
T.
Mančal
,
Y.-C.
Cheng
,
R. E.
Blankenship
, and
G. R.
Fleming
, “
Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems
,”
Nature
446
,
782–786
(
2007
).
https://doi.org/10.1038/nature05678
Google Scholar
Crossref
Search ADS
PubMed
 
18.
M.
Mohseni
,
P.
Rebentrost
,
S.
Lloyd
, and
A.
Aspuru-Guzik
, “
Environment-assisted quantum walks in photosynthetic energy transfer
,”
J. Chem. Phys.
129
,
11B603
(
2008
).
Google Scholar
Crossref
Search ADS
 
19.
D.
Abbott
,
P. C.
Davies
, and
A. K.
Pati
,
Quantum Aspects of Life
(
World Scientific
,
Singapore
,
2008
).
Google Scholar
Crossref
Search ADS
 
20.
D.
Bleh
,
T.
Calarco
, and
S.
Montangero
, “
Quantum game of life
,”
Europhys. Lett.
97
,
20012
(
2012
).
https://doi.org/10.1209/0295-5075/97/20012
Google Scholar
Crossref
Search ADS
 
21.
F.
Bagarello
,
R. Di
Salvo
,
F.
Gargano
, and
F.
Oliveri
, “
(H, ρ)-induced dynamics and the quantum game of life
,”
Appl. Math. Model.
43
,
15
–
32
(
2017
).
https://doi.org/10.1016/j.apm.2016.10.043
Google Scholar
Crossref
Search ADS
 
22.
P.
Arrighi
 and
J.
Grattage
, “
The quantum game of life
,”
Phys. World
25
,
23–39
(
2012
).
https://doi.org/10.1088/2058-7058/25/06/37
Google Scholar
Crossref
Search ADS
 
23.
T.
Ceccherini-Silberstein
 and
M.
Coornaert
,
Cellular Automata and Groups
(
Springer
,
Berlin
,
2013
).
Google Scholar
 
24.
“
Boost releases on github
,” <https://github.com/boostorg/boost/releases>, accessed on December 5, 2019.
25.
B. M.
Boghosian
,
P. V.
Coveney
, and
H.
Wang
, “
A new pathology in the simulation of chaotic dynamical systems on digital computers
,”
Adv. Theory Simul.
2
,
1900125
(
2019
).
https://doi.org/10.1002/adts.201900125
Google Scholar
Crossref
Search ADS
PubMed
 
26.
The authors in Ref. 25 illustrate error-some calculations using the Bernoulli map x i + 1 = mod ( β x i , 1 ) with β = 2. The repeating sequence 0.6, 0.2, 0.4, 0.8, 0.6… has ⟨ x ⟩ = 0.5⁠, but an eight-line code in matlab yields ⟨ x ⟩ → 0.0 when executed for a large number of iterations.
27.
See supplementary material at http://dx.doi.org/10.1119/10.0000666 for the full videos for each of the evolutions represented in figure 4, plus a spreadsheet that demonstrates the propagation of floating-point representation error that is responsible for the loss of symmetry in qutub evolution.
© 2020 American Association of Physics Teachers.
2020
American Association of Physics Teachers

Supplementary Material

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