# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# Growth Phenomena in Cellular Automata

• Janko Gravner
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_266-5

## Definition of the Subject

In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., (Meakin 1998) and its more than 1,300 references). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation (Toffoli and Margolus 1997; Wójtowicz 2001), have become particularly popular in the last 30 years in many fields, such as physics (Chopard and Droz 1998; Toffoli and Margolus 1997; Vichniac 1984), biology (Deutsch and Dormann 2005), chemistry (Chopard and Droz 1998; Kier et al. 2005), social sciences (Bäck et al. 1996), and artificial life (Lindgren and Nordahl 1994). In contrast to voluminous empirical literature on CA in general and their growth properties in particular, precise mathematical results are rather scarce. A general CA theory is out of the question, since a Turing machine can be embedded in a CA, so that examples as “simple” as elementary one-dimensional CA (Cook 2005...

## Keywords

Cellular Automaton Cellular Automaton Cellular Automaton Modeling Asymptotic Density Moore Neighborhood
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

## Primary Literature

1. Adamatzky A, Martínez GJ, Mora JCST (2006) Phenomenology of reaction-diffusion binary-state cellular automata. Int J Bifurc Chaos Appl Sci Eng 16:2985–3005
2. Adler J (1991) Bootstrap percolation. Phys A 171:453–4170
3. Adler J, Staufer D, Aharony A (1989) Comparison of bootstrap percolation models. J Phys A: Math Gen 22:L279–L301
4. Aizenman M, Lebowitz J (1988) Metastability effects in bootstrap percolation. J Phys A: Math Gen 21:3801–3813
5. Allouche J-P, Shallit J (2003) Automatic sequences: theory, applications, generalizations. Cambridge University Press, Cambridge
6. Andjel E, Mountford TS, Schonmann RH (1995) Equivalence of decay rates for bootstrap percolation like cellular automata. Ann Inst H Poincaré 31:13–25
7. Berlekamp ER, Conway JH, Guy RK (2004) Winning ways for your mathematical plays, vol 4, 2nd edn. Peters, Natick
8. Bohman T (1999) Discrete threshold growth dynamics are omnivorous for box neighborhoods. Trans Am Math Soc 351:947–983
9. Bohman T, Gravner J (1999) Random threshold growth dynamics. Random Struct Algorithms 15:93–111
10. Bramson M, Neuhauser C (1994) Survival of one-dimensional cellular automata under random perturbations. Ann Probab 22:244–263
11. Brummitt CD, Delventhal H, Retzlaff M (2008) Packard snowflakes on the von Neumann neighborhood. J Cell Autom 3:57–80
12. Bäck T, Dörnemann H, Hammel U, Frankhauser P (1996) Modeling urban growth by cellular automata. In: Lecture notes in computer science. Proceedings of the 4th international conference on parallel problem solving from nature, vol 1141. Springer, Berlin, pp 636–645Google Scholar
13. Cerf R, Cirillo ENM (1999) Finite size scaling in three-dimensional bootstrap percolation. Ann Probab 27:1837–1850
14. Cerf R, Manzo F (2002) The threshold regime of finite volume bootstrap percolation. Stoch Process Appl 101:69–82
15. Chopard B, Droz M (1998) Cellular automata modeling of physical systems. Cambridge University Press, Cambridge
16. Cobham A (1972) Uniform tag sequences. Math Syst Theory 6:164–192
17. Cook M (2005) Universality in elementary cellular automata. Complex Syst 15:1–40Google Scholar
18. Deutsch A, Dormann S (2005) Cellular automata modeling of biological pattern formation. Birkhäuser, BostonGoogle Scholar
19. Durrett R, Steif JE (1991) Some rigorous results for the Greenberg-Hastings model. J Theor Probab 4:669–690
20. Durrett R, Steif JE (1993) Fixation results for threshold voter systems. Ann Probab 21:232–247
21. Eppstein D (2002) Searching for spaceships. In: More games of no chance (Berkeley, CA, 2000). Cambridge University Press, Cambridge, pp 351–360Google Scholar
22. Evans KM (2001) Larger than life: digital creatures in a family of two-dimensional cellular automata. In: Cori R, Mazoyer J, Morvan M, Mosseri R (eds) Discrete mathematics and theoretical computer science, vol AA. pp 177–192Google Scholar
23. Evans KM (2003) Replicators and larger than life examples. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 119–159Google Scholar
24. Fisch R, Gravner J, Griffeath D (1991) Threshold-range scaling for the excitable cellular automata. Stat Comput 1:23–39
25. Fisch R, Gravner J, Griffeath D (1993) Metastability in the Greenberg-Hastings model. Ann Appl Probab 3:935–967
26. Gardner M (1976) Mathematical games. Sci Am 133:124–128
27. Goles E, Martinez S (1990) Neural and automata networks. Kluwer, Dordrecht
28. Gotts NM (2003) Self-organized construction in sparse random arrays of Conway’s game of life. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 1–53Google Scholar
29. Gravner J, Griffeath D (1993) Threshold growth dynamics. Trans Am Math Soc 340:837–870
30. Gravner J, Griffeath D (1996) First passage times for the threshold growth dynamics on. Ann Probab 24:1752–1778
31. Gravner J, Griffeath D (1997a) Multitype threshold voter model and convergence to Poisson-Voronoi tessellation. Ann Appl Probab 7:615–647
32. Gravner J, Griffeath D (1997b) Nucleation parameters in discrete threshold growth dynamics. Exp Math 6:207–220
33. Gravner J, Griffeath D (1998) Cellular automaton growth on: theorems, examples and problems. Adv Appl Math 21:241–304
34. Gravner J, Griffeath D (1999a) Reverse shapes in first-passage percolation and related growth models. In: Bramson M, Durrett R (eds) Perplexing problems in probability. Festschrift in honor of Harry Kesten. Birkhäuser, Boston, pp 121–142
35. Gravner J, Griffeath D (1999b) Scaling laws for a class of critical cellular automaton growth rules. In: Révész P, Tóth B (eds) Random walks. János Bolyai Mathematical Society, Budapest, pp 167–186Google Scholar
36. Gravner J, Griffeath D (2006a) Modeling snow crystal growth. I. Rigorous results for Packard’s digit snowflakes. Exp Math 15:421–444
37. Gravner J, Griffeath D (2006b) Random growth models with polygonal shapes. Ann Probab 34:181–218
38. Gravner J, Holroyd AE (2008) Slow convergence in bootstrap percolation. Ann Appl Probab 18:909–928
39. Gravner J, Mastronarde N Shapes in deterministic and random growth models (in preparation)Google Scholar
40. Gravner J, McDonald E (1997) Bootstrap percolation in a polluted environment. J Stat Phys 87:915–927
41. Gravner J, Tracy C, Widom H (2002) A growth model in a random environment. Ann Probab 30:1340–1368
42. Greenberg J, Hastings S (1978) Spatial patterns for discrete models of diffusion in excitable media. SIAM J Appl Math 4:515–523
43. Griffeath D (1994) Self-organization of random cellular automata: four snapshots. In: Grimmett G (ed) Probability and phase transition. Kluwer, Dordrecht, pp 49–67
44. Griffeath D, Hickerson D (2003) A two-dimensional cellular automaton with irrational density. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, Oxford, pp 119–159Google Scholar
45. Griffeath D, Moore C (1996) Life without death is P-complete. Complex Syst 10:437–447
46. Holroyd AE (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab Theory Relat Fields 125:195–224
47. Holroyd AE (2006) The metastability threshold for modified bootstrap percolation in d dimensions. Electron J Probab 11:418–433
48. Holroyd AE, Liggett TM, Romik D (2004) Integrals, partitions, and cellular automata. Trans Am Math Soc 356:3349–3368
49. Jen E (1991) Exact solvability and quasiperiodicity of one-dimensional cellular automata. Nonlinearity 4:251–276
50. Kier LB, Seybold PG, Cheng C-K (2005) Cellular automata modeling of chemical systems. Springer, DordrechtGoogle Scholar
51. Lindgren K, Nordahl MG (1994) Evolutionary dynamics of spatial games. Phys D 75:292–309
52. Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge
53. Packard NH (1984) Lattice models for solidification and aggregation. Institute for advanced study preprint. Reprinted in: Wolfram S (ed) (1986) Theory and application of cellular automata. World Scientific, Singapore, pp 305–310Google Scholar
54. Packard NH, Wolfram S (1985) Two-dimensional cellular automata. J Stat Phys 38:901–946
55. Pimpinelli A, Villain J (1999) Physics of crystal growth. Cambridge University Press, CambridgeGoogle Scholar
56. Schonmann RH (1992) On the behavior of some cellular automata related to bootstrap percolation. Ann Probab 20:174–193
57. Schonmann RH (1990) Finite size scaling behavior of a biased majority rule cellular automaton. Phys A 167:619–627
58. Song M (2005) Geometric evolutions driven by threshold dynamics. Interfaces Free Bound 7:303–318
59. Toffoli T, Margolus N (1997) Cellular automata machines. MIT Press, CambridgeGoogle Scholar
60. van Enter ACD (1987) Proof of Straley’s argument for bootstrap percolation. J Stat Phys 48:943–945
61. van Enter ACD, Hulshof T (2007) Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections. J Stat Phys 128:1383–1389
62. Vichniac GY (1984) Simulating physics with cellular automata. Phys D 10:96–116
63. Vichniac GY (1986) Cellular automata models of disorder and organization. In: Bienenstock E, Fogelman-Soulie F, Weisbuch G (eds) Disordered systems and biological organization. Springer, Berlin, pp 1–20Google Scholar
64. Wiener N, Rosenblueth A (1946) The math foundation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiol Mex 16:205–265
65. Willson SJ (1978) On convergence of configurations. Discret Math 23:279–300
66. Willson SJ (1984) Cellular automata can generate fractals. Discret Appl Math 8:91–99
67. Willson SJ (1987) Computing fractal dimensions for additive cellular automata. Phys D 24:190–206
68. Wójtowicz M (2001) Mirek’s celebration: a 1D and 2D cellular automata explorer, version 4.20. http://www.mirwoj.opus.chelm.pl/ca/

## Books and Reviews

1. Adamatzky A (1995) Identification of cellular automata. Taylor & Francis, LondonGoogle Scholar
2. Allouche J-P, Courbage M, Kung J, Skordev G (2001) Cellular automata. In: Encyclopedia of physical science and technology, vol 2, 3rd edn. Academic Press, San Diego, pp 555–567Google Scholar
3. Allouche J-P, Courbage M, Skordev G (2001b) Notes on cellular automata. Cubo, Matemática Educ 3:213–244
4. Durrett R (1988) Lecture notes on particle systems and percolation. Wadsworth & Brooks/Cole, Pacific Grove
5. Durrett R (1999) Stochastic spatial models. SIAM Rev 41:677–718
6. Gravner J (2003) Growth phenomena in cellular automata. In: Griffeath D, Moore C (eds) New constructions in cellular automata. Oxford University Press, New York, pp 161–181Google Scholar
7. Holroyd AE (2007) Astonishing cellular automata. Bull Centre Rech Math 10:10–13Google Scholar
8. Ilachinsky A (2001) Cellular automata: a discrete universe. World Scientific, SingaporeGoogle Scholar
9. Liggett TM (1985) Interacting particle systems. Springer, New York
10. Liggett TM (1999) Stochastic interacting systems: contact, voter and exclusion processes. Springer, New York
11. Rothman DH, Zaleski S (1997) Lattice-gas cellular automata. Cambridge University Press, Cambridge
12. Toom A (1995) Cellular automata with errors: problems for students of probability. In: Snell JL (ed) Topics in contemporary probability and its applications. CRC Press, Boca Raton, pp 117–157Google Scholar
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9078948497772217, "perplexity": 27760.40485975701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376826800.31/warc/CC-MAIN-20181215061532-20181215083532-00244.warc.gz"}