Computing fractal dimensions for additive cellular automata

Willson, Stephen J.

doi:10.1016/0167-2789(87)90074-1

Cited by 47 publications

(31 citation statements)

References 7 publications

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“…Seeds such as (1a) and (1b), which produce self-similar structure reminiscent of the additive case [24,25], are known as replicators [2,5,6,10]. The replicating elements are identical in the former case, but slightly different in the latter (see Section 2 for details).…”

Section: Examplementioning

confidence: 99%

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The One-Dimensional Exactly 1 Cellular Automaton: Replication, Periodicity, and Chaos from Finite Seeds

Gravner

Griffeath²

2010

J Stat Phys

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In the Exactly 1 cellular automaton (also known as Rule 22), every site of the onedimensional lattice is either in state 0 or in state 1, and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three-site neighborhood at the current time. We analyze this rule started from finite seeds, i.e., those initial configurations that have only finitely many 1's. Three qualitatively different types of evolution are observed: replication, periodicity, and chaos. We focus on rigorous results, assisted by algorithmic searches, for the first two behaviors. In particular, we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds. Some empirical observations about chaotic seeds are also presented.

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Section: Examplementioning

confidence: 99%

“…2 as t . Another instance is Rule 150, in which the addition is over the same three-site neighborhood that Exactly 1 uses, and many others have been studied [8,24,25,27].…”

Section: Introductionmentioning

confidence: 99%

The One-Dimensional Exactly 1 Cellular Automaton: Replication, Periodicity, and Chaos from Finite Seeds

Gravner

Griffeath²

2010

J Stat Phys

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“…There is a direct connection between the space-time output patterns of additive cellular automata and self-similar fractal patterns [82,83,84,85,86,87,88]. The simplest examples are elementary rules 102 and 90.…”

Section: Additive Cellular Automata and Fractalsmentioning

confidence: 98%

“…[52], which deals extensively with the use of additive cellular automata in computing applications and VLSI chip design, is of particular value. [82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101] provide a good background in the relation between cellular automata and fractal patterns. [81,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,…”

Section: Bibliographymentioning

confidence: 99%

Additive Cellular Automata

Voorhees¹

2012

Computational Complexity

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“…Additive rules commute with addition modulo 2 and seem to be the only class of CA rules amenable to general mathematical theory (e.g., [7,2,[8][9][10][11]), thus they play a role analogous to linear dynamical systems, and are for this reason sometimes called linear.…”

Section: Introductionmentioning

confidence: 99%