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'''Replicator''' is a [[cellular automaton#Well-known Life-like cellular automata|Life-like cellular automaton]] where a cell survives or is born if there are an odd number of neighbors. It is one of two Life-like Fredkin replicator rules. Under this ruleset, every pattern self-replicates; furthermore, every pattern will eventually produce an arbitrary number of copies of itself, all arbitrarily far away from each other.
'''Replicator''' is a [[cellular automaton#Well-known Life-like cellular automata|Life-like cellular automaton]] where a cell survives or is born if there are an odd number of neighbors. It is one of two Life-like Fredkin replicator rules. Under this ruleset, every pattern self-replicates; furthermore, every pattern will eventually produce an arbitrary number of copies of itself, all arbitrarily far away from each other.


==Replication Property==
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The replication property follows from a property of Fredkin replicator rules, in which patterns can be modelled as an infinite grid whose entries are elements of the [http://en.wikipedia.org/wiki/Cyclic_group cyclic group] Z<sub>n</sub>, where n is the number of states. In this case, n=2, 0 is the off state, and 1 is the on state. The rule can be expressed equivalently as assigning a new value to a cell by summing all neighboring cells. Since Z<sub>n</sub> is an abelian group, addition is commutative and associative; hence applying the rule to a sum (XOR) of two patterns is the same as summing the two patterns after the rule is applied to each one.
 
Thus, to find the nth generation of a pattern, it suffices to XOR together the nth generation of each of the single cells which compose the pattern. A single cell is a replicator. More specifically, an on-cell at (0,0) at time 0 will produce, at time 2<sup>n</sup>, 8 on-cells at all positions (b,c) where b and c are any of -2<sup>n</sup>, 0, or 2<sup>n</sup>, and b and c are not both 0 (this can be proven using induction on n). When n is large enough, the 8 cells are arbitrarily far away, and thus, for a pattern, the XOR sum of the (2<sup>n</sup>)th generation of each of its cells forms the pattern's (2<sup>n</sup>)th generation, 8 copies of the original. Repeating this process produces an arbitrary number of copies, all at arbitrary distance.


==Replicator 2==
==Replicator 2==
{{Rule|name=Fredkin|s=02468|b=1357|char=Explosive}}
{{Rule|name=Fredkin|s=02468|b=1357|char=Explosive}}
'''Replicator 2''', also known as '''Fredkin''', is a related totalistic Fredkin replicator rule, where a cell survives or is born if the number of neighbors, including itself, is odd. It has the totalistic rulestring 13579. Like Replicator, every pattern self-replicates.
'''Replicator 2''', also known as '''Fredkin''', is a related totalistic Fredkin replicator rule, where a cell survives or is born if the number of neighbors, including itself, is odd. It has the totalistic rulestring 13579. Like Replicator, every pattern self-replicates.

Revision as of 02:44, 17 August 2010

Replicator
x=64, y = 64, rule = B1357/S1357 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
LifeViewer-generated pseudorandom soup
View static image
Rulestring 1357/1357
B1357/S1357
Character Explosive

Replicator is a Life-like cellular automaton where a cell survives or is born if there are an odd number of neighbors. It is one of two Life-like Fredkin replicator rules. Under this ruleset, every pattern self-replicates; furthermore, every pattern will eventually produce an arbitrary number of copies of itself, all arbitrarily far away from each other.

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Replicator 2

Fredkin
x=64, y = 64, rule = B1357/S02468 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
LifeViewer-generated pseudorandom soup
Rulestring 02468/1357
B1357/S02468
Character Explosive

Replicator 2, also known as Fredkin, is a related totalistic Fredkin replicator rule, where a cell survives or is born if the number of neighbors, including itself, is odd. It has the totalistic rulestring 13579. Like Replicator, every pattern self-replicates.

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