CellularAutomaton

t+1 fixed-width generations of an elementary 1-D cellular automaton, from init.

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CellularAutomaton(rule, init, t)t+1 generations of Wolfram's elementary (k = 2 colors, radius 1) rule number rule, from initial condition init

Typed signatures

Type	Package	Applies
`(integer<0..255>, integer \| list<integer> \| tuple<list<integer>, integer>, integer<0..>) -> list<list<integer>>`	`combinatorics`	always

Domain: Collections

Details

init is an explicit {list, background} (or {list}, background 0) -- a bare integer like 1 is not valid init (Wolfram raises initn).
Every generation has the same width: init's non-background cells span [minPos, maxPos], and the displayed window is [minPos - t, maxPos + t] -- the widest region t steps could possibly reach.
A cell of init that already equals background doesn't widen that span -- {{1, 0, 0}, 0} behaves like a single seed at position 0, not like a 3-wide active region.
Only the elementary (k = 2, radius 1) rule form is answered; totalistic and multi-color rule specs are left unevaluated.

Examples

Basic #

The classic Rule 30 triangle, from a single seed cell, at its final fixed width

\mathrm{CellularAutomaton}(30, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, 0\bigr\rbrack, 3)

Out

\bigl\lbrack\bigl\lbrack0, 0, 0, 1, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 1, 1, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 1, 0, 0, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 1, 1, 1, 1\bigr\rbrack\bigr\rbrack

Rule 30 from a single cell, spreading as a triangle

\mathrm{CellularAutomaton}(30, \bigl\lbrack0, 0, 0, 1, 0, 0, 0\bigr\rbrack, 2)

Out

\bigl\lbrack\bigl\lbrack0, 0, 0, 1, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 1, 1, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 1, 0, 0, 1, 0\bigr\rbrack\bigr\rbrack

Scope #

A 3-cell init with only one cell differing from background behaves like a single-seed automaton

\mathrm{CellularAutomaton}(90, \bigl\lbrack\bigl\lbrack1, 0, 0\bigr\rbrack, 0\bigr\rbrack, 2)

Out

\bigl\lbrack\bigl\lbrack0, 0, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 0, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 0, 0, 1\bigr\rbrack\bigr\rbrack

Seed with background $0$ expands the lattice each step

\mathrm{CellularAutomaton}(30, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, 0\bigr\rbrack, 2)

Out

\bigl\lbrack\bigl\lbrack0, 0, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 0, 1\bigr\rbrack\bigr\rbrack

A cyclic boundary wraps the growth around the edge

\mathrm{CellularAutomaton}(30, \bigl\lbrack1, 0, 0, 0, 0, 0\bigr\rbrack, 2)

Out

\bigl\lbrack\bigl\lbrack1, 0, 0, 0, 0, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 0, 0, 1\bigr\rbrack, \bigl\lbrack0, 0, 1, 0, 1, 1\bigr\rbrack\bigr\rbrack

\mathrm{Grid}(\mathrm{CellularAutomaton}(73, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, 0\bigr\rbrack, 4))

Out

0

0

0

0

1

0

0

0

0

1

1

1

0

0

0

1

1

1

0

0

1

0

1

0

1

0

0

1

0

0

0

0

0

0

0

1

1

0

1

1

1

1

1

0

1

The same evolution rendered as a grid

\mathrm{Grid}(\mathrm{CellularAutomaton}(30, \bigl\lbrack1, 0, 0, 0, 0, 0\bigr\rbrack, 2))

Out

1

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

Properties #

Rule 30 on a small cyclic row

\mathrm{CellularAutomaton}(30, \bigl\lbrack1, 0, 1, 1\bigr\rbrack, 3)

Out

\bigl\lbrack\bigl\lbrack1, 0, 1, 1\bigr\rbrack, \bigl\lbrack0, 0, 1, 0\bigr\rbrack, \bigl\lbrack0, 1, 1, 1\bigr\rbrack, \bigl\lbrack0, 1, 0, 0\bigr\rbrack\bigr\rbrack

Possible issues #

Rule 90 gives the Sierpinski pattern

\mathrm{CellularAutomaton}(90, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, 0\bigr\rbrack, 2)

Out

\bigl\lbrack\bigl\lbrack0, 0, 1, 0, 0\bigr\rbrack, \bigl\lbrack0, 1, 0, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 0, 0, 1\bigr\rbrack\bigr\rbrack

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