## Numbers of cellular automata

All cellular automata I have considered this far have used the standard Moore neighbourhood, a 3×3 square with the current cell in the middle and eight neighbours. Yesterday I extended my program to support larger neighbourhoods. Now it supports neighbourhoods of arbitrary range. If the range is r, the neighbourhood consists of all cells you can reach in no more than r steps from the current cell, each step being horizontal, vertical, or diagonal. Hence, the range-r Moore neighbourhood is a square consisting of (2r + 1)2 cells, with the current cell in the middle. r = 1 yields the standard eight-cell Moore neighbourhood.

The number of possible extended-range cellular automata is nearly ungraspable. Let us do some simple math.

• The number of life-like cellular automata in standard Moore neighbourhood is only 218 = 262144, and I have investigated 216 = 65536 of these individually (although extremely superficially).

• The number of binary cellular automata in standard Moore neighbourhood is 229 ≈ 1.34⋅10154.

• The number of N-state cellular automata in standard Moore neighbourhood is NN9. For N = 3, this yields approximately 1.51⋅109391; you can imagine what happens for N = 24...

• The number of life-like cellular automata with range r (that is, the binary automata that cares only about the cell itself and the number of living neighbours) is 22(2r + 1)2; for r = 2 and 3, this is equal to 1.13⋅1015 and 3.17⋅1029, respectively.

• The number of binary range-r automata is 22(2r + 1)2. For r = 2, this is about 3.31⋅1010100890. Imagine what happens for r = 3 and beyond.

• The number of N-state range-r cellular automata is NN(2r + 1)2. Already N = 3 and r = 2 yields 1.32⋅10404259404447.

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