## 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 (2`r` + 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 2

^{18}= 262144, and I have investigated 2^{16}= 65536 of these individually (although*extremely*superficially).The number of binary cellular automata in standard Moore neighbourhood is 2

^{29}≈ 1.34⋅10^{154}.The number of

`N`-state cellular automata in standard Moore neighbourhood is`N`^{N9}. For`N`= 3, this yields approximately 1.51⋅10^{9391}; 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 2^{2(2r + 1)2}; for`r`= 2 and 3, this is equal to 1.13⋅10^{15}and 3.17⋅10^{29}, respectively.The number of binary range-

`r`automata is 2^{2(2r + 1)2}. For`r`= 2, this is about 3.31⋅10^{10100890}. Imagine what happens for`r`= 3 and beyond.The number of

`N`-state range-`r`cellular automata is`N`^{N(2r + 1)2}. Already`N`= 3 and`r`= 2 yields 1.32⋅10^{404259404447}.