Consider cellular automaton rules for a two-dimensional universe with two states, where a cell’s new state can depend on its previous state and the states of the cells in its Moore neighborhood. Such a rule can be modeled as a function that takes as input a 3 by 3 matrix of bits, and outputs a bit.

Such a rule can be represented easily using a string of 512 bits representing the output for each of the $ 2^{3 \times 3}$ input states. Since this representation is bijective, a rule can also be randomly generated by sampling 512 bits.

However, it is sometimes preferable for reasons of aesthetics and comprehensibility that the output of the rule be the same when the input matrix is rotated or flipped horizontally or vertically. The rule for Conway’s Game of Life is an example of a function satisfying this restriction.

Given a function $ r : 2^9 \to 2$ which does not necessarily obey this restriction, we can obtain a function that does by mapping the input to the lexicographically smallest matrix obtainable by reflecting or rotating it before passing to $ r$ . However, this provides little insight into the questions I am interested in:

- How many functions obey this restriction?
- How can they be compactly representing using a bit string?
- How can they be efficiently randomly generated?