Extrementhusiast linked to this post from another thread, and I decided to revive it because I was thinking of making a very similar post.

I think it would be good to classify all possible symmetries that Life patterns might have. Extrementhusiast's list above only describes the possible symmetries of oscillators, but we should also include things like time-translation symmetry for spaceships. This would give us a unified way to talk about oscillators, spaceships, still lives, wicks, waves and agars, since each of these objects can be classified by their symmetry group. The projects of finding new oscillator periods and spaceship speeds can be thought of as attempts to determine exactly which symmetries, out of all possible symmetries, are actually achieved by patterns in Life.

(From a mathematical point of view, we can let G be the set of permutations of ℤ^3 (two dimensions of space and one of time) such that the induced action on functions ℤ^3 ⟶ {Dead, Alive} preserves whether or not a pattern obeys the rules of Life. The task of naming all the symmetries is then the same as classifying all the subgroups of G up to conjugation. The

*elements* of G are easy to describe. Each element of G applies some spatial symmetry (a rotation, reflection, translation, or glide reflection) and a translation in time.)

The beginning of the classification would be to note that every symmetry group has a

*mod*. This is defined to be the least nonzero number of generations moved in time by a transformation in that group, or zero if the symmetry isn't time periodic. (I would have preferred to call this the period, but for some reason in Life the word "period" is used to describe the amount of time needed for a pattern to return to its original configuration with a

*translation* applied, whereas "mod" allows any spatial transformation in place of the translation.) A symmetry can be specified by giving its mod, its spacial symmetry, and a spatial transformation that describes how the pattern changes between generation 0 and the time when it first repeats.

So to complete the classification we need to describe all the possible spatial symmetries, and for each possible combination of spatial symmetry and spatial transformation between generations pick a canonical such transformation (for example, the zebra stripes agar can be thought of as translating 2 cells to the left every generation, but it's more canonical to describe it as staying still every generation).

The finite spatial symmetry groups have already been named, they're C1, C2_1, C2_2, C2_4, C4_1, C4_4, D2_+1, D2_+2, D2_x, D4_+1, D4_+2, D4_+4, D4_x1, D4_x4, D8_1 and D8_4. But if we also want to describe the symmetries of agars then we would need to describe spatial symmetry groups containing translations. Presumably there is already a list of such groups somewhere. They're analogous to the

wallpaper groups, but for ℤ^2 rather than ℝ^2.