ConwayLife.com

Posted: **July 19th, 2015, 5:18 pm**

Now that we've established names for space symmetries (via Catagolue), I think it's about time to name the time symmetries as well, along with combinations of both types. Here is a list of the 43 possibilities that I made, sorted by space symmetry and operation through time:

C1, identity
C1, half turn around center
C1, half turn around edge
C1, half turn around corner
C1, quarter turn around center
C1, quarter turn around corner
C1, orthogonal reflection through center
C1, orthogonal reflection through corner
C1, diagonal reflection
C2_1, identity
C2_1, quarter turn
C2_1, orthogonal reflection
C2_1, diagonal reflection
C2_2, identity
C2_2, orthogonal reflection
C2_4, identity
C2_4, quarter turn
C2_4, orthogonal reflection
C2_4, diagonal reflection
C4_1, identity
C4_1, orthogonal/diagonal reflection
C4_4, identity
C4_4, orthogonal/diagonal reflection
D2_+1, identity
D2_+1, half turn/other orthogonal reflection around/through center
D2_+1, half turn/other orthogonal reflection around/through edge
D2_+2, identity
D2_+2, half turn/other orthogonal reflection around/through edge
D2_+2, half turn/other orthogonal reflection around/through corner
D2_x, identity
D2_x, half turn/other diagonal reflection around/through center
D2_x, half turn/other diagonal reflection around/through corner
D4_+1, identity
D4_+1, quarter turn/diagonal reflection
D4_+2, identity
D4_+4, identity
D4_+4, quarter turn/diagonal reflection
D4_x1, identity
D4_x1, quarter turn/orthogonal reflection
D4_x4, identity
D4_x4, quarter turn/orthogonal reflection
D8_1, identity
D8_4, identity

If it makes it any easier, one can also think of these as edges between the 16 symmetry groups:

C1 => C1
C1 => C2_1
C1 => C2_2
C1 => C2_4
C1 => C4_1
C1 => C4_4
C1 => D2_+1
C1 => D2_+2
C1 => D2_x
C2_1 => C2_1
C2_1 => C4_1
C2_1 => D4_+1
C2_1 => D4_x1
C2_2 => C2_2
C2_2 => D4_+2
C2_4 => C2_4
C2_4 => C4_4
C2_4 => D4_+1
C2_4 => D4_x1
C4_1 => C4_1
C4_1 => D8_1
C4_4 => C4_4
C4_4 => D8_4
D2_+1 => D2_+1
D2_+1 => D4_+1
D2_+1 => D4_+2
D2_+2 => D2_+2
D2_+2 => D4_+2
D2_+2 => D4_+4
D2_x => D2_x
D2_x => D4_x1
D2_x => D4_x4
D4_+1 => D4_+1
D4_+1 => D8_1
D4_+2 => D4_+2
D4_+4 => D4_+4
D4_+4 => D8_4
D4_x1 => D4_x1
D4_x1 => D8_1
D4_x4 => D4_x4
D4_x4 => D8_4
D8_1 => D8_1
D8_4 => D8_4

Any ideas? I was thinking about a notation like D2_+1_h2, which gives the initial symmetry and has the the operation used tacked on at the end.

Posted: **July 19th, 2015, 7:55 pm**

The glider, since it's asymmetric yet undergoes a glide reflection, would be in my notation C1S (for slope or diagonal reflection). The symmetrical champagne glass would be D2_+1H (horizontal) and so on, appending a letter to denote time symmetry.

I have so far F for identity (since I looks too much like 1 sometimes), H and S for ortho/diag reflection parallel to the axis of reflection, J and T for the same perpendicular to axis (next available letters), P for a half turn (point) and Q for a quarter turn. Any finer details can be specified with more letters; the C1 symmetries listed would then be C1F, C1P, C1Q, C1H, C1S, C1J and C1T. There is no need to specify where the centre of rotation or reflection is; a displacement would suffice but that's out of the context.

Posted: **February 11th, 2019, 7:29 am**

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.

Posted: **February 11th, 2019, 4:58 pm**

Macbi wrote: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.)

I myself would probably call such a concept a comma, as after a comma, the cells are allowed to differ in relative position, if only because of the possibility of a different orientation, whereas after a period, the cells must be in the same relative position and the same orientation. (Or something like that; the idea was much clearer in my head.)

Posted: **February 11th, 2019, 7:45 pm**

Some Life-like CA (though not specifically B3/S23) also have a larger set of spatial symmetry groups with the addition of e.g. skew symmetries or D8_2.

Posted: **February 12th, 2019, 4:06 am**

77topaz wrote:Some Life-like CA (though not specifically B3/S23) also have a larger set of spatial symmetry groups with the addition of e.g. skew symmetries or D8_2.

For example in B/S the only valid pattern is the vacuum, and so the cells can be permuted in literally any way.

Posted: **February 12th, 2019, 8:22 pm**

Well, that's true, though the rules I was referring were a lot less trivial/more interesting than B/S. But, that does raise the question - what would be the group of all those possible permutations of a two-dimensional grid?

Posted: **February 13th, 2019, 5:34 am**

My point was that some cellular automata can have a lot of symmetries not present in Life, so the project of classifying all symmetries that a cellular automaton can have is probably too complex. It woild be easier just to start with Life and then work out from there into the isotopic range-1 rules.

But, that does raise the question - what would be the group of all those possible permutations of a two-dimensional grid?

The group of all permutations of a set is called the symmetric group on that set.

Posted: **February 13th, 2019, 4:50 pm**

That makes sense. But, since you already listed all the symmetry groups for CGoL, my comments was simply a suggestion about extension/generalisation that would fall under "then work out from there into the isotopic range-1 rules.", since skew symmetries and D8_2 are observed in certain isotropic range-1 rules.

Posted: **February 20th, 2019, 7:54 am**

vedika31 wrote:I want to start with the basic question.
What is Time symmetries ?

When a pattern repeats itself (possibly reflected, rotated and translated) after some number of generations.

Posted: **February 20th, 2019, 9:22 am**

I feel that temporal glide-symmetric would be the name for some of these symmetries;
Champagne glass (with a house) would be D2+1(TGS11), because it reflects along a line of cells every 11 generations.

Certain RROs would be C4_4(C4TRS45), if their fourfold quarter period versions had C4_4 symmetry, and they would make a 90-degree (π/2 radian) turns every 45 generations. The fist part, C4_4 symmetry, is indicated by the C4_4 before the parentheses, and the second part would be indicated by the contents of the parentheses (C4 meaning 90° turns, TRS meaning temporal rotational symmetry, and 45 indicating the time it takes). Of course these would almost certainly be in OCA but this was just to give an example.

EDIT, on Feb 19, 2020-- hey, wow, I've been here for over a year-- this post is almost 365 days old

Posted: **December 14th, 2021, 9:07 am**

Here are examples of each of the 43 square grid symmetry types, using a slightly modified version of Dean Hickerson's naming system (I use f instead of c for "face", and v instead of k for "vertex"), picked from DRH-oscillators.

Complementary blinker as well as 2.4.4 appear to have been miscategorized and are therefore moved to a more fitting category - these use the Poison theme. I've also added examples for spaceships, oscillators with only spatial symmetry, as well as extra still life examples I myself deduced - these use the Generations theme.

Space Only

n

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