## Names of Time Symmetries

For general discussion about Conway's Game of Life.
Extrementhusiast
Posts: 1796
Joined: June 16th, 2009, 11:24 pm
Location: USA

### Names of Time Symmetries

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.
I Like My Heisenburps! (and others)

Freywa
Posts: 589
Joined: June 23rd, 2011, 3:20 am
Location: Singapore
Contact:

### Re: Names of Time Symmetries

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.
Princess of Science, Parcly Taxel

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

### Re: Names of Time Symmetries

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.

Extrementhusiast
Posts: 1796
Joined: June 16th, 2009, 11:24 pm
Location: USA

### Re: Names of Time Symmetries

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.)
I Like My Heisenburps! (and others)

77topaz
Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Names of Time Symmetries

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.

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

### Re: Names of Time Symmetries

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.

77topaz
Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Names of Time Symmetries

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?

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

### Re: Names of Time Symmetries

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.

77topaz
Posts: 1345
Joined: January 12th, 2018, 9:19 pm

### Re: Names of Time Symmetries

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.

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

### Re: Names of Time Symmetries

What is Time symmetries ?
When a pattern repeats itself (possibly reflected, rotated and translated) after some number of generations.

Moosey
Posts: 2479
Joined: January 27th, 2019, 5:54 pm
Location: A house, or perhaps the OCA board.
Contact:

### Re: Names of Time Symmetries

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.
I am a prolific creator of many rather pathetic googological functions

My CA rules can be found here

Also, the tree game
Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"