Kinetic symmetry

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A kinetic symmetry (contrast static symmetry) describes the spatial and temporal symmetries of still lifes, oscillators and spaceships. It combines a pattern's spatial (rotational and reflectional) symmetries from the more general static symmetry with symmetrical transformations of said pattern arising from its evolution.[1]

On a square grid

There are a total of 43 different kinetic symmetries possible on a usual square grid, comprised of the 16 static symmetries (D8_2 is excluded) with 27 possible time transformations.

Oscillators have a much wider range of possible kinetic symmetries than still lifes. It is very easy to see that the 27 time transformations cannot apply to still lifes by definition, as they require the pattern to have distinct phases which can be compared to each other, and therefore for the pattern to evolve, which still lifes do not. As such, oscillators can only have 16 of the possible 43 kinetic symmetry types, which therefore corresponds with the 16 different static symmetry types. The symmetry class is the symmetry class of the oscillator in a single generation followed by the symmetry class of the union of the generation and its congruent successors.[2]

Both still lifes and oscillators can exhibit a wider range than spaceships can, at least as far as isotropic rules are concerned. Many higher kinetic symmetries, notably those involving rotation or with reflection happening on more than one axis, would forbid the pattern from having a nonzero displacement, as the symmetry would either force it to move in two directly opposing directions or redirect it back to its starting point. Many spaceships can have glide symmetry, which oscillators cannot have due to having no overall displacement, however glide symmetry very closely resembles certain mirror symmetries which oscillators do exhibit.

The ratio of a pattern's mod to its period, for rules on a square grid, can only be 1, 2 or 4.

Kinetic symmetry naming system

Dean Hickerson invented a compact naming system for kinetic symmetries.

For still lifes, as well as oscillators and spaceships which have identical mods to periods, an initial symbol stand for a kind of transformation, and a symbol following it refers to the type of region where said transformation is centered.

Oscillators and spaceships of unequal period and mod will follow this string with another string detailing how the pattern is transformed after cycling through its mod.

Hickerson's naming scheme initially used "c" for the center of a cell and "k" for the vertex of a cell - these have been revised to "f" for the face of a cell and "v" for the vertex of a cell to clarify things and reduce any confusion that may arise due to homophones.

Symbols

Hickerson name Wiki name Meaning
n No symmetry
- One line of orthogonal mirror symmetry
/ One line of diagonal mirror symmetry
+ Two lines of orthogonal mirror symmetry
x Two lines of diagonal mirror symmetry
* Two lines each of orthogonal and diagonal mirror symmetry
r 90-degree rotational symmetry
. 180-degree rotational symmetry
c f Transformation is centered on the center of a cell
e Transformation is centered on the edge of a cell
k v Transformation is centered on the vertex of a cell

Still lifes

These are equivalent to static symmetries (excluding D8_2). The corresponding static symmetries are detailed in the table for each type.

Equivalent static symmetry Hickerson name Wiki name Description Diagram Example
C1 n No symmetry

Eater 1
D2_+1 -c -f One line of orthogonal mirror symmetry
Line passes through cell centers and edges

Hat
D2_+2 -e One line of orthogonal mirror symmetry
Line passes through cell edges and vertices

Cap and table
D2_x / One line of diagonal mirror symmetry

Boat
C2_1 .c .f 180-degree rotation
Rotation is centered on the center of a cell

Long snake
C2_2 .e 180-degree rotation
Rotation is centered on the edge of a cell

Aircraft carrier
C2_4 .k .v 180-degree rotation
Rotation is centered on the vertex of a cell

Snake
D4_+1 +c +f Two lines of orthogonal mirror symmetry
Both lines pass through cell centers and edges

Hat cis-siamese hat
D4_+2 +e Two lines of orthogonal mirror symmetry
One line passes through cell centers and edges
One line passes through cell edges and vertices

Beehive
D4_+4 +k +v Two lines of orthogonal mirror symmetry
Both lines pass through cell edges and vertices

unnamed
D4_x1 xc xf Two lines of diagonal mirror symmetry
Lines meet at the center of a cell

Ship
D4_x4 xk xv Two lines of diagonal mirror symmetry
Lines meet at the vertex of a cell

Barge
C4_1 rc rf 90-degree rotation
Rotation is centered on the center of a cell

Spiral
C4_4 rk rv 90-degree rotation
Rotation is centered on the vertex of a cell

unnamed
D8_1 *c *f Two lines each of orthogonal and diagonal mirror symmetry
Orthogonal lines pass through cell centers and edges

Tub
D8_4 *k *v Two lines each of orthogonal and diagonal mirror symmetry
Orthogonal lines pass through cell edges and vertices

Block

Oscillators

For oscillators which have a mod identical to their period, refer to the still lifes table above.

"Composite symmetry" refers to the resulting symmetry of the pattern created from each of the oscillator's "identical" phases:

• for patterns with a mod half their period, the union of the pattern's initial state and the state it appears in at half its period
• for patterns with a mod a quarter their period, the union of the pattern's initial phase, generation (period/4), generation (period/2) and generation (3period/4)
Hickerson name Wiki name Static symmetry Composite symmetry period/mod Description Example
n-c n-f C1 D2_+1 2 Pattern is asymmetric
Appears flipped across an orthogonal line during (period/2)
Line passes through cell centers and edges

xp2_466t186z6961696
n-e C1 D2_+2 2 Pattern is asymmetric
Appears flipped across an orthogonal line during (period/2)
Line passes through cell edges and vertices

Block on griddle
n/ C1 D2_x 2 Pattern is asymmetric
Appears flipped across a diagonal line during (period/2)

Muttering moat 1
n.c n.f C1 C2_1 2 Pattern is asymmetric
Appears rotated 180 degrees during (period/2)
Rotation is centered on the center of a cell

xp2_0ml1ik8z1259a6
n.e C1 C2_2 2 Pattern is asymmetric
Appears rotated 180 degrees during (period/2)
Rotation is centered on the edge of a cell

Laputa
n.k n.v C1 C2_4 2 Pattern is asymmetric
Appears rotated 180 degrees during (period/2)
Rotation is centered on the vertex of a cell

2.3.3
nrc nrf C1 C4_1 4 Pattern is asymmetric
Appears rotated 90 degrees every (period/4)
Rotation is centered on the center of a cell

Sixty-nine
nrk nrv C1 C4_4 4 Pattern is asymmetric
Appears rotated 90 degrees every (period/4)
Rotation is centered on the vertex of a cell

Dinner table
-c+c -f+f D2_+1 D4_+1 2 Pattern has D2_+1 symmetry
Appears flipped across a perpendicular orthogonal line during (period/2)
Line passes through cell centers and edges

Piston
-c+e -f+e D2_+1 D4_+2 2 Pattern has D2_+1 symmetry
Appears flipped across a perpendicular orthogonal line during (period/2)
Line passes through cell edges and vertices

by flops
-e+e -e+e D2_+2 D4_+2 2 Pattern has D2_+2 symmetry
Appears flipped across a perpendicular orthogonal line during (period/2)
Line passes through cell centers and edges

xp2_0giligz344k743zw121
-e+k -e+v D2_+2 D4_+4 2 Pattern has D2_+2 symmetry
Appears flipped across a perpendicular orthogonal line during (period/2)
Line passes through cell edges and vertices

unnamed
/xc /xf D2_x D4_x1 2 Pattern has D2_x symmetry
Appears flipped across a perpendicular diagonal line during (period/2)
Lines meet at the center of a cell

xp2_066oo4g53zc8502046
/xk /xv D2_x D4_x4 2 Pattern has D2_x symmetry
Appears flipped across a perpendicular diagonal line during (period/2)
Lines meet at the vertex of a cell

Tripole
.c+c .f+f C2_1 D4_+1 2 Pattern has C2_1 symmetry
Appears flipped across one of two perpendicular orthogonal lines during (period/2)
Both lines pass through cell centers and edges

xp2_g8j1cdj8gz01cb38c1
.cxc .fxf C2_1 D4_x1 2 Pattern has C2_1 symmetry
Appears flipped across one of two diagonal lines during (period/2)
Lines meet at the center of a cell

Bipole
.crc .frf C2_1 C4_1 2 Pattern has C2_1 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Rotation is centered on the center of a cell

xp2_2aa08060922zgg50p050lkgzw1
.e+e .e+e C2_2 D4_+2 2 Pattern has C2_2 symmetry
Appears flipped across one of two perpendicular orthogonal lines during (period/2)
Line may pass through either cell centers and edges, or cell edges and vertices

unnamed
.k+k .v+v C2_4 D4_+4 2 Pattern has C2_4 symmetry
Appears flipped across one of two perpendicular orthogonal lines during (period/2)
Both lines pass through cell edges and vertices

xp2_0e0j0944z44i0p0e
.kxk .vxv C2_4 D4_x4 2 Pattern has C2_4 symmetry
Appears flipped across one of two diagonal lines during (period/2)
Lines meet at the vertex of a cell

Clock
.krk .vrv C2_4 C4_4 2 Pattern has C2_4 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Rotation is centered on the vertex of a cell

unnamed
+c*c +f*f D4_+1 D8_1 2 Pattern has D4_+1 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Could also be interpreted as diagonal flipping on one of two lines
Rotation is centered on/lines intersect at the center of a cell

+k*k +v*v D4_+4 D8_4 2 Pattern has D4_+4 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Could also be interpreted as diagonal flipping on one of two lines
Rotation is centered on/lines intersect at the vertex of a cell

unnamed
xc*c xf*f D4_x1 D8_1 2 Pattern has D4_x1 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Could also be interpreted as horizontal flipping on one of two lines
Rotation is centered on/lines intersect at the center of a cell

Washing machine
xk*k xv*v D4_x4 D8_4 2 Pattern has D4_x4 symmetry
Appears rotated 90 degrees either clockwise or anticlockwise during (period/2)
Could also be interpreted as horizontal flipping on one of two lines
Rotation is centered on/lines intersect at the vertex of a cell

unnamed
rc*c rf*f C4_1 D8_1 2 Pattern has C4_1 symmetry
Appears flipped across one of two perpendicular orthogonal lines during (period/2)
Both lines pass through cell centers and edges

unnamed
rk*k rv*v C4_4 D8_4 2 Pattern has C4_4 symmetry
Appears flipped across one of two perpendicular orthogonal lines during (period/2)
Both lines pass through cell edges and vertices

The following shows oscillators displaying each of the 43 temporal symmetry types:

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On other grids

See here for a list of all oscillator time symmetries on {6,3} or {3,6}.

References

1. GUYTU6J (December 13, 2021). Re: Help with symmetries (discussion thread) at the ConwayLife.com forums
2. Dean Hickerson's oscillator stamp collection. Retrieved on December 13, 2021.