Kinetic symmetry

A kinetic symmetry (contrast static symmetry) describes the spatial and temporal symmetries of a periodic pattern such as a still life, oscillator or spaceship. 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.

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.

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

List

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	Example	Image
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 siamese hat
D4_+2	+e		Two lines of orthogonal mirror symmetry One lines 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	(1.5.4)
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	(1.5.5)
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	Image
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	(2.2.4)	pending
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	Griddle and block	pending
n/		C1	D2_x	2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2)	Muttering moat 1	pending
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	(2.2.7)	pending
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	pending
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	(2.2.0)	pending
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	pending
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	pending
-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	pending
-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	pending
-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	(2.5.0)	pending
-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	(2.5.2)	pending
/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	(2.4.2)	pending
/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	pending
.c+c	.f+f	C2_1	D4_+1	2	Pattern has C2_1 symmetry Appears rotated 180 degrees during (period/2) Rotation is centered on the center of a cell	(2.4.0)	pending
.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	pending
.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	(2.5.5)	pending
.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	(2.6.1)	pending
.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	(2.3.3)	pending
.kxk	.vxv	C2_4	D4_x4	2	Pattern has C2_4 symmetry	Clock	pending
.krk	.vrv	C2_4	C4_4	2	Pattern has C2_4 symmetry		pending
+c*c	+f*f	D4_+1	D8_1	2	Pattern has D4_+1 symmetry	Blinker	pending
+k*k	+v*v	D4_+4	D8_4	2	Pattern has D4_+4symmetry		pending
xc*c	xf*f	D4_x1	D8_1	2	Pattern has D4_x1 symmetry	Washing machine	pending
xk*k	xv*v	D4_x4	D8_4	2	Pattern has D4_x4 symmetry		pending
rc*c	rf*f	C4_1	D8_1	2	Pattern has C4_1 symmetry		pending
rk*k	rv*v	C4_4	D8_4	2	Pattern has C4_4 symmetry	Quad	pending

On a hexagonal grid

There does not seem to have been an attempt at describing all possible oscillator and spaceship time symmetries on {6,3} and {3,6} so far.