Kinetic symmetry

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.

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

Oscillators have a much wider range of possible kinetic symmetries than still lifes. It is easy to see that the 27 time transformations cannot apply to still lifes by definition, as they require the pattern to have distinct phases that can be compared to each other, and therefore the pattern has to evolve over time (which still lifes do not).

Both still lifes and oscillators can exhibit a wider range of symmetries 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.

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 stands 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's symmetry changes if all phases of the pattern are taken into account.

Symbols

Symbol	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	Transformation is centered on the center of a cell
e	Transformation is centered on the edge of a cell
k	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.

Name (Hickerson notation)	Catagolue equivalent	Description	Diagram	Example
n	C1	No symmetry		Eater 1
-c	D2_+1	One line of orthogonal mirror symmetry Line passes through cell centers and edges		Hat
-e	D2_+2	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
.c	C2_1	180-degree rotation Rotation is centered on the center of a cell		Long snake
.e	C2_2	180-degree rotation Rotation is centered on the edge of a cell		Aircraft carrier
.k	C2_4	180-degree rotation Rotation is centered on the vertex of a cell		Snake
+c	D4_+1	Two lines of orthogonal mirror symmetry Both lines pass through cell centers and edges		Hat cis-siamese hat
+e	D4_+2	Two lines of orthogonal mirror symmetry One line passes through cell centers and edges One line passes through cell edges and vertices		Beehive
+k	D4_+4	Two lines of orthogonal mirror symmetry Both lines pass through cell edges and vertices		unnamed
xc	D4_x1	Two lines of diagonal mirror symmetry Lines meet at the center of a cell		Ship
xk	D4_x4	Two lines of diagonal mirror symmetry Lines meet at the vertex of a cell		Barge
rc	C4_1	90-degree rotation Rotation is centered on the center of a cell		Spiral
rk	C4_4	90-degree rotation Rotation is centered on the vertex of a cell		Four snakes around block
*c	D8_1	Two lines each of orthogonal and diagonal mirror symmetry Orthogonal lines pass through cell centers and edges		Tub
*k	D8_4	Two lines each of orthogonal and diagonal mirror symmetry Orthogonal lines pass through cell edges and vertices		Block

Oscillators

There are 43 oscillator symmetry types. In Hickerson's notation, each of those 43 types has a two-part identifier starting with generation 0's symmetry type, and appending the symmetry type of the full set of all phases of the oscillator.

16 out of those oscillator symmetry types consist of two identical parts, as in "xkxk" for example: the identifier is a doubled version of the single-phase symmetry type from the table above ("xk" in this case). An oscillator with any of these 16 symmetry types will have a mod that is equal to its period.

The remaining (27 = 43 - 16) oscillator symmetry types consist of two different parts (as in "-c+e" or "nrk").

Note that "xk" by itself refers to one of the 16 single-generation symmetry types, while "xkxk" refers to one of the 43 oscillator symmetry types. The sets of identifiers for still lifes and for oscillators are completely disjoint. This helpful property makes it easy to tell whether a descriptor refers to a single-generation symmetry type or an oscillator symmetry type.

In the table below, "composite symmetry" refers to the symmetry type of the collection of phases of the oscillator that can be matched up to each other:

• for patterns with a mod equal to 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 equal to a quarter their period, the union of the pattern's initial phase, generation [period/4], generation [period/2] and generation [3period/4]

Name	Symmetry		Transform names	p/m	Description	Example	Gutteroids
	Static	Composite
n-c	n C1	-c D2_+1	FlipX FlipY	2	Pattern is asymmetric Appears flipped across an orthogonal line during (period/2) Line passes through cell centers and edges	unnamed	1 orthogonal p/2
n-e	n C1	-e 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	none
n/	n C1	/ D2_x	Flip⟍ Flip⟋	2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2)	Muttering moat 1	1 diagonal p/2
n.c	n C1	.c C2_1	Rot180	2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on the center of a cell	unnamed	central cell p/2
n.e	n C1	.e C2_2		2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on the edge of a cell	Laputa	none
n.k	n C1	.k C2_4		2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on the vertex of a cell	unnamed	none
nrc	n C1	rc C4_1	Rot90CW Rot90CCW	4	Pattern is asymmetric Appears rotated 90 degrees every (period/4) Rotation is centered on the center of a cell	Dinner table	central cell p/4
nrk	n C1	rk C4_4		4	Pattern is asymmetric Appears rotated 90 degrees every (period/4) Rotation is centered on the vertex of a cell	Sixty-nine	none
-c+c	-c D2_+1	+c D4_+1	FlipXOrRot180 FlipYOrRot180	2	Pattern has D2_+1 symmetry Appears flipped across a perpendicular orthogonal line during (period/2) Line passes through cell centers and edges	Piston	1 orthogonal p/2
-c+e	-c D2_+1	+e 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	none
-e+e	-e D2_+2	+e 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	unnamed	1 orthogonal p/2
-e+k	-e D2_+2	+k 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	none
/xc	/ D2_x	xc D4_x1	Flip⟍OrRot180 Flip⟋OrRot180	2	Pattern has D2_x symmetry Appears flipped across a perpendicular diagonal line during (period/2) Lines meet at the center of a cell	unnamed	1 diagonal p/2
/xk	/ D2_x	xk 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	1 diagonal p/2
.c+c	.c C2_1	+c D4_+1	FlipOrth	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	unnamed	2 orthogonal p/2
.cxc	.c C2_1	xc D4_x1	FlipDiag	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	2 diagonal p/2
.crc	.c C2_1	rc C4_1	Rot90	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	unnamed	central cell p/2
.e+e	.e C2_2	+e D4_+2	FlipOrth	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	1 orthogonal p/2
.k+k	.k C2_4	+k 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	unnamed	none
.kxk	.k C2_4	xk D4_x4	FlipDiag	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	2 diagonal p/2
.krk	.k C2_4	rk C4_4	Rot90	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	none
+c*c	+c D4_+1	*c D8_1	FlipDiagOrRot90	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	Blinker	2 diagonal p/2
+k*k	+k D4_+4	*k 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	2 diagonal p/2
xc*c	xc D4_x1	*c D8_1	FlipOrthOrRot90	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	2 orthogonal p/2
xk*k	xk D4_x4	*k 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	none
rc*c	rc C4_1	*c D8_1	FlipOrthOrDiag	2	Pattern has C4_1 symmetry Appears flipped across one of two perpendicular orthogonal lines or across one of two perpendicular diagonal lines during (period/2) Both orthogonal lines pass through cell centers and edges	unnamed	2 orthogonal 2 diagonal p/2
rk*k	rk C4_4	*k D8_4		2	Pattern has C4_4 symmetry Appears flipped across one of two perpendicular orthogonal lines or across one of two perpendicular diagonal lines during (period/2) Both orthogonal lines pass through cell edges and vertices	Quad	2 diagonal p/2

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

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(click above to open LifeViewer) RLE: here Plaintext: here

row 1: Caterer, Honey thieves, Beluchenko's p40, 22P36, Kok's galaxy, 48P22.1, 1-2-3-4, Short keys, Heart, Gray counter, Pentadecathlon, 101, Merzenich's p11, Jason's p6, Diamond ring, Octagon 2
row 2: Baker's dozen, Merzenich's p64, Achim's p144, Windmill, Achim's p16, 44P10, Tumbler, Heavyweight emulator, 46P10, 68P32.1, A for all, Washing machine, Unicycle
row 3: Trans-queen bee shuttle, 2.2.6, Two pre-L hasslers, Eureka, p30 traffic light hassler, p24 shuttle
row 4: Blocker, Achim's p8
row 5: Champagne glass, p196 pi-heptomino hassler
row 6: Rob's p16
row 7: 30P6.1
row 8: Four eaters hassling lumps of muck
row 9: Twirling T-tetsons 2

Array

composite\static

C1

D2_+1

D2_+2

C2_2

D4_+2

D2_x

C2_1

C2_4

D4_+1

D4_+4

D4_x1

D4_x4

C4_1

C4_4

D8_1

D8_4

C1

D2_+1

D2_+2

C2_2

D4_+2

D2_x

C2_1

C2_4

D4_+1

D4_+4

D4_x1

D4_x4

C4_1

C4_4

D8_1

D8_4

Spaceships

Due to the constraints of isotropy, spaceships in 2D cannot have rotational symmetry any higher than C1. This limits the possible symmetries for a spaceship to eight. The following four symmetries describe spaceships with no time symmetry:

Name	Catagolue equivalent	Description	Diagram	Example
n	C1	No symmetry		25P3H1V0.1
-c	D2_+1	One line of orthogonal mirror symmetry Line passes through cell centers and edges		Dart
-e	D2_+2	One line of orthogonal mirror symmetry Line passes through cell edges and vertices		56P6H1V0
/	D2_x	One line of diagonal mirror symmetry		37P4H1V1.2

The remaining four symmetries describe spaceships which are temporally symmetric:

Name	Static symmetry	Composite symmetry	period/mod	Description	Example
n-c	n (C1)	-c (D2_+1)	2	Pattern is asymmetric Appears flipped across an orthogonal line during (period/2) Line passes through cell centers and edges	Lightweight spaceship
n-e	n (C1)	-e (D2_+2)	2	Pattern is asymmetric Appears flipped across an orthogonal line during (period/2) Line passes through cell edges and vertices	Two HWSS dragging boat
n/	n (C1)	/ (D2_x)	2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2) Line passes through cell centers and vertices	p8 swan tagalong
n/e	n (C1)	/e (no static equivalent)	2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2) Line passes through cell edges	Glider

n/e is a kinetic symmetry exclusive to spaceships in which the diagonal line of reflection passes through the midpoints of the edges of cells, but never the vertices or cell centers. Only spaceships which move an odd number of cells diagonally in a period cycle can have this kinetic symmetry; those which move an even distance will have standard n/ symmetry. Indeed, oscillators with n/ symmetry translate by a total of 0 cells diagonally, an even number.

Related terms

Flipper

'Flipper' can refer to any oscillator that appears reflected across an orthogonal or diagonal line halfway through its period cycle. There are many kinetic symmetries in which an oscillator flips halfway through its period:

• n-c, n-e, n/, -c+c, -c+e, -e+e, -e+k, /xc and /xk: flip across one line
• .c+c, .cxc, .e+e, .k+k and .kxk: flip across one of two perpendicular lines
• rc*c and rk*k: flip across one of four lines
• +c*c, +k*k, xc*c and xk*k: can be considered as either flipping across one of two perpendicular lines or as rotating 90 degrees around their center

Glide symmetry

A spaceship is said to be glide symmetric if it exhibits glide reflection - that is, it becomes its mirror image halfway through its period cycle, alongside moving in its direction of travel. In practice, this means that the spaceship has either the n-c, n-e, n/ or n/e kinetic symmetries.

The term "flipper" is also sometimes used for these spaceships. The two terms are equivalent to some extent, as any glide symmetric spaceship is a flipper and any spaceship that is a flipper is glide symmetric.

On a hexagonal or triangular grid

This table details the gist of the kinetic symmetries on both {6,3} or {3,6}; it is recommended that the specific tiling's page be visited for more specific information on each symmetry.

Still lifes

Name	Catagolue equivalent	Description
n	C1	No symmetry
-	D2_xo	Orthogonal mirror symmetry
/	D2_x	Diagonal mirror symmetry
.c	C2_1	180-degree rotation around a hexagonal cell or triangle vertex
.e	C2_4	180-degree rotation around an edge
rc	C3_1	120-degree rotation around a hexagonal cell or triangle vertex
rk	C3_3	120-degree rotation around a triangular cell or hexagon vertex
+	D4_x1	A line of orthogonal mirror symmetry and a line of diagonal mirror symmetry meeting at a hexagonal cell or triangle vertex
x	D4_x4	A line of orthogonal mirror symmetry and a line of diagonal mirror symmetry meeting at an edge
*-	D6_1o	Three lines of orthogonal mirror symmetry
*/	D6_1	Three lines of diagonal mirror symmetry meeting at a hexagonal cell or triangle vertex
*k	D6_3	Three lines of diagonal mirror symmetry meeting at a triangular cell or hexagon vertex
r	C6	60-degree rotation around a hexagonal cell or triangle vertex
*	D12	Three lines of orthogonal mirror symmetry and three lines of diagonal mirror symmetry

Oscillators

Name	Symmetry		Transform names	p/m	Description	Gutteroids
	Static	Composite				Hexagonal	Triangular
n-	n C1	- D2_xo		2	Pattern is asymmetric Appears flipped across an orthogonal line during (period/2)	1 orthogonal p/2	none
n/	n C1	/ D2_x		2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2)	1 diagonal p/2	1 diagonal p/2
n.c	n C1	.c C2_1		2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on the center of a cell	central cell p/2	none
n.e	n C1	.e C2_4		2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on the edge of a cell	none	none
nrc	n C1	rc C3_1		3	Pattern is asymmetric Appears rotated 120 degrees every (period/3) Rotation is centered on the center of a cell	central cell p/3	none
nrk	n C1	rk C3_3		3	Pattern is asymmetric Appears rotated 120 degrees every (period/3) Rotation is centered on the vertex of a cell	none	central cell p/3
nr	n C1	r C6		6	Pattern is asymmetric Appears rotated 60 degrees every (period/6) Rotation is centered on the center of a cell	central cell p/6	none
-+	- D2_xo	+ D4_x1		2	Pattern has D2_xo symmetry Appears	1 diagonal p/2	1 diagonal p/2
-x	- D2_xo	x D4_x4		2	Pattern has D2_xo symmetry Appears	1 diagonal p/2	1 diagonal p/2
/+	/ D2_x	+ D4_x1		2	Pattern has D2_x symmetry Appears	1 orthogonal p/2	none
/x	/ D2_x	x D4_x4		2	Pattern has D2_x symmetry Appears	1 orthogonal p/2	none
.c+	.c C2_1	+ D4_x1		2	Pattern has C2_1 symmetry Appears	1 orthogonal 1 diagonal p/2	1 diagonal p/2
.cr	.c C2_1	r C6		3	Pattern has C2_1 symmetry Appears	central cell p/3	none
.ex	.e C2_4	x D4_x4		2	Pattern has C2_4 symmetry Appears	1 orthogonal 1 diagonal p/2	1 diagonal p/2
rc*-	rc C3_1	*- D6_1o		2	Pattern has C3_1 symmetry Appears	3 orthogonal p/2	none
rc*/	rc C3_1	*/ D6_1		2	Pattern has C3_1 symmetry Appears	3 diagonal p/2	3 diagonals p/2
rcr	rc C3_1	r C6		2	Pattern has C3_1 symmetry Appears	central cell p/2	none
rk*k	rk C3_3	*k D6_3		2	Pattern has C3_3 symmetry Appears	3 diagonal p/2	3 diagonals p/2
*-*	*- D6_1o	* D12		2	Pattern has D6_1o symmetry Appears	3 diagonal p/2	3 diagonals p/2
*/*	*/ D6_1	* D12		2	Pattern has D6_1 symmetry Appears	1 orthogonal p/2	none
r*	r C6	* D12		2	Pattern has C6 symmetry Appears flipped across one of three orthogonal lines during (period/2) Equivalently could be considered as flipping across their three diagonal perpendiculates	3 orthogonal 3 diagonal p/2	3 diagonals p/2

Spaceships

you should put a table here that would be useful i think

On other grids

Euclidean

The time symmetries on {4,3,4} are listed here.

Symmetries on {4,3,3,4}, {3,3,4,3} and {3,4,3,3} have been enumerated and given quaternion-based names, but have not been assigned human-readable names so far, likely due to the sheer quantity of even the static symmetries.

Symmetries on {4,3,3,3,4} and higher are yet to be investigated at all.

Rules and symmetries on dense Euclidean tilings such as {5/2,10} and {8/3,8} have not been investigated so far due to their chaotic nature.

Symmetry types per grid

Schläfli symbol	Static	Kinetic						Spaceship				Totals
		p/2	p/3	p/4	p/6	p/8	p/12	p	p/2	p/3	p/4	Osc	Ship
Exceptional regular tilings
{3,6}	14	17	3	-	1	-	-	3	4	-	-	35	7
{6,3}
{3,3,4,3}	501	1569	77	143	70	4	1	?	?	?	?	2365	?
{3,4,3,3}
n-hypercubic honeycombs
{∞}	3	2	-	-	-	-	-	1	-	-	-	5	1
{4,4}	16	25	-	2	-	-	-	4	4	-	-	43	8
{4,3,4}	92	220	7	14	4	-	-	22	39	2	2	337	65
{4,3,3,4}	686	2535	38	235	46	2	-	?	?	?	?	3542	?
{4,3,3,3,4}	unknown
{4,3,3,3,3,4}	unknown

In general, an n-dimensional cubic honeycomb will have kinetic symmetries where the mod is the period divided by n as well as symmetries where it is divided by 2n.

Hyperbolic

Cellular automata have been investigated on compact hyperbolic tilings; paracompact and noncompact tilings are generally not considered due to the existence of ideal and ultra-ideal elements. There are an infinite number of possible compact tilings in 2D hyperbolic space.

For a tiling {p,q} where both p and q are prime, there are eight possible symmetries.

See also

• LifeWiki:Project OmniOsci

External links

• Flipper at the Life Lexicon