# 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.

#### 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.

Equivalent static symmetry | Name | Description | Diagram | Example |
---|---|---|---|---|

C1 | n | No symmetry | Eater 1 | |

D2_+1 | -c | 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 | 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 | 180-degree rotation Rotation is centered on the vertex of a cell |
Snake | |

D4_+1 | +c | 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 | Two lines of orthogonal mirror symmetry Both lines pass through cell edges and vertices |
unnamed | |

D4_x1 | xc | Two lines of diagonal mirror symmetry Lines meet at the center of a cell |
Ship | |

D4_x4 | xk | Two lines of diagonal mirror symmetry Lines meet at the vertex of a cell |
Barge | |

C4_1 | rc | 90-degree rotation Rotation is centered on the center of a cell |
Spiral | |

C4_4 | rk | 90-degree rotation Rotation is centered on the vertex of a cell |
unnamed | |

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

D8_4 | *k | 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)

Name | Static symmetry | Composite symmetry | period/mod | Description | Example |
---|---|---|---|---|---|

n-c | 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 | 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 | 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 | C1 | C4_1 | 4 | Pattern is asymmetric Appears rotated 90 degrees every (period/4) Rotation is centered on the center of a cell |
Dinner table |

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

-c+c | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 |
Blinker |

+k*k | 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 | 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 | 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 | 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 | 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 |
Quad |

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

(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:

- n
- -c
- -e
- /
- n-c
- n-e
- n/
- n/e

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.^{[3]} Indeed, oscillators with n/ symmetry translate by a total of 0 cells diagonally, an even number.

### Related terms

#### Flipper

A **flipper** can refer to any oscillator which appears reflected across an orthogonal or diagonal line halways 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; both are interchangeable, as any glide symmetric spaceship is a flipper and any spaceship which is a flipper is glide symmetric.

## On other grids

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

The time symmetries on {4,3,4} do not appear to have been fully enumerated so far, as the static symmetries are still yet to be named and visually classified. Symmetries on {4,3,3,4}, {3,3,4,3} and {3,4,3,3} are yet to be investigated at all.

## See also

## References

- ↑ GUYTU6J (December 13, 2021). Re: Help with symmetries (discussion thread) at the ConwayLife.com forums
- ↑ Dean Hickerson's oscillator stamp collection. Retrieved on December 13, 2021.
- ↑ https://conwaylife.com/forums/viewtopic.php?f=7&t=1898&p=158648#p158648

## External links

- Flipper at the Life Lexicon