Cellular automaton: Symmetrica

8 Feb 2022

Symmetrica is an isotropic, non-totalistic cellular automaton. Like the Game of Life, it has two states (alive and dead) and is played on a square grid. The future state of a cell is determined not only by the number of live cells in its Moore neighborhood, but also by the relative configuration of those neighbors.

A dead cell becomes alive in the following conditions:

  • It has two or three live neighbors, and the configuration of the cell along with its neighbors has reflectional symmetry.
  • It has four live neighbors, and the neighbors are either all orthogonal or all diagonal.

The conditions for survival are the same as those in Life: a cell needs two or three live neighbors to survive, regardless of configuration. In the Hensel notation, Symmetrica is written B2-ak3-jnqr4ce/S23. The name "Symmetrica" comes from the word "symmetry" and the letters "ca" for cellular automaton.

Unlike my other automaton Longevity, random patterns in Symmetrica do not explode indefinitely. Also, stable patterns can interact much more easily. Below is a list of notable patterns I have discovered so far. For oscillators where the mod is different from the period, the period and the mod are written with a slash, e.g. 4/2.

Tugboat
spaceship
period 2
tugboat pattern
Schooner
spaceship
period 2
schooner pattern
Flare
oscillator
period 2/1
volatility 1
flare pattern
Blinker
oscillator
period 2/1
volatility 0.8
blinker pattern
Chomp
oscillator
period 2
volatility 1
chomp pattern
Long chomp
oscillator
period 2
volatility 1
long chomp pattern
Throw
oscillator
period 4
volatility 0.67
throw pattern
Long throw
oscillator
period 4
volatility 0.79
long throw pattern
Bacon
oscillator
period 4
volatility 0.33
bacon pattern
Long bacon
oscillator
period 4
volatility 0.57
long bacon pattern
Quartet
oscillator
period 3
volatility 0.37
quartet pattern
Boomerang
oscillator
period 8
volatility 0.68
boomerang pattern
Pinwheel
oscillator
period 8/2
volatility 1
pinwheel pattern
Victory
oscillator
period 20/10
volatility 1
victory pattern
Shellshock
oscillator
period 4
volatility 0.59
shellshock pattern
Double shellshock
oscillator
period 4
volatility 0.76
double shellshock pattern
Insect
oscillator
period 3
volatility 0.16
insect pattern
Long insect
oscillator
period 3
volatility 0.21
long insect pattern
Block
still life
block pattern
Carrier
still life
carrier pattern
Arcade
still life
arcade pattern
Shell
still life
shell pattern
Chip
induction coil
chip pattern
Table
induction coil
table pattern
Long table
induction coil
long table pattern
Horn
induction coil
horn pattern
Long horn
induction coil
long horn pattern

The names "blinker", "block", "carrier", and "table" are taken from the Game of Life. All of the other names are original. Pinwheel is named for its rotating behavior; it is the only oscillator yet discovered whose mod is a quarter of its period. The period 8 oscillator is called "boomerang" because it looks like a blinker being thrown back and forth by two blocks.

All of the patterns with a "long" variant are infinitely extensible. For example, the family of oscillators begginging with insect and long insect can be extended to form very long insect, extra long insect, remarkably long insect, and so on by adding additional components to the chain. However, the schooner, despite looking like an extended tugboat, cannot be extended further. Adding 3 cells behind a schooner will create a tugboat and a bunch of ash after 1149 generations.

If you add a cell diagonally adjecent to a block, the block will absorb the new cell and restabilise after 2 generations. This spark-eating ability allows the block to interact with other patterns in interesting ways. Several of the oscillators above contain blocks (throw, bacon, quartet, and boomerang). You can also create an oscillator by placing a block near a table or by placing two blocks on opposite sides of a long horn.