## 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

**Schooner**

spaceship

period 2

**Flare**

oscillator

period 2/1

volatility 1

**Blinker**

oscillator

period 2/1

volatility 0.8

**Chomp**

oscillator

period 2

volatility 1

**Long chomp**

oscillator

period 2

volatility 1

**Throw**

oscillator

period 4

volatility 0.67

**Long throw**

oscillator

period 4

volatility 0.79

**Bacon**

oscillator

period 4

volatility 0.33

**Long bacon**

oscillator

period 4

volatility 0.57

**Quartet**

oscillator

period 3

volatility 0.37

**Boomerang**

oscillator

period 8

volatility 0.68

**Pinwheel**

oscillator

period 8/2

volatility 1

**Victory**

oscillator

period 20/10

volatility 1

**Shellshock**

oscillator

period 4

volatility 0.59

**Double shellshock**

oscillator

period 4

volatility 0.76

**Insect**

oscillator

period 3

volatility 0.16

**Long insect**

oscillator

period 3

volatility 0.21

**Block**

still life

**Carrier**

still life

**Arcade**

still life

**Shell**

still life

**Chip**

induction coil

**Table**

induction coil

**Long table**

induction coil

**Horn**

induction coil

**Long horn**

induction coil

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.