Symmetrica is a cellular automaton with two states. (The name "Symmetrica" comes from the word "symmetry" combined with "CA" for cellular automaton.) As in the Game of Life, the states are referred to as "alive" and "dead." Each generation, every cell changes state in accordance with the following rules:

1. If the cell is dead and has 2 or 3 live neighbors in a symmetrical configuration, it becomes alive.
2. If the cell is dead and has 4 live neighbors which are all orthogonal or all diagonal, it becomes alive.
3. If the cell is alive and has 2 or 3 live neighbors, it stays alive.
4. Otherwise, the cell will be dead in the next generation.

In Hensel notation, these rules are abbreviated as "B2-ak3-jnqr4ce/S23."

In February 2022, I wrote a (now outdated) article with a list of patterns I had discovered the previous summer. Since then, I have investigated Symmetrica further with the help of apgsearch and Catagolue. The list of known patterns is now much more extensive, and includes puffers, rakes, and large spaceships.

This website is a catalogue of named objects and their properties. You can click on any image to run it in LifeViewer.

Arcade

• Period-48 puffer
• Debris: 6 blocks in an arch formation
• Front section is identical to that of the symmetrical byzantine ship

Asymmetrical byzantine ship (ABS)

• Period-32 spaceship generated by two byzantines
• Produces beehive sparks

• The ABS can eat blocks, carriers, steeples, and other small patterns.

• Two puffers made of ABSs are shown below: a nova puffer (left) and a feast puffer (right).

• See also: symmetrical byzantine ship

Beacon

• Period-4 oscillator consisting of two blocks
• Most common period-4 oscillator
• 5th most common oscillator
• Named after a similar pattern from the Game of Life

Beehive spark

• 12-cell spark
• Extremely common

• The pattern known as beehive in the Game of Life creates a beehive spark after 5 generations. Other ancestors of the beehive spark include the J-tetromino and the U-pentomino.

Beekeeper

• Period-23 oscillator
• Produces beehive sparks
• The three most common variants are shown

Big insect

• Period-6 oscillator stabilised by two shells
• Most common period-6 oscillator

Blinker

• Period-2 oscillator
• 3rd most common object
• 2nd most common oscillator
• Named after an identical pattern from the Game of Life

• Here are some interesting blinker-tugboat collisions:

Block

• 4-cell still life
• Most common object
• Smallest still life

• The block can "eat" a cell placed at one of its corners. This behavior appears in many oscillators, including beacon, dinner, and quartet.

Boomerang

• Period-8 oscillator generated by two blocks and a blinker
• Two variants: cis boomerang (left) and trans boomerang (right)

Boron

• Period-5 oscillator consisting of three blocks and a steeple

• See also: quartet

Bowtie

• Period-2 phoenix
• One phase consists of two flares, and the other phase consists of two flares and a dot
• 4th most common period-2 oscillator
• 10th most common oscillator

Byzantine

• Period-192 puffer
• Debris: 12 blocks, 5 flares, 5 blinkers, 1 carrier, 1 shell
• Front section repeats at a period of 16
• Powered by a tango

• Although the byzantine is fairly dirty on its own, multiple byzantines can combine to produce a variety of clean puffers. Six examples are shown below. (From left to right: flare puffer, shell puffer, block puffer, block on table puffer, beacon puffer, victory puffer).

• The byzantine can combine with a tugboat to make a period-16 block puffer.

• See also: symmetrical byzantine ship, asymmetrical byzantine ship, varangian guard

Carrier

• 6-cell still life consisting of two chips
• 4th most common still life
• 2nd smallest still life
• Named after an identical pattern from the Game of Life.

Cat

• Period-4 spaceship
• One phase contains a 32-cell polyomino

• The tugboats on either side can be replaced with schooners or tugboat chains.

Chicken wire superstring

• Period-2 superstring
• Extends chicken wire agar

Chip

• 3-cell induction coil
• Evolves into a block if not stabilised
• Found in steeple and carrier
• The majority of distinct still lifes listed on Catagolue contain at least one chip

Cyclops

• Period-4 tagalong for two tugboats or schooners
• Six variants are shown

Dinner

• Period-4 oscillator consisting of a block and a table
• 3rd most common period-4 oscillator
• 8th most common oscillator

Dot fuse

• Lightspeed diagonal fuse
• Stabilised by a chip

• Two variants of the dot fuse are shown below: blinker puller (left) and wildfire (right).

Double boomerang

• Period-12 oscillator similar to the boomerang
• Consists of 4 blocks hassling a blinker
• Most common period-12 oscillator

Double flare

• Period-2 phoenix consisting of two flares
• 3rd most common period-2 oscillator
• 9th most common oscillator

Epsilon fuse

• Period-18 block-producing fuse
• Speed: (10, 5)c/18
• Stabilised by a chip
• Named for the "ε" shape that appears at generation 15

Feast

• 32-cell still life
• Consists of two long^3 tables and four chips

• Feast can be generated by two beehive sparks.

Flare

• Period-2 phoenix
• 2nd most common pattern
• Most common oscillator
• Smallest pattern by population

Flint and steel

• Period-4 oscillator consisting of a flare and block
• 2nd most common period-4 oscillator
• 7th most common oscillator

Glidermaker

• Period-8 spaceship
• Named after the sparks it produces, which resemble the glider from the Game of Life

Goalpost

• Period-4 spaceship
• Powered by a tango
• Back section has a period of 2, and resembles two tugboats fused together

Half-squid

• Period-4 spaceship

• See also: squid

Hexadecimal 1

• Period-16 tagalong for two tugboats
• Only period-16 spaceship to have occurred naturally

Hexadecimal 2

• Period-16 tagalong for two tugboats

Hyperion

• Period-26 blinker puffer
• 2nd most common puffer

• Here is a nova-producing puffer made of two hyperions and two tugboats:

• See also: titan

Insect

• Period-3 oscillator generated by two shells

Marathon

• Period-262 oscillator
• Produces beehive sparks
• Highest-period oscillator known (excluding LCMs)

• The marathon is one of the only patterns known to be capable of reflecting tugboats.

• It can be generated by four blocks and two tangos, or by six blocks a beehive:

Mason

• Period-19 block puffer
• Speed: 5c/19
• Most common puffer

• See also: slowpoke

Moonshine

• Period-18 spaceship
• Produces tub sparks

Moose

• Period-9 oscillator
• Consists of two blocks hassling a shell predecessor
• Most common period-9 oscillator
• Three variants are shown

Nova

• Period-10 statorless oscillator
• Made of two beehive sparks colliding
• Most common period-10 oscillator

Oscillators

• The ten most common oscillators are:
1. Flare
2. Blinker
3. Pinwheel
4. Victory
5. Beacon
6. Quartet
7. Flint and steel
8. Dinner
9. Double flare
10. Bowtie
• 25 is the lowest period for which no oscillators are known.

Owl

• Period-4 tagalong for two tugboats or schooners
• Nine variants are shown

Parade

• Period-16 blinker puffer
• 3rd most common puffer

• The parade can combine with a tugboat to create a spaceship, a block puffer, or a period-160 puffer that outputs pinwheels. The last of these is the basis for schooner rake 1.

Pinwheel

• Period-8 statorless oscillator
• Only known oscillator whose mod is a quarter of its period
• Most common period-8 oscillator
• 3rd most common oscillator

• Here are some interesting pinwheel-tugboat collisions:

Plow

• Period-140 puffer
• Debris: 12 blocks, 4 flares
• Front section repeats at a period of 20

• Two plows can combine to make a puffer that produces shells and pinwheels.

Quartet

• Period-3 oscillator consisting of four blocks
• Most common period-3 oscillator
• 6th most common oscillator

• A 5-cell long diagonal line results in a quartet.

Rake 1

• Period-80 tugboat rake
• Contains 2 sea urchins and 3 SBSs

Rake 2

• Period-64 tugboat rake
• Contains 2 byzantines, 1 tugboat, 1 schooner, and 4 SBSs

Rake 3

• Period-160 tugboat rake
• Contains 2 byzantines, 1 schooner, and 5 SBSs

Rake 4

• Period-32 tugboat rake
• (Barrel is period-32, tail end is period-64)

Schooner

• Period-2 spaceship
• 2nd most common spaceship

• The schooner can eat blinkers and flares, and can be eaten by a block.

Schooner Rake 1

• Period-160 schooner backrake
• Uses rake 3 to convert pinwheels into schooners

Schooner Rake 2

• Period-64 schooner rake
• Uses rake 2 to destroy half of the victories left behind by a victory puffer, and another copy of rake 2 to convert the remaining victories into schooners

Sea urchin

• Period-4 spaceship
• Powered by a tango

• Two sea urchins can combine to produce a period-80 puffer. This puffer is the basis for rake 1.

Shell

• 14-cell still life
• 3rd most common still life

• The shell can eat a tugboat. It can also "reflect" a tugboat (while getting destroyed in the process).

• The table and its descendants (such as the two-blinker pattern on the right) are the most common ancestors of the shell.

Shellshock p4

• Period-4 oscillator
• Consists of a shell hassling a tango
• One of the first oscillators discovered, despite being fairly rare

Shellshock p19

• Period-19 oscillator
• Generated by a shell and a beehive spark

Slowpoke

• Period-19 spaceship
• Speed: 5c/19

• The slowpoke can eat blocks:

• Two slowpokes can combine to produce a period-19 blinker puffer.

• See also: mason

Spaceships

• The vast majority of spaceships discovered so far travel at a speed of c/2 orthogonal. The slowpoke is the primary exception, with a speed of 5c/19. These are the only known spaceship speeds. No diagonal spaceships have been discovered.
• The tugboat and the schooner are the most common spaceships to occur naturally.

Squid

• Period-4 spaceship

• See also: half-squid

Staircase

• Period-5 oscillator
• Generated by a shell and a block
• Most common period-5 oscillator

Steeple

• 7-cell still life consisting of a block and a chip
• 2nd most common still life
• 3rd smallest still life

Still lifes

• The four most common still lifes are:
1. Block
2. Steeple
3. Shell
4. Carrier
• Still lifes exist with every possible population except for 1, 2, 3, 5, and 8.

Symmetrical byzantine ship (SBS)

• Period-16 spaceship generated by two byzantines

• The SBS is one of the few spaceships capable of eating blocks. It can also eat carriers, steeples, and other small patterns. This property makes it useful for cleaning up the debris left behind by puffers.

• The SBS is one of the only patterns known to be capable of reflecting tugboats.

• See also: asymmetrical byzantine ship, arcade

Table

• 6-cell induction coil
• Evolves into a shell if not stabilised

• The table can be stabilised by a block, a flare, or a chip.

Tango

• 3-cell methuselah
• Stabilises at generation 290
• Debris: 4 blocks, 3 blinkers, 2 flares, 2 victories
• Descendants include the T-tetromino and the T-pentomino (both of which are also referred to as "tango")

• A spaceship or puffer is said to be "powered by a tango" if it has a T-tetromino at its fore in any of its phases. Examples include the sea urchin, the goalpost, and the byzantine.

Titan

• Period-26 spaceship
• 5th most common spaceship
• Emits one tub spark and two beehive sparks every 26 generations

• The smallest known predecessor of the titan has only 11 cells.

• Two titans can combine, resulting in a puffer that produces flares and blocks.

• See also: hyperion

Transformers

• Period-4 spaceship
• Front section appears to alternate between tugboats and schooners
• Two variants are shown

Trawler

• Period-6 tagalong for two tugboats

Tub spark

• 8-cell spark

• The pattern known as tub in the Game of Life creates a tub spark after 4 generations. Other ancestors of the tub spark include the boat and the X-pentomino.

Tugboat

• Period-2 spaceship
• Smallest and most common spaceship

• Tugboats can be stacked indefinitely as shown below. The resulting spaceships are known as "tugboat chains."

• Tugboats can eat blinkers:

• Here is a view of every head-on tugboat collision:

Turbulence

• Period-8 spaceship

• Separating the two halves by an additional cell results in a period-196 puffer.

Varangian guard

• Period-16 spaceship consisting of a byzantine and a schooner
• Produces beehive sparks

Victory

• Period-20 statorless oscillator
• 4th most common oscillator

• Here are some interesting victory-tugboat collisions:

• Here is a remarkable period-102 oscillator consisting of four blocks hassling two victories:

Voyager

• Period-16 blinker puffer