Polyomino

From LifeWiki
Jump to: navigation, search

A polyomino (or simply omino) is a finite collection of orthogonally connected cells. The mathematical study of polyominoes was initiated by Solomon Golomb in 1953. Conway's early investigations of Life and other cellular automata involved tracking the histories of small polyominoes, this being a reasonable way to ascertain the typical behaviour of different cellular automata when the patterns had to be evolved by hand rather than by computer. Polyominoes have no special significance in Life, but their extensive study during the early years lead to a number of important discoveries and has influenced the terminology of Life.

A cross.
During the course of its evolution a polyomino could regain its original form, thereby constituting one phase of an oscillator, as seen in the (infinite) cross family. The only other known examples are the block (which has period 1), the blinker, the toad, the star and (in two different phases) the pentadecathlon. Were a recurring polyomino displaced, it would be participating in a spaceship, though the only known examples are the lightweight spaceship, the middleweight spaceship, and the heavyweight spaceship. Of course, its evolution could follow other trajectories, possibly very long lived.


Sizes of polyominoes

Polyominoes of with n cells for n = 1, 2, 3, 4, ... are called haplominoes, dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, octominoes, and n-ominoes in general. The number of distinct polyominoes with n cells for n = 1, 2, 3, ... is given by the sequence 1, 1, 2, 5, 12, 35, 108, 369, 1285, ... (Sloane's A000105).

Haplominoes

There is only one haplomino (also called monomino or dot) and by itself it dies after one generation. Several objects, such as the middleweight spaceship, produce dot sparks.

Dominoes

There is also only one domino and by itself it too dies after one generation. A number of objects, such as the heavyweight spaceship and the pentadecathlon, produce domino sparks.

Based upon that instant dying property of the domino, Squaredance is a low density dominos agar and phoenix.

Triominoes

There are exactly two distinct triominoes. The term is rarely used in Life, because the two objects in question are simply the blinker and the pre-block.

Tetrominoes

There are five distinct tetrominoes, each of which is shown below. The first is the block, the second is the T-tetromino, and the remaining three rapidly evolve into beehives. The fourth is commonly referred to as a tail and is often attached to small still lifes.

The five distinct tetrominoes. From left to right: O, T, I, J, Z.
Manipulate via Java: click here
Download RLE: click here

Pentominoes

There are 12 distinct pentominoes. John Conway assigned them all letters in the range O to Z, loosely based on their shapes, and they are all shown below in order.

  • The O-pentomino is a traffic light predecessor, though not one of the more common ones.
  • The P-pentomino is a common spark that dies in generation 4.
  • The Q-pentomino is a traffic light predecessor.
  • The R-pentomino is a methuselah and by far the most well-known pentomino.
  • The S-pentomino dies in generation 5.
  • The T-pentomino is a common parent of the T-tetromino.
  • The U-pentomino dies in generation 4.
  • The V-pentomino evolves into a loaf in generation 3.
  • The W-pentomino is a common loaf grandparent.
  • The X-pentomino is a traffic light predecessor.
  • The Y-pentomino dies in generation 3.
  • The Z-pentomino dies in generation 3.
O
P
Q
R
S
T
U
V
W
X
Y
Z

Hexominoes

There are 35 distinct hexominoes, the majority of which behave in uninteresting ways. The most interesting and well-known examples are century, stairstep hexomino, table, toad and Z-hexomino.

Heptominoes

There are 108 distinct heptominoes. Those with names in common use are the B-heptomino, the bullet heptomino, the C-heptomino, the Herschel and the pi-heptomino.

Octominoes

There are 369 distinct octominoes. Despite the abundance of octominoes, the following fairly common octomino (or its two-generation successor), which stabilizes after 386 generations into two traffic lights and four beehives, is often referred to as simply the octomino:

The octomino and its two-generation successor.

Switch Engine

Charles Corderman discovered the switch engine by running an exhaustive computer search on all decominoes. The machine that discovered the decomino seed was unique, possessing an unusual architecture.

Nomenclature

The name is a back-formation from 'domino', which explicitly refers to a pair of orthogonally connected live cells. According to the Life Lexicon, the pluralised name can be spelt two different ways: 'polyominos' and 'polyominoes' are both equally acceptable. The same applies to the systematic names of each of the polyomino sizes.

See also

External links