Spaceship (cellular automaton)

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Orthogonal spaceships in Conway's Game of Life of varying speeds. Note some spaceships “overtake” others due to speed differences.

In a cellular automaton, a finite pattern is called a spaceship if it reappears after a certain number of generations in the same orientation but in a different position. The smallest such number of generations is called the period of the spaceship.

Description

The speed of a spaceship is often expressed in terms of c, the metaphorical speed of light (one cell per generation) which in many cellular automata is the fastest that an effect can spread. For example, a glider in Conway's Game of Life is said to have a speed of ${\displaystyle c/4}$, as it takes four generations for a given state to be translated by one cell. Similarly, the lightweight spaceship is said to have a speed of ${\displaystyle c/2}$, as it takes four generations for a given state to be translated by two cells. More generally, if a spaceship in a 2D automaton is translated by ${\displaystyle (x,y)}$ after ${\displaystyle n}$ generations, then the speed ${\displaystyle v}$ is defined as:

${\displaystyle v={\frac {\max \left(|x|,|y|\right)}{n}}\,c}$

This notation can be readily generalised to cellular automata with dimensionality other than two.

A tagalong is a pattern that is not a spaceship in itself but that can be attached to the back of a spaceship to form a larger spaceship. Similarly, a pushalong is placed at the front.

A pattern that, when a spaceship is input, outputs a copy of the spaceship travelling in a different direction is called a reflector.

Spaceships are important because they can sometimes be modified to produce puffers. Spaceships can also be used to transmit information. For example, in Conway's Game of Life, the ability of the glider (Life's simplest spaceship) to transmit information is part of a proof that Life is Turing-complete.

In March 2016, the unexpected discovery of a small but high-period spaceship enthused the Game of Life community. It was named "copperhead".[1]

In March 2018, the first elementary spaceship with displacement (2,1) (knightwise) was discovered and named Sir Robin.[citation needed]

References

1. ^ "New Spaceship Speed in Conway's Game of Life". 7 March 2016.