Speed

The speed of a pattern is a measure of the number of generations that it takes for some effect to travel some given distance. Speeds are almost always measured in reference to the speed of light (or speed of life), denoted by c which, the fastest possible speed at which any effect can propagate. William Poundstone, in The Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge, writes (p. 79; our emphasis)

Orthogonal speeds of spaceships in Conway Life. Note: This image is outdated as it does not include 31c/240, c/10 nor 3c/7 orthogonal.

Thus for first neighbor neighborhoods, such as the Moore and von Neumann neighbourhoods of range one, the speed of light is a rate of one cell per generation. Rules with extended neighbourhoods, such as Larger than Life rules, have higher speeds of light; other classes of cellular automata may have different speed of lights yet.

Spaceships
The lightweight spaceship has speed c/2 because it moves 2 cells in 4 generations

For spaceships, the speed describes the number of cells that it has been displaced by after it has gone through one period. Speed is reported in the form dc/p where d is the displacement and p is the period. It is most common to reduce this "fraction" to lowest terms. For example, even though the period of a lightweight spaceship is 4, it moves 2 cells during those generations, giving it a speed of 2c/4 = c/2.

For spaceships that move diagonally, speed is defined the same as above, but where "the number of cells that is has been displaced by" refers to the maximum of the x or y displacement; not their sum. So, for example, a glider has a speed of c/4, since it takes 4 generations to move one cell in the x direction and one cell in the y direction.

More formally, if a spaceship in any 2D cellular automaton is translated by (x,y) after n generations then its speed v may be defined as:

This definition can be generalized in a straightforward manner to cellular automata with dimension other than two.

gollark: That's exactly as hard as pi.
gollark: Imaginary numbers of apples were trickier, since all the apples are imaginary, but I just made those apples especially imaginary.
gollark: Imagining negative apples, obviously.
gollark: This was a bit difficult when I learned about negative numbers but I eventually found a resolution.
gollark: This is why when I see any number, I simply imagine that many apples.

See also
  • List of important speeds
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