Superlinear growth
Superlinear growth is an infinite growth faster than any rate proportional to T, where T is the number of ticks that a pattern has been run. This term usually applies to a pattern's population growth, rather than diametric growth or bounding-box growth. For example, breeders' and spacefillers' population asymptotically grows faster than any linear-growth pattern.
The term may also be used to describe the rate of increase in the number of subpatterns present in a pattern, such as when describing a replicator's rate of reproduction.
Patterns have been created that display a wide range of different growth rates strictly between linear and quadratic. Due to limits enforced by the speed of light, no pattern's population can grow at an asymptotic rate faster than quadratic growth. Specifically, if the bounding box is x-by-y to begin with, after k generations it is at most (x+2k)-by-(y+2k), which has area xy+2k(x+y)+4k^2, which is quadratic in k. Since the area of the bounding box is an upper bound on the number of alive cells in the pattern, the number of alive cells can't grow any faster than quadratically.
Temporary faster-than-quadratic growth
Although sustained faster-than-quadratic growth cannot be reached in a two-dimensional Euclidean universe, patterns temporarily exhibiting higher-than-quadratic population growth rates can be constructed, for instance a cubic growth. They will eventually return to quadratic due to running out of space.[1]
References
- ↑ toroidalet (September 20, 2020). Re: random cellular automata concepts (discussion thread) at the ConwayLife.com forums
- ↑ toroidalet (July 3, 2022). Re: super breeders? (discussion thread) at the ConwayLife.com forums
- ↑ praosylen (January 7, 2018). Re: Rules with interesting dynamics (discussion thread) at the ConwayLife.com forums
External links
- Superlinear growth at the Life Lexicon