Checkerboard patterns are topological restrictions created by symmetry that have a limited life. They are set up in a closed torus universe of a life-like state symmetrical cellular automata. The surface of the torus is initialised as identical sized rectangles of alternate state with a few cells added near one boundary to create asymmetry. They have the following fate.
All seven wave rules support checkerboard patterns to some degree. The SymCACP places seeds in the centre of one of the boundries of one square so that seeds that have been selected for not been symmetrical do not generate symmetrical initial patterns.
The behavior of the rules is related to Blowoff and Seed.
| Rule | Result of Blowoff | Effect |
| H068 | complex pattern | Any '1' or '2' digits in the seed must be separated from other digits by '0's otherwise mirror symmetry does not occur. Solid islands of each state emerge from the complex patterns and sometimes these are checkerboard patterns sometimes more than two "rectangles" in each half. |
| H0C8 | single cell checkerboard | The seed must be mirror symmetric i.e. made up of '0's and '3's. Chaos spreads from the seeded rectangle and solid islands of each state emerge from this. Sometimes these form checkerboard patterns. |
| H0E8 | target checkerboard | Any '1' or '2' digits in the seed must be separated from other digits by '0's otherwise mirror symmetry does not occur. Complex waves on the vertical boundary die out quite quickly leaving low period oscillators. |
| H168 | complex pattern in target checkerboard | Any '1' or '2' digits in the seed must be separated from other digits by '0's otherwise mirror symmetry does not occur. The complex pattern formed in the unseeded rectangles can stops checkerboards from reforming after Blowoff. This occurs in Universes smaller than 228 cells square and other specific sizes e.g. 1000 cells square does not form checkerboards with somec seeds e,g, '01'. |
| H1A8 | Resonating | Checkerboards do not form easily in universes smaller than 130 cells square because of the depth of the boundaries. Some seeds result in mirror symmetric patterns like '01' and '0333' but some don't like '03' or '033'. After Blowoff chaos spreads from the seeded area and condense into solid islands which may form checkerboard patterns. |
| H1C8 | single cell checkerboard | Checkerboards do not form easily in universes smaller than 130 cells square because of the depth of the boundaries. The seed must be mirror symmetric i.e. made up of '0's and '3's. Chaos spreads from the seeded rectangle and solid islands of each state emerge from this. Sometimes these form checkerboard patterns. |
| H1E8 | target checkerboard | Any '1' or '2' digits in the seed must be separated from other digits by '0's otherwise mirror symmetry does not occur. Complex waves on the vertical boundary die out quite quickly leaving low period oscillators, however the corners do not stabilise and react to the mirror symmetric waves on the horizontal boundary. This sends ripples up the vertical boundary which die out but over time cause the vertical boundary to perform a random walk. |
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Site by Paul Rendell. |
Last Update 29/June/26 |
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