Checkerboard dual: Difference between revisions
Apple Bottom (talk | contribs) New (?) concept, based on Calcyman's post. Any errors are due to a lack of understanding on my part. Please edit this as you deem necessary and/or proper. |
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where ⊕ denotes the [https://en.wikipedia.org/wiki/Exclusive_or XOR operation]. | where ⊕ denotes the [https://en.wikipedia.org/wiki/Exclusive_or XOR operation]. | ||
Checkerboard duality is similar to [[strobing duality]], but applies to isotropic rules; the checkerboard equivalent of a [[semi-totalistic]] rule (such as [[Conway's Game of Life|Life]]) is not generally semi-totalistic. | Checkerboard duality is similar to [[strobing duality]], but applies to isotropic rules; the checkerboard equivalent of a [[semi-totalistic]] rule (such as [[Conway's Game of Life|Life]]) is not generally semi-totalistic. The checkerboard dual of an isotropic non-totalistic rule is also isotropic non-totalistic if and only if the original rule is [[self-complementary]]. | ||
The checkerboard dual of [[Day & Night]] is B1e2cn3acjkr4cny5einqy6ei7c/S01e2-ei3acjkr4-ejr5einqy6-cn7c8 <ref name="post48009" />. | |||
==Finding the checkerboard dual of a given rule== | ==Finding the checkerboard dual of a given rule== | ||
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The correct sister rule (in your case) is obtained by XORing each B/S condition with a 5-cell X, analogous to how the strobing dual is obtained by XORing each B/S condition with a 9-cell solid box. | The correct sister rule (in your case) is obtained by XORing each B/S condition with a 5-cell X, analogous to how the strobing dual is obtained by XORing each B/S condition with a 9-cell solid box. | ||
==References== | |||
<references> | |||
<ref name="post48009">{{LinkForumThread | |||
|p = 48009 | |||
|format = ref | |||
|title = Re: Day & Night (B3678/S34678) | |||
|author = AforAmpere | |||
|accessdate = January 16, 2018 | |||
|date = August 4, 2017 | |||
}}</ref> | |||
</references> | |||
==External links== | ==External links== | ||
{{LinkForumThread|p=55132|author=Adam P. Goucher|title=Re: Thread for basic questions|accessdate=January 16, 2018|date=January 16, 2018}} | {{LinkForumThread|p=55132|author=Adam P. Goucher|title=Re: Thread for basic questions|accessdate=January 16, 2018|date=January 16, 2018}} | ||
{{LinkForumThread|p=47992|author=Aidan F. Pierce|title=Re: Day & Night (B3678/S34678)|accessdate=January 16, 2018|date=January 16, 2018}} | |||
Revision as of 20:59, 16 January 2018
The checkerboard dual of an isotropic non-totalistic Life-like cellular automaton is another such automaton such that any finite arrangement of live cells in an empty universe in the original rule will behave like finite perturbations in a checkerboard universe in the dual.
That is to say, for a finite pattern P, a rule R and its checkerboard dual C(R):
- R(P ⊕ checkerboard) = S(P) ⊕ checkerboard
where ⊕ denotes the XOR operation.
Checkerboard duality is similar to strobing duality, but applies to isotropic rules; the checkerboard equivalent of a semi-totalistic rule (such as Life) is not generally semi-totalistic. The checkerboard dual of an isotropic non-totalistic rule is also isotropic non-totalistic if and only if the original rule is self-complementary.
The checkerboard dual of Day & Night is B1e2cn3acjkr4cny5einqy6ei7c/S01e2-ei3acjkr4-ejr5einqy6-cn7c8 [1].
Finding the checkerboard dual of a given rule
- FIXME: reword/clear up
Let the 'B/S conditions' of a rule be the set of 3-by-3 boxes of cells such that, after iterating by one generation, result in a live centre cell. Clearly, a rule is determined uniquely by its B/S conditions. These black/white symmetric rules have exactly 256 B/S conditions (out of a possible 512).
The correct sister rule (in your case) is obtained by XORing each B/S condition with a 5-cell X, analogous to how the strobing dual is obtained by XORing each B/S condition with a 9-cell solid box.
References
- ↑ AforAmpere (August 4, 2017). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums
External links
- Adam P. Goucher (January 16, 2018). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
- Aidan F. Pierce (January 16, 2018). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums