Isotropic non-totalistic rule

Non-totalistic Life-like cellular automata are a generalization of Life-like cellular automata in which any transition function which is isotropic (that is, invariant under rotations and reflections) is allowed.

Non-totalistic rules are described using Hensel notation, an extension of B/S notation developed by Alan Hensel additionally describing allowed or forbidden configurations. Each digit in the rule's birth and survival conditions is followed by an optional suffix, with each allowed configuration described by a specific letter; a minus sign may be used to forbid configurations rather than allow them. If no configurations are specified, all are considered to be allowed, as in the totalistic case. This notation is not used by non-isotropic Life-like cellular automata.

For instance, B2-a/S12 (the "Just Friends" rule) indicates that a live cell will survive on 1 or 2 neighbors, or a dead cell get born on 2 neighbors, except when they are adjacent.

This notation has the following symmetry: For any letter x and number n≠4, nx is defined if and only x(8-n) is defined and moreover x(8-n) is the complement (change live cells to dead and dead cells to live; ignore the center cell) of xn.

The following table describes all possible neighborhood configurations for the Moore neighbourhood; where appropriate, the same configurations apply to the von Neumann neighbourhood:

	0	1	2	3	4	5	6	7	8
— (no letter)	File:Neighborhood 0.png								File:Neighborhood 8.png
c (corner)			File:Neighborhood 2c.png	File:Neighborhood 3c.png	File:Neighborhood 4c.png	File:Neighborhood 5c.png	File:Neighborhood 6c.png	File:Neighborhood 7c.png
e (edge)		File:Neighborhood 1e.png	File:Neighborhood 2e.png	File:Neighborhood 3e.png	File:Neighborhood 4e.png	File:Neighborhood 5e.png	File:Neighborhood 6e.png	File:Neighborhood 7e.png
k (knight)			File:Neighborhood 2k.png	File:Neighborhood 3k.png	File:Neighborhood 4k.png	File:Neighborhood 5k.png	File:Neighborhood 6k.png
a (adjacent)			File:Neighborhood 2a.png	File:Neighborhood 3a.png	File:Neighborhood 4a.png	File:Neighborhood 5a.png	File:Neighborhood 6a.png
i			File:Neighborhood 2i.png	File:Neighborhood 3i.png	File:Neighborhood 4i.png	File:Neighborhood 5i.png	File:Neighborhood 6i.png
n			File:Neighborhood 2n.png	File:Neighborhood 3n.png	File:Neighborhood 4n.png	File:Neighborhood 5n.png	File:Neighborhood 6n.png
y				File:Neighborhood 3y.png	File:Neighborhood 4y.png	File:Neighborhood 5y.png
q				File:Neighborhood 3q.png	File:Neighborhood 4q.png	File:Neighborhood 5q.png
j				File:Neighborhood 3j.png	File:Neighborhood 4j.png	File:Neighborhood 5j.png
r				File:Neighborhood 3r.png	File:Neighborhood 4r.png	File:Neighborhood 5r.png
t					File:Neighborhood 4t.png
w					File:Neighborhood 4w.png
z					File:Neighborhood 4z.png

Hexagonal neighbourhood

It is possible to define non-totalistic Life-like CAs on a hexagonal grid as well. The following table describes all possible neighborhood configurations for the hexagonal neighbourhood, using notation due to Paul Callahan:

	0	1	2	3	4	5	6
— (no letter)	File:Hexagonal neighborhood 0.png	File:Hexagonal neighborhood 1.png				File:Hexagonal neighborhood 5.png	File:Hexagonal neighborhood 6.png
o			File:Hexagonal neighborhood 2o.png	File:Hexagonal neighborhood 3o.png	File:Hexagonal neighborhood 4o.png
m			File:Hexagonal neighborhood 2m.png	File:Hexagonal neighborhood 3m.png	File:Hexagonal neighborhood 4m.png
p			File:Hexagonal neighborhood 2p.png	File:Hexagonal neighborhood 3p.png	File:Hexagonal neighborhood 4p.png

Soup-searching non-totalistic rules

Adam P. Goucher's apgsearch was modified to support non-totalistic rules by Aidan F. Pierce on December 17, 2015. Catagolue gained the ability to census non-totalistic rules in late January 2016.

See also

• Totalistic Life-like cellular automaton
• Non-isotropic Life-like cellular automaton
• Generations
• Larger than Life

External links

• Alan Hensel. "Table of non-totalistic neighborhoods". Retrieved on 2016-06-12.
• Alan Hensel. "Rule notation". Retrieved on 2016-06-12. (note that the table on this page describes an earlier version of Hensel notation that has fallen into disuse)