Isotropic non-totalistic rule

An isotropic non-totalistic Life-like cellular automaton is a generalization of the concept of a Life-like cellular automaton in which transitions take into account not only the total number of live neighbors of a cell, but also the relative configuration of those neighbors.

Isotropic 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 dead cell will be born with 2 neighbors, except when they are adjacent (indicated by the "-a"), and that a live cell will survive with 1 or 2 neighbors in any configuration. This exclusion of the "B2a" transition prevents the rule from exploding in a similar manner as Seeds.

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

Moore neighbourhood

The following table describes all possible neighborhood configurations for the Moore neighbourhood of range 1:

	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

A proposed notation for range-2 von Neumann isotropic non-totalistic rules is based on this.

Rules using the von Neumann neighbourhood can be simulated via isotropic non-totalistic rules on the Moore neighbourhood; for example, B1/SV becomes B1e2ak3inqy4ny5e/S.

Hexagonal neighbourhood

Main article: Hexagonal neighbourhood

It is possible to define isotropic 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; the names ortho, meta and para were chosen in analogy to arene substitution patterns in aromatic chemistry:

	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 (ortho)			File:Hexagonal neighborhood 2o.png	File:Hexagonal neighborhood 3o.png	File:Hexagonal neighborhood 4o.png
m (meta)			File:Hexagonal neighborhood 2m.png	File:Hexagonal neighborhood 3m.png	File:Hexagonal neighborhood 4m.png
p (para)			File:Hexagonal neighborhood 2p.png	File:Hexagonal neighborhood 3p.png	File:Hexagonal neighborhood 4p.png

Golly does not support isotropic non-totalistic hexagonal rules using this syntax, so they must instead be simulated using either rule tables or MAP strings. LifeViewer and lifelib support them natively.

Range-2 von Neumann neighbourhood

There are 618 different transitions possible in the range-2 von Neumann neighbourhood. Four notations have been proposed as of February 2018.

Soup-searching non-totalistic rules

Adam P. Goucher's apgsearch was modified to support isotropic non-totalistic rules by Aidan F. Pierce on December 17, 2015. This hacked version was later modified in late January 2016 to be able to upload the search results to Catagolue. However, apgsearch did not gain native support for these rules until v4.2, released on September 10, 2017, which can search isotropic non-totalistic rules without B0. v4.66 and above also support the searching of isotropic hexagonal neighborhood rules.

See also

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

• Non-totalistic Rules - notations, projects, & discussion (discussion thread) at the ConwayLife.com forums