# Isotropic non-totalistic Life-like cellular automaton

**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) |
|||||||||

c (corner) |
|||||||||

e (edge) |
|||||||||

k (knight) |
|||||||||

a (adjacent) |
|||||||||

i | |||||||||

n | |||||||||

y | |||||||||

q | |||||||||

j | |||||||||

r | |||||||||

t | |||||||||

w | |||||||||

z |

All von Neumann rules can be simulated via non-totalistic rules; for example, B1/SV becomes B1e/S.

## 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;^{[1]} 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) |
|||||||

o (ortho) |
|||||||

m (meta) |
|||||||

p (para) |

## 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.^{[2]} Catagolue gained the ability to census non-totalistic rules in late January 2016.^{[3]}

## See also

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

## References

- ↑ http://www.conwaylife.com/forums/download/file.php?id=261
- ↑ Aidan F. Pierce (December 17, 2015). "Re: Hacking apgsearch".
*ConwayLife.com forums*. Retrieved on June 12, 2016. - ↑ Adam P. Goucher (January 21, 2016). "Re: apgsearch v2.2".
*ConwayLife.com forums*. Retrieved on June 12, 2016.

## 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)