Higher-range isotropic non-totalistic rule

Higher-range isotropic non-totalistic (abbreviated as HRINT) are a generalization of range-1 isotropic non-totalistic rules and of higher-range outer-totalistic rules; a rulespace with a range greater than 1 where isotropy is observed, but transitions are not necessarily totalistic.

There are a wide variety of such rules, with limited software support.

Classes

"Exploded" Moore neighbourhood

The range-1, 8-cell Moore neighbourhood can be "exploded" into other neighbourhoods consisting of four orthogonal cells of distance E and four diagonal cells of distance C. Isotropic non-totalistic rules can be run using this neighbourhood using standard Hensel notation.

Neighbourhoods with orthogonal cells of distance 1 and diagonal cells of distance 2, as well as with orthogonal cells of distance 3 and diagonal cells of distance 1, can be run using CAViewer.

Range-2 cross neighbourhood

A proposal for a notation for this neighbourhood was created in 2017 as a result of a misunderstanding of how the range-2 von Neumann neighbourhood was to be defined. A second notation for this neighbourhood was created in mid-2020, however a 3-cell transition and its corresponding 5-cell transition was mistakenly omitted.

CAViewer supports this rulespace, however the exact notation it uses (accounting for the omitted transition) is unknown.

Range-2 knight neighbourhood

A proposal for a notation for this neighbourhood was created in mid-2020. However, the proposal duplicated one of the 4-cell transitions.

CAViewer supports this rulespace, however the exact notation it uses (accounting for the duplicated transition) is unknown.

4-cell extensions

A notation system exists that can be used for extending existing isotropic non-totalistic rules by adding four equidistant cells, either orthogonally or diagonally aligned with the center. This can be used for notating isoptropic non-totalistic rules with the range-2 von Neumann neighbourhood, or any of the other three equivalent neighbourhoods below. It is also used for notating the range-3 cross neighbourhood by extending the range-2 cross neighbourhood.

x = 29, y = 5, rule = B/S01234V 2bo5bo3bo3bobobo3bobobo$b3o5b3o6bo6bobo$2ob2o4bobo4b2ob2o3bo3bo$b3o5b 3o6bo6bobo$2bo5bo3bo3bobobo3bobobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ NOGUI ]]

These four neighbourhoods are effectively equivalent (click above to open LifeViewer)

The extension-based notation was initially proposed on February 24, 2019 by AforAmpere and Milo Jacquet for range-2 von Neumann isotropic non-totalistic rules. Prior to this, a two-letter system was proposed for said rulespace, in which two letters were used for each of the 618 possible transitions; this ultimately was not adopted.

A table for the extensions notation is as follows:

a	c	d	e	f	g	i	j
k	l	m	n	o	p	q	r

For each transition, use the transition corresponding to the configuration of the center 8 cells, aligned with the canonical directions in the table above. Then take the outside 4 cells, and depending on the configurations in the table below, add that letter.

For transitions that are the same under reflections or rotations, the canonical transition is the lowest alphabetically.

'x' is used after a totalistic number. An example range-2 von Neumann rulestring is 'B3x2ic1ei5x-3kr/S0x8x'.

bubblegum has also proposed to use the 'v' character to represent outer totalistic transitions for the inner 8 cells for range-2 von Neumann rules.

CAViewer supports this notation for the range-2 von Neumann neighbourhood.

3-state, range-1 Moore neighbourhood

Several notations have been proposed historically. The currently accepted notation, which does not currently have software support, was published by Cyclotrons in 2023.

Table of isotropic non-totalistic rulespaces

The number of unique transitions for higher ranges tends to be extremely large, and the development of notations for such rules is generally infeasible. The following table documents notable examples (grids are notated with Schläfli symbols as to simultaneously convey grid type and dimension):

Common name	Rulespace definitions				Transitions	Notated?	Major software support
	Grid	Range	States	Neighbourhood			Golly	LifeViewer	lifelib
Block cellular automata	{4,4}	1/2	2	Moore	6	Yes	(Unnecessary)
-	{4,4}	1	2	von Neumann	6	Trivial	(Unnecessary)
3D vN non-totalistic	{4,3,4}	1	2	von Neumann	10	No	No	No	No
Non-totalistic hexagonal	{6,3}	1	2	hexagonal	13	Yes	No	Yes	Yes
Knight INT	{4,4}	2	2	knight	43	Yes	No	No	No
Isotropic non-totalistic	{4,4}	1	2	Moore	51	Yes	Yes	Yes	Yes
Exploded	{4,4}	≥1	2	variable	51	Yes	No	Some	Some
Cross INT	{4,4}	2	2	cross	55	Yes	No	No	No
-	{4,4}	2	2	nonstandard	570	No	No	No	No
R2 von Neumann INT	{4,4}	2	2	von Neumann	618	Yes	No	No	Yes
-	{4,4}	2	2	nonstandard	618	No	No	No	No
3-state knight INT	{4,4}	2	3	knight	873	No	No	No	No
3-state INT	{4,4}	1	3	Moore	954	Yes	No	No	No
3-state cross INT	{4,4}	2	3	cross	1035	No	No	No	No
4-state knight INT	{4,4}	2	4	knight	8356	No	No	No	No
4-state INT	{4,4}	1	4	Moore	8740	Yes	No	No	No
4-state cross INT	{4,4}	2	4	cross	9316	No	No	No	No
R2 hexagonal INT	{6,3}	2	2	hexagonal	22668	No	No	No	No
3-state R2 vN INT	{4,4}	2	3	von Neumann	68715	No	No	No	No
R2 Moore-without-corners	{4,4}	2	2	circular	132744	No	No	No	No
3D non-totalistic	{4,3,4}	1	2	Moore	1426144	No	No	No	No
R2 Moore INT	{4,4}	2	2	Moore	2105872	No	No	No	No
4D non-totalistic	{4,3,3,4}	1	2	Moore	3148244699232062849152	No	No	No	No

A rule with an n-dimensional range-k Moore neighbourhood has  A361870(n,2*k+1) transitions.

Soup-searching non-totalistic rules

Range 2 von Neumann isotropic rules can be searched using Adam P. Goucher's apgsearch by means of a ruletable using a custom neighbourhood. However, note that GPU censuses support only outer-totalistic rules.

See also

• Generations
• Larger than Life