View source for 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.<ref name="post128801" /> 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.<ref name="post50386" /> A second notation for this neighbourhood was created in mid-2020,<ref name="post101003" /> however a 3-cell transition and its corresponding 5-cell transition was mistakenly omitted.<ref name="p128945">{{LinkForumThread|f=3|t=1622|p=128945|title=Re: Pattern viewer for forum threads|format=ref}}</ref>

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.<ref name="post101003"/> However, the proposal duplicated one of the 4-cell transitions.<ref name="p128945"/>

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.

{{EmbedViewer
|rle          = x = 29, y = 5, rule = B/S01234V
2bo5bo3bo3bobobo3bobobo$b3o5b3o6bo6bobo$2ob2o4bobo4b2ob2o3bo3bo$b3o5b
3o6bo6bobo$2bo5bo3bo3bobobo3bobobo!
|viewerconfig = #C [[ NOGUI ]]
|position     = center
|caption      = These four neighbourhoods are effectively equivalent
}}

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.<ref name="post71574" /> 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.<ref name="post70180" /><ref name="post71574" />

A table for the extensions notation is as follows:
{{Isotropic non-totalistic transitions (square grid extensions)}}

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.<ref name="post102838" />

CAViewer supports this notation for the range-2 von Neumann neighbourhood.<ref name="post113480"/>

=== 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.<ref>https://conwaylife.com/forums/viewtopic.php?p=164994#p164994</ref>

== Table of isotropic non-totalistic rulespaces ==
{{split|List of 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 symbol]]s as to simultaneously convey grid type and dimension):

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;"
! rowspan="2" | Common name
! colspan="4" | Rulespace definitions
! rowspan="2" | Transitions
! rowspan="2" | Notated?
! colspan="3" | Major software support
|-
! Grid
! Range
! States
! Neighbourhood
! Golly
! LifeViewer
! lifelib
|-
! Block cellular automata
| {4,4}
| 1/2
| 2
| Moore
| 6
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?f=11&t=3243&p=55536#p55536 Yes]
| colspan="3" style="background-color:#c6efce;" | (Unnecessary)
|-
! -
| {4,4}
| 1
| 2
| von Neumann
| 6
| style="background-color:#c6efce;" | Trivial
| colspan="3" style="background-color:#c6efce;" | (Unnecessary)
|-
! 3D vN non-totalistic
| {4,3,4}
| 1
| 2
| von Neumann
| 10
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! Non-totalistic hexagonal
| {6,3}
| 1
| 2
| hexagonal
| 13
| style="background-color:#c6efce;" | [https://conwaylife.com/ref/lifepage/hexrule.txt Yes]
| style="background-color:#ffc7ce;" | No<ref group="n">The rulespace can be simulated via MAP strings, or, less preferably, ruletables, however no direct support for the notation exists.</ref>
| style="background-color:#c6efce;" | Yes
| style="background-color:#c6efce;" | Yes
|-
! Knight INT
| {4,4}
| 2
| 2
| knight
| 43
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?f=3&t=1622&p=101003#p101003 Yes]
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! Isotropic non-totalistic
| {4,4}
| 1
| 2
| Moore
| 51
| style="background-color:#c6efce;" | [http://www.ibiblio.org/lifepatterns/neighbors2.html Yes]
| style="background-color:#c6efce;" | Yes
| style="background-color:#c6efce;" | Yes
| style="background-color:#c6efce;" | Yes
|-
! Exploded
| {4,4}
| ≥1
| 2
| variable<ref group="n">Based on the Moore neighbourhood, with the four edge cells and/or the four corner cells moved away from the origin a defined distance.</ref>
| 51
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?f=11&t=4763#p128801 Yes]
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffffdd;" | Some<ref group="n">The range-N "far corners" + range-M "far edges" cases are supported, but the general case of exploded Moore neighbourhoods are not.</ref>
| style="background-color:#ffffdd;" | Some<ref group="n">The range-2 "far corners" and range-3 "far edges" cases are supported, but the general case of exploded Moore neighbourhoods are not.</ref>
|-
! Cross INT
| {4,4}
| 2
| 2
| cross
| 55
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?f=3&t=1622&p=101003#p101003 Yes]
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! -
| {4,4}
| 2
| 2
| nonstandard<ref group="n">The number is the same for a "circle" formed of four three-cell lines at distance 2 from the center, and four "pre-blocks" with the gaps all facing the center or all facing outwards (or any configuration of eight eightfold neighbours and two fourfold ones)</ref>
| 570
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! R2 von Neumann INT
| {4,4}
| 2
| 2
| von Neumann
| 618
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?t=&p=70180#p70180 Yes]
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#c6efce;" | Yes
|-
! -
| {4,4}
| 2
| 2
| nonstandard<ref group="n">Range-1 Moore, with added range-2 corners</ref>
| 618
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 3-state knight INT
| {4,4}
| 2
| 3
| knight
| 873
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 3-state INT
| {4,4}
| 1
| 3
| Moore
| 954
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?p=164994#p164994 Yes]
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3">The rulespace can be simulated via ruletables, however no direct support for the notation exists.</ref>
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3"/>
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3"/>
|-
! 3-state cross INT
| {4,4}
| 2
| 3
| cross
| 1035
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 4-state knight INT
| {4,4}
| 2
| 4
| knight
| 8356
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 4-state INT
| {4,4}
| 1
| 4
| Moore
| 8740
| style="background-color:#c6efce;" | [https://conwaylife.com/forums/viewtopic.php?f=11&t=6126 Yes]
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3"/>
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3"/>
| style="background-color:#ffc7ce;" | No<ref group="n" name="no3"/>
|-
! 4-state cross INT
| {4,4}
| 2
| 4
| cross
| 9316
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! R2 hexagonal INT
| {6,3}
| 2
| 2
| hexagonal
| 22668
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 3-state R2 vN INT
| {4,4}
| 2
| 3
| von Neumann
| 68715
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! R2 Moore-without-corners
| {4,4}
| 2
| 2
| circular
| 132744
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 3D non-totalistic
| {4,3,4}
| 1
| 2
| Moore
| 1426144<ref group="n">first calculated by Milo Jacquet</ref>
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! R2 Moore INT
| {4,4}
| 2
| 2
| Moore
| 2105872
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|-
! 4D non-totalistic
| {4,3,3,4}
| 1
| 2
| Moore
| 3148244699232062849152<ref group="n">first calculated using the [[Pólya enumeration theorem]] by [https://gist.github.com/DroneBetter/90318584a0ad6a0f4d1c418d3da132cd DroneBetter's <tt>dimensional_INT_enumerator.py</tt>] (explained [[User:DroneBetter/dimensional INT enumerator|here]])</ref>
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
| style="background-color:#ffc7ce;" | No
|}
<references group="n"/>

<!--The number of unique transitions in rulespaces with range-1 von Neumann neighbourhoods (or equivalent) with <tt>n</tt> states follows the {{LinkWikipedia|Doubly_triangular_number|name=doubly-triangular numbers|nobullet=nobullet|nosuffix=nosuffix}}, <tt>{{OEIS|A002817}}(n)</tt>.-->

A rule with an <tt>n</tt>-dimensional range-<tt>k</tt> Moore neighbourhood has <tt>{{OEIS|A361870}}(n,2*k+1)</tt> 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.<ref name="post99110" /> However, note that GPU censuses support only outer-totalistic rules.

== See also ==
* [[Generations]]
* [[Larger than Life]]

== References ==
<references>
<ref name="post128801">{{LinkForumThread
|format = ref
|title  = Re: Lemon41625's Cellular Automaton
|p      = 128801
|author = Connor Steppie
|date   = April 23, 2021
}}</ref>
<ref name="post71574">{{LinkForumThread
|format = ref
|title  = Re: Range-2 von Neumann isotropic non-totalistic rulespace
|p      = 71574
|author = AforAmpere
|date   = February 23, 2019
}}</ref>
<ref name="post50386">{{LinkForumThread
|format = ref
|title  = Re: Golly suggestions
|p      = 50386
|author = toroidalet
|date   = September 2, 2017
}}</ref>
<ref name="post101003">{{LinkForumThread
|format = ref
|title  = Re: Pattern viewer for forum threads
|p      = 101003
|author = lemon41625
|date   = July 23, 2020
}}</ref>
<ref name="post70180">{{LinkForumThread
|format = ref
|title  = Range-2 von Neumann isotropic non-totalistic rulespace
|p      = 70180
|author = Connor Steppie
|date   = February 9, 2019
}}</ref>
<ref name="post102838">{{LinkForumThread
|format = ref
|title  = Re: Range-2 von Neumann isotropic non-totalistic rulespace
|p      = 102838
|author = bubblegum
|date   = August 26, 2020
}}</ref>
<ref name="post113480">{{LinkForumThread
|format = ref
|title  = Re: CAViewer - A Cellular Automaton Simulator written in Java
|p      = 113480
|author = lemon41625
|date   = November 29, 2020
}}</ref>
<ref name="post99110">{{LinkForumThread
|format = ref
|title  = Re: Range-2 von Neumann isotropic non-totalistic rulespace
|p      = 99110
|author = lemon41625
|date   = June 19, 2020
}}</ref>
</references>

[[Category:Algorithms]]
[[Category:Cellular automata]]
__NOTOC__