Difference between revisions of "Triangular neighbourhood"

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{{Glossary}}
{{Glossary}}
The '''triangular neighbourhood''' is the set of all [[cell]]s that are adjacent to the region of interest in a grid tiled with triangles (the region of interest itself may or may not be considered part of the triangular neighbourhood, depending on context).
The '''triangular neighbourhood''' is the set of all [[cell]]s that are adjacent to the region of interest in [[Triangular tiling|a grid tiled with triangles]] (the region of interest itself may or may not be considered part of the triangular neighbourhood, depending on context).


The triangular neigbourhood can either refer to:
== Triangular neighbourhoods ==
* The 12-cell triangular neighbourhood (sometimes referred to as the Triangular Moore neighbourhood)
The triangular neighbourhood can either refer to:
[[File:Triangular12.png|150px]]
* The 9-cell triangular vertices neighbourhood
[[File:Triangular9.png|150px]]
* The 3-cell triangular edges neighbourhood (sometimes referred to as the Triangular von Neumann neighbourhood).
[[File:Triangular3.png|150px]]


The triangular tiling shares its symmetries with that of the hexagonal tiling.
{|style="margin:auto;"
|-
| {{Nbh-demo |r = 1 |n = L |z = 8 |w = 100 |h = 100}}
| The 12-cell triangular neighbourhood (sometimes referred to as the Triangular Moore neighbourhood)
|-
| {{Nbh-demo |r = 2 |n = @06E5C0L |z = 8 |w = 100 |h = 100}}
| The 9-cell triangular vertices neighbourhood
|-
| {{Nbh-demo |r = 2 |n = @02a540L |z = 8 |w = 100 |h = 100}}
| The 6-cell triangular outer neighbourhood
|-
| {{Nbh-demo |r = 2 |n = @055880L |z = 8 |w = 100 |h = 100}}
| The 6-cell triangular inner neighbourhood
|-
| {{Nbh-demo |r = 1 |n = @58L |z = 8 |w = 100 |h = 100}}
| The 3-cell triangular edges neighbourhood (sometimes referred to as the Triangular von Neumann neighbourhood)
|-
| {{Nbh-demo |r = 2 |n = @03bd40L |z = 8 |w = 100 |h = 100}}
| The 9-cell triangular biohazard neighbourhood
|-
| {{Nbh-demo |r = 2 |n = @044080L |z = 8 |w = 100 |h = 100}}
| The 3-cell triangular radiation neighbourhood
|}


==Software support==
== Software support ==
[[LifeViewer]] natively supports the three aforementioned triangular neighbourhoods using 2 states. For triangles pointing down the following neighbourhood is used on a square tiling:
[[LifeViewer]] natively supports the seven aforementioned triangular neighbourhoods using 2 states. For triangles pointing up the following neighbourhood is used on a square tiling:


[[File:Triangular neighbourhood (radius 1).png|150px]]
[[File:Emulated triangular Moore neighbourhood (range 1).png]]


For triangles pointing up the above neighbourhood is reflected through the y-axis.
For triangles pointing down the above neighbourhood is flipped vertically.


B0 emulation, [[Alternating rule]]s and [[Generations]] are also supported.
B0 emulation, [[Alternating rule]]s and [[Generations]] are also supported.


Triangular rules are notated with an <tt><nowiki>L</nowiki></tt>, <tt><nowiki>LE</nowiki></tt> or <tt><nowiki>LV</nowiki></tt> suffix for Triangular, Triangular Edges and Triangular Vertices neighbourhoods respectively (e.g. <tt><nowiki>B456/S34L</nowiki></tt>). X is used for 10, Y for 11 and Z for 12. This notation avoids conflicts with [[Isotropic_non-totalistic_Life-like_cellular_automaton|Isotropic non-totalistic Moore neighbourhood]] rules.
In the birth/survival notation, Triangular rules are notated with an <tt><nowiki>L</nowiki></tt>, <tt><nowiki>LE</nowiki></tt>, <tt><nowiki>LV</nowiki></tt>, <tt><nowiki>LI</nowiki></tt>, <tt><nowiki>LO</nowiki></tt>, <tt><nowiki>LB</nowiki></tt> or <tt><nowiki>LR</nowiki></tt> suffix for Triangular, Triangular Edges, Triangular Vertices, Triangular Inner, Triangular Outer, Triangular Biohazard or Triangular Radiation neighbourhoods respectively (e.g. <tt><nowiki>B456/S34L</nowiki></tt>). X is used for 10, Y for 11 and Z for 12. This notation avoids conflicts with [[isotropic non-totalistic]] rules on the Moore neighbourhood. However, in [[higher-range outer-totalistic]] notation, only <tt>L</tt> is supported to follow N for Triangular; <tt>LE</tt>, <tt>LV</tt>, <tt>LI</tt>, <tt>LO</tt>, <tt>LB</tt> or <tt>LR</tt> should instead be replaced with weighted or custom neighbourhoods.<ref name="post137325" />


{{filename|TriLife.zip}} is available on Golly's online pattern archive<ref>Golly's [https://github.com/gollygang/ruletablerepository online pattern archive]</ref>, and simulates 2-state triangular outer-totalistic Moore rules using 4 states, dividing each square cell into two triangles. A is used for 10, B for 11 and C for 12.


TriLife.zip is available on Golly's online pattern archive, and simulates 2-state triangular outer-totalistic moore rules using 4 states, dividing each square cell into two triangles. A is used for 10, B for 11 and C for 12.
[[Tim Hutton]]'s [[Fredkin]] replicator rule generation script also divides cells into two triangles, and uses T for triangular von Neumann and TM for triangular Moore.


 
== Symmetries ==
[[Tim Hutton]]'s Fredkin replicator rule generation script also divides cells into two triangles, and uses T for triangular von Neumann and TM for triangular Moore.
{{related|Static symmetry|Kinetic symmetry}}
 
The triangular tiling shares its symmetries with that of the hexagonal tiling, and thus features a different set of inherent symmetries than the [[Moore neighbourhood|Moore]] and [[Von Neumann neighbourhood|von Neumann]] neighbourhoods when dealing with [[isotropic]] rules:
==Symmetries==
{{main|Symmetry}}
The triangular neighbourhood relies on a different grid than the [[Moore]] and [[von Neumann]] neighborhoods and thus features a different set of inherent symmetries when dealing with [[isotropic]] rules:


* Asymmetric (C1, 8x32, 4x64, 2x128, 1x256)
* Asymmetric (C1, 8x32, 4x64, 2x128, 1x256)
Line 48: Line 63:
* D12
* D12


==See also==
== See also ==
* [[Neighbourhood]]
* [[Neighbourhood]]
** [[Hexagonal neighbourhood]]
** [[Hexagonal neighbourhood]]
Line 54: Line 69:
** [[Moore neighbourhood]]
** [[Moore neighbourhood]]
** [[von Neumann neighbourhood]]
** [[von Neumann neighbourhood]]
** [[Euclidean neighbourhood]]
** [[Circular neighbourhood]]
* [[Zone of influence]]
* [[Zone of influence]]


==External links==
== References ==
{{LinkWikipedia|Triangular_tiling|name=Triangular tiling}}
<references>
<ref name="post137325">{{LinkForumThread
|format    = ref
|p          = 137325
|title      = Re: Pattern viewer for forum threads
|author    = Chris Rowett
|date      = November 5, 2021
}}</ref>
</references>
 
== External links ==
* {{LinkWikipedia|Triangular_tiling|name=Triangular tiling}}


[[Category:Neighbourhoods]]
__NOTOC__
__NOTOC__

Latest revision as of 08:33, 27 March 2023

The triangular neighbourhood is the set of all cells that are adjacent to the region of interest in a grid tiled with triangles (the region of interest itself may or may not be considered part of the triangular neighbourhood, depending on context).

Triangular neighbourhoods

The triangular neighbourhood can either refer to:

x=1, y = 1, rule = R1,C0,S,B1,NL o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 12-cell triangular neighbourhood (sometimes referred to as the Triangular Moore neighbourhood)
x=1, y = 1, rule = R2,C0,S,B1,N@06E5C0L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 9-cell triangular vertices neighbourhood
x=1, y = 1, rule = R2,C0,S,B1,N@02a540L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 6-cell triangular outer neighbourhood
x=1, y = 1, rule = R2,C0,S,B1,N@055880L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 6-cell triangular inner neighbourhood
x=1, y = 1, rule = R1,C0,S,B1,N@58L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 3-cell triangular edges neighbourhood (sometimes referred to as the Triangular von Neumann neighbourhood)
x=1, y = 1, rule = R2,C0,S,B1,N@03bd40L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 9-cell triangular biohazard neighbourhood
x=1, y = 1, rule = R2,C0,S,B1,N@044080L o! #C [[ GRID GRIDMAJOR 0 STARTFROM 1 NOREPORT NOGUI ZOOM 8 WIDTH 100 HEIGHT 100 ]] #C [[ COLOR GRID 192 192 192 COLOR BACKGROUND 255 255 255 COLOR ALIVE 192 239 192 COLOR DEAD 0 0 0 ]]
 The 3-cell triangular radiation neighbourhood

Software support

LifeViewer natively supports the seven aforementioned triangular neighbourhoods using 2 states. For triangles pointing up the following neighbourhood is used on a square tiling:

Emulated triangular Moore neighbourhood (range 1).png

For triangles pointing down the above neighbourhood is flipped vertically.

B0 emulation, Alternating rules and Generations are also supported.

In the birth/survival notation, Triangular rules are notated with an L, LE, LV, LI, LO, LB or LR suffix for Triangular, Triangular Edges, Triangular Vertices, Triangular Inner, Triangular Outer, Triangular Biohazard or Triangular Radiation neighbourhoods respectively (e.g. B456/S34L). X is used for 10, Y for 11 and Z for 12. This notation avoids conflicts with isotropic non-totalistic rules on the Moore neighbourhood. However, in higher-range outer-totalistic notation, only L is supported to follow N for Triangular; LE, LV, LI, LO, LB or LR should instead be replaced with weighted or custom neighbourhoods.[1]

TriLife.zip is available on Golly's online pattern archive[2], and simulates 2-state triangular outer-totalistic Moore rules using 4 states, dividing each square cell into two triangles. A is used for 10, B for 11 and C for 12.

Tim Hutton's Fredkin replicator rule generation script also divides cells into two triangles, and uses T for triangular von Neumann and TM for triangular Moore.

Symmetries

See also: Static symmetry, Kinetic symmetry

The triangular tiling shares its symmetries with that of the hexagonal tiling, and thus features a different set of inherent symmetries than the Moore and von Neumann neighbourhoods when dealing with isotropic rules:

  • Asymmetric (C1, 8x32, 4x64, 2x128, 1x256)
  • C2_1
  • C2_4
  • C3_1
  • C3_3
  • C6
  • D2_xo
  • D2_x
  • D4_x1
  • D4_x4
  • D6_1
  • D6_1o
  • D6_3
  • D12

See also

References

  1. ↑ Chris Rowett (November 5, 2021). Re: Pattern viewer for forum threads (discussion thread) at the ConwayLife.com forums
  2. ↑ Golly's online pattern archive

External links

Retrieved from "https://conwaylife.com/w/index.php?title=Triangular_neighbourhood&oldid=127800"