Triangular neighbourhood
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:
| The 12-cell triangular neighbourhood (sometimes referred to as the Triangular Moore neighbourhood) | ||
| The 9-cell triangular vertices neighbourhood | ||
| The 6-cell triangular outer neighbourhood | ||
| The 6-cell triangular inner neighbourhood | ||
| The 3-cell triangular edges neighbourhood (sometimes referred to as the Triangular von Neumann neighbourhood) | ||
| The 9-cell triangular biohazard neighbourhood | ||
| 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:
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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
- ↑ Chris Rowett (November 5, 2021). Re: Pattern viewer for forum threads (discussion thread) at the ConwayLife.com forums
- ↑ Golly's online pattern archive
External links
- Triangular tiling at Wikipedia