# 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).

The triangular neigbourhood can either refer to:

- The 12-cell triangular neighbourhood (sometimes referred to as the Triangular Moore neighbourhood)

- The 9-cell triangular vertices neighbourhood

- The 3-cell triangular edges neighbourhood (sometimes referred to as the Triangular von Neumann neighbourhood).

The triangular tiling shares its symmetries with that of the hexagonal tiling.

## 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:

For triangles pointing up the above neighbourhood is flipped vertically.

B0 emulation, Alternating rules and Generations are also supported.

Triangular rules are notated with an `L`, `LE` or `LV` suffix for Triangular, Triangular Edges and Triangular Vertices 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 Moore neighbourhood rules.

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

*Main article: 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)
- 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

## External links

- Triangular tiling at Wikipedia