Triangular tiling
The triangular tiling, triangular grid, triangular tessellation or triangular lattice (Schläfli symbol {3,6}) is one of the three possible regular tilings of the plane, alongside the square tiling and hexagonal tiling. It is constructed with six triangles being placed at each vertex.
Of the three regular 2D tilings, it is by far the least investigated, with very little native software support. Native support for triangular rules is offered by some general-purpose cellular automaton simulation programs, although said support is seldom without flaw.
Coordinates, directions and displacements
While Cartesian coordinates can be used to describe the positions of triangles, this does not preserve symmetry well, as it effectively treats the triangular tiling as a square tiling and fails to consider its unique symmetries. While there exist many coordinate systems which can be used to describe the coordinates of triangles in a symmetrical way, such as cube coordinates, none seem to be in use in cellular automaton simulation programs.
Orthogonal and diagonal directions
While the existence of distinct orthogonal and diagonal directions on a triangular grid is not immediately as obvious as on a square grid, they are distinct and have been defined.[1] By analogy to the hexagonal grid, if we consider a vertex of the triangular grid, the six edges radiating out from the vertex are considered as pointing in the six orthogonal directions, whereas the six rays directly between these (passing through cell centers) are considered to be the six diagonal directions.
Neighbourhoods on the triangular tiling
For what is considered to be a range of 1, the triangular tiling has the greatest diversity of neighbourhoods: there are effectively three types of cells adjacent to the central cell, giving a total of seven unique neighbourhoods (excluding the null case). These are listed on the Triangular neighbourhood page. However, the six-cell "outer" neighbourhood is effectively trivial, as it emulates two non-interacting hexagonal grids.[2] The "radiation" neighbourhood similarly splits the grid into four non-interacting grids which can be described with the simpler "edges" neighbourhood.
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| von Neumann |
outer (trivial) |
biohazard | radiation (trivial) |
inner | vertices | Moore |
|---|
For higher ranges, only the 12-cell "triangular Moore" neighbourhood is usually considered.
| Range \ Type | triangular Moore |
|---|---|
| 1 |
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| 2 | .png)
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| 3 | .png)
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Symmetries
All of the static and kinetic symmetries which are present on the triangular tiling also apply to the hexagonal tiling.
The ratio of a pattern's mod to its period, for rules on these grids, can only be 1, 2, 3 or 6.
Static
Isotropic rules on the triangular tiling will always confirm to one of 14 different static symmetries. Its sixfold rotational and reflectional symmetry permits asymmetry, twofold cyclic symmetry, threefold cyclic symmetry, sixfold cyclic symmetry, twofold mirror symmetry, fourfold mirror symmetry, sixfold mirror symmetry and twelvefold mirror symmetry, with further variants depending on the positioning of the center of rotation or the positioning and/or rotation of the planes of reflection.
| Asymmetry | Rotational symmetries | ||||
|---|---|---|---|---|---|
| Onefold | Twofold | Threefold | Sixfold | ||
| Cell-centered | Edge-centered | Cell-centered | Vertex-centered | ||
| C1 | C2_1 | C2_4 | C3_1 | C3_3 | C6 |
| Reflectional symmetries | Rotational and reflectional symmetries | ||||||
|---|---|---|---|---|---|---|---|
| Twofold | Fourfold | Sixfold | Twelvefold | ||||
| Diagonal | Orthogonal | Vertex-centered | Edge-centered | Vertex-centered | Cell-centered | ||
| Diagonal | Orthogonal | ||||||
| D2_x | D2_xo | D4_x1 | D4_x4 | D6_1 | D6_1o | D6_3 | D12 |
Kinetic
See also
References
Further reading
- Forum threads
- Triangular neighborhood (Carter Bays' TriLife rules) (discussion thread) at the ConwayLife.com forums
- Triangular & Tripod von Neuman neighborhoods (discussion thread) at the ConwayLife.com forums
- Triangular neighbourhood rules (discussion thread) at the ConwayLife.com forums
- Defining coordinates and directions on hexagonal/triangular grids (discussion thread) at the ConwayLife.com forums
- Other
- Triangular tiling at Wikipedia
- "Grid parts and relationships: Triangle Grids".