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. 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. The "radiation" neighbourhood similarly splits the grid into four non-interacting grids which can be described with the simpler "edges" neighbourhood.

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
2
3

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

In addition to these 14 static symmetries, periodic patterns can also conform to one of 35 possible distinct kinetic symmetries. 14 of these duplicate the static symmetries above, with the other 21 arising for periodic patterns which have a mod unequal to their period.

"Composite symmetry" refers to the resulting symmetry of the pattern created from each of the oscillator's phases.

Static symmetry	Composite symmetry	period/mod	Description
C1	C2_1	2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on a vertex
C1	C2_4	2	Pattern is asymmetric Appears rotated 180 degrees during (period/2) Rotation is centered on an edge
C1	C3_1	3	Pattern is asymmetric Appears rotated 120 degrees during (period/3) Rotation is centered on a vertex
C1	C3_3	3	Pattern is asymmetric Appears rotated 120 degrees during (period/3) Rotation is centered on a cell
C1	C6	6	Pattern is asymmetric Appears rotated 60 degrees during (period/6) Rotation is centered on a vertex
C2_1	C6	3	Pattern has C2_1 symmetry Appears rotated 60 degrees during (period/3) Rotation is centered on a vertex
C3_1	C6	2	Pattern has C3_1 symmetry Appears rotated 60 degrees during (period/2) Rotation is centered on a vertex
C1	D2_x	2	Pattern is asymmetric Appears flipped across a diagonal line during (period/2) Line passes through vertices and cell centers
C1	D2_xo	2	Pattern is asymmetric Appears flipped across an orthogonal line during (period/2) Line passes through edges only
C2_1	D4_x1	2	Pattern has C2_1 symmetry Appears flipped across both a diagonal and orthogonal line during (period/2) Lines intersect at a vertex
C2_4	D4_x4	2	Pattern has C2_4 symmetry Appears flipped across both a diagonal and orthogonal line during (period/2) Lines intersect at the center of an edge
D2_x	D4_x1	2	Pattern has D2_x symmetry Appears flipped across an orthogonal line during (period/2) Lines intersect at a vertex
D2_x	D4_x4	2	Pattern has D2_x symmetry Appears flipped across an orthogonal line during (period/2) Lines intersect at the center of an edge
D2_xo	D4_x1	2	Pattern has D2_xo symmetry Appears flipped across a diagonal line during (period/2) Lines intersect at a vertex
D2_xo	D4_x4	2	Pattern has D2_xo symmetry Appears flipped across a diagonal line during (period/2) Lines intersect at the center of an edge
C3_1	D6_1	2	Pattern has C3_1 symmetry Appears flipped across one of three diagonal lines during (period/2) Lines intersect at a vertex
C3_1	D6_1o	2	Pattern has C3_1 symmetry Appears flipped across one of three orthogonal lines during (period/2) Lines intersect at a vertex
C3_3	D6_3	2	Pattern has C3_3 symmetry Appears flipped across one of three diagonal lines during (period/2) Lines intersect at the center of a cell
C6	D12	2	Pattern has C6 symmetry Appears flipped across one of three diagonal lines and one of three orthogonal lines during (period/2) Lines intersect at a vertex
D6_1	D12	2	Pattern has D6_1 symmetry Appears rotated 60 degrees during (period/2) Could also be interpreted as flipping on one of three orthogonal lines Rotation is centered on/lines intersect at a vertex
D6_1o	D12	2	Pattern has D6_1o symmetry Appears rotated 60 degrees during (period/2) Could also be interpreted as flipping on one of three diagonal lines Rotation is centered on/lines intersect at a vertex

See also

• Triangular neighbourhood
• Java Triangular Cell

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".