Triangular tiling
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Triangular tiling | |
---|---|
Rank | 3 |
Type | Regular |
Space | Euclidean |
Notation | |
Bowers style acronym | Trat |
Coxeter diagram | o6o3x () |
Schläfli symbol | {3,6} |
Elements | |
Faces | 2N triangles |
Edges | 3N |
Vertices | N |
Vertex figure | Hexagon, edge length 1 |
Petrie polygons | Zigzags |
Holes | Hexagons |
Measures (edge length 1) | |
Vertex density | |
Related polytopes | |
Army | Trat |
Regiment | Trat |
Dual | Hexagonal tiling |
Petrie dual | Petrial triangular tiling |
φ 2 | Hexagonal tiling |
Conjugate | None |
Abstract & topological properties | |
Flag count | 12N |
Surface | Sphere |
Orientable | Yes |
Genus | 0 |
Properties | |
Symmetry | V3 |
Convex | Yes |
Nature | Tame |
The triangular tiling, or trat, is one of the three regular tilings of the Euclidean plane. 6 triangles join at each vertex of this tiling. It is also the 2-dimensional simplicial honeycomb. It is also the alternation of the hexagonal tiling.
Vertex coordinates[edit | edit source]
The vertices of a triangular tiling of edge length 1 are given by
- ,
where i and j range over the integers.
Integral vertex coordinates for the triangular tiling can be given in 3D as:
- ,
where i and j range over the integers.
Representations[edit | edit source]
A triangular tiling has the following Coxeter diagrams:
- o6o3x () (full symmetry)
- x3o3o3*a () (P3 symmetry, triangles considered of two types)
- s6o3o () (alternated hexagonal tilling)
- o6s3s ()
- s3s3s3*a ()
- xdoo3xodo3xood&#zx (as hull of hexagonal tiling and three larger triangular tilings)
In vertex figures[edit | edit source]
Name | Picture | Schläfli symbol |
---|---|---|
Tetrahedral honeycomb | {3,3,6} | |
Order-6 cubic honeycomb | {4,3,6} | |
Order-6 dodecahedral honeycomb | {5,3,6} | |
Order-6 hexagonal tiling honeycomb | {6,3,6} |
Related tiling[edit | edit source]
The triangular tiling is the colonel of a two-member regiment that also includes the ditrigonary triangular-hemiapeirogonal tiling. Also in this regiment is a compound of three hexagonal tilings.
External links[edit | edit source]
- Klitzing, Richard. "trat".
- Wikipedia contributors. "Triangular tiling".