# Triheptagonal tiling

Triheptagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (3.7)$^2$ |
| Schläfli symbol | r{7,3} or ${\begin{Bmatrix}7\\3\end{Bmatrix}}$ |
| Wythoff symbol | 2 | 7 3 |
| Coxeter diagram | or |
| Symmetry group | [7,3], (*732) |
| Dual | Order-7-3 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.
Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

## Images

| Klein disk model of this tiling preserves straight lines, but distorts angles | The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex. |

## 7-3 Rhombille

In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.
7-3 rhombile tiling in band model

## Related polyhedra and tilings

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
Quasiregular tilings: (3.n)$^2$ vte :
| Sym. *n32 [n,3] | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n32 [n,3] | *332 [3,3] T$_d$ | *432 [4,3] O$_h$ | *532 [5,3] I$_h$ | *632 [6,3] p6m | *732 [7,3] | *832 [8,3]... | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] |
| Figure | | | | | | | | | | |
| Figure | | | | | | | | | | |
| Vertex | (3.3)$^2$ | (3.4)$^2$ | (3.5)$^2$ | (3.6)$^2$ | (3.7)$^2$ | (3.8)$^2$ | (3.∞)$^2$ | (3.12i)$^2$ | (3.9i)$^2$ | (3.6i)$^2$ |
| Schläfli | r{3,3} | r{3,4} | r{3,5} | r{3,6} | r{3,7} | r{3,8} | r{3,∞} | r{3,12i} | r{3,9i} | r{3,6i} |
| Coxeter | | | | | | | | | | |
| Dual uniform figures ||||||||||||
| Dual conf. | V(3.3)$^2$ | V(3.4)$^2$ | V(3.5)$^2$ | V(3.6)$^2$ | V(3.7)$^2$ | V(3.8)$^2$ | V(3.∞)$^2$ | | | |
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform heptagonal/triangular tilings vte :
| Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | Symmetry: [7,3], (*732) | [7,3]$^+$, (732) |
| {7,3} | t{7,3} | r{7,3} | t{3,7} | {3,7} | rr{7,3} | tr{7,3} | sr{7,3} |
| Uniform duals |||||||||||
| V7$^3$ | V3.14.14 | V3.7.3.7 | V6.6.7 | V3$^7$ | V3.4.7.4 | V4.6.14 | V3.3.3.3.7 |
Dimensional family of quasiregular polyhedra and tilings: (7.n)$^2$ vte :
| Symmetry *7n2 [n,7] | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Paracompact | Noncompact |
| Symmetry *7n2 [n,7] | *732 [3,7] | *742 [4,7] | *752 [5,7] | *762 [6,7] | *772 [7,7] | *872 [8,7]... | *∞72 [∞,7] | [iπ/λ,7] |
| Coxeter | | | | | | | | |
| Quasiregular figures configuration | 3.7.3.7 | 4.7.4.7 | 7.5.7.5 | 7.6.7.6 | 7.7.7.7 | 7.8.7.8 | 7.∞.7.∞ | 7.∞.7.∞ |