| Truncated triheptagonal tiling | |
|---|---|
 Poincaré disk model of thehyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.6.14 |
| Schläfli symbol | tr{7,3} or |
| Wythoff symbol | 2 7 3 | |
| Coxeter diagram |    or  |
| Symmetry group | [7,3], (*732) |
| Dual | Order 3-7 kisrhombille |
| Properties | Vertex-transitive |
Ingeometry, thetruncated triheptagonal tiling is a semiregulartiling of thehyperbolic plane. There is onesquare, onehexagon, and onetetradecagon (14-sides) on eachvertex. It hasSchläfli symbol oftr{7,3}.
There is only oneuniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)
Eachtriangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of theWythoff construction for the symmetry group [7,3].
 |  | |
| The dual tiling is called anorder-3 bisected heptagonal tiling, made as a complete bisection of theheptagonal tiling, here shown with triangles with alternating colors. | ||
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) andCoxeter-Dynkin diagram
. Forp < 6, the members of the sequence areomnitruncated polyhedra (zonohedrons), shown below as spherical tilings. Forp > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
| *n32 symmetry mutation of omnitruncated tilings: 4.6.2n | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sym. *n32 [n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| *232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3] | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] | [3i,3] | |
| Figures |  |  |  |  |  | |  |  |  |  |  |  |
| Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |
| Duals |  |  |  |  |  | |  |  |  |  |  |  |
| Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |
From aWythoff construction there are eight hyperbolicuniform 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 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | |||||||||||
 | | | | | | |  | ||||
|  |  |  | |  |  |  | ||||
| V73 | V3.14.14 | V3.7.3.7 | V6.6.7 | V37 | V3.4.7.4 | V4.6.14 | V3.3.3.3.7 | ||||
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