# Truncated order-8 triangular tiling

Truncated order-8 triangular tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 8.6.6 |
| Schläfli symbol | t{3,8} |
| Wythoff symbol | 2 8 | 3 4 3 3 | |
| Coxeter diagram | |
| Symmetry group | [8,3], (*832) [(4,3,3)], (*433) |
| Dual | Octakis octagonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

## Uniform colors

| The half symmetry [1$^+$,8,3] = [(4,3,3)] can be shown with alternating two colors of hexagons | Dual tiling |

## Symmetry

The dual of this tiling represents the fundamental domains of *443 symmetry. It only has one subgroup 443, replacing mirrors with gyration points.
This symmetry can be doubled to 832 symmetry by adding a bisecting mirror to the fundamental domain.
Small index subgroups of [(4,3,3)], (*433) :
| Type | Reflectional | Rotational |
| Index | 1 | 2 |
| Diagram | | |
| Coxeter (orbifold) | [(4,3,3)] = (*433) | [(4,3,3)]$^+$ = (433) |

## Related tilings

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.
Uniform octagonal/triangular tilings vte :
| Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | Symmetry: [8,3], (*832) | [8,3]$^+$ (832) | [1$^+$,8,3] (*443) | [1$^+$,8,3] (*443) | [8,3$^+$] (3*4) |
| {8,3} | t{8,3} | r{8,3} | t{3,8} | {3,8} | rr{8,3} s$_2${3,8} | tr{8,3} | sr{8,3} | h{8,3} | h$_2${8,3} | s{3,8} |
| | | | | | | | | or | or | |
| Uniform duals |||||||||||
| V8$^3$ | V3.16.16 | V3.8.3.8 | V6.6.8 | V3$^8$ | V3.4.8.4 | V4.6.16 | V3$^4$.8 | V(3.4)$^3$ | V8.6.6 | V3$^5$.4 |
It can also be generated from the (4 3 3) hyperbolic tilings:
Uniform (4,3,3) tilings vte :
| Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | Symmetry: [(4,3,3)], (*433) | [(4,3,3)]$^+$, (433) |
| h{8,3} t$_0$(4,3,3) | r{3,8}$^1$/$_2$ t$_{0,1}$(4,3,3) | h{8,3} t$_1$(4,3,3) | h$_2${8,3} t$_{1,2}$(4,3,3) | {3,8}$^1$/$_2$ t$_2$(4,3,3) | h$_2${8,3} t$_{0,2}$(4,3,3) | t{3,8}$^1$/$_2$ t$_{0,1,2}$(4,3,3) | s{3,8}$^1$/$_2$ s(4,3,3) |
| Uniform duals ||||||||
| V(3.4)$^3$ | V3.8.3.8 | V(3.4)$^3$ | V3.6.4.6 | V(3.3)$^4$ | V3.6.4.6 | V6.6.8 | V3.3.3.3.3.4 |
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated tilings: n.6.6 vte :
| Sym. *n42 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact | Compact | Parac. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n42 [n,3] | *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] |
| Truncated figures | | | | | | | | | | | |
| Config. | 2.6.6 | 3.6.6 | 4.6.6 | 5.6.6 | 6.6.6 | 7.6.6 | 8.6.6 | ∞.6.6 | 12i.6.6 | 9i.6.6 | 6i.6.6 |
| n-kis figures | | | | | | | | | | | |
| Config. | V2.6.6 | V3.6.6 | V4.6.6 | V5.6.6 | V6.6.6 | V7.6.6 | V8.6.6 | V∞.6.6 | V12i.6.6 | V9i.6.6 | V6i.6.6 |
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n vte :
| Sym. *n43 [(n,4,3)] | Spherical | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paraco. |
| Sym. *n43 [(n,4,3)] | *243 [4,3] | *343 [(3,4,3)] | *443 [(4,4,3)] | *543 [(5,4,3)] | *643 [(6,4,3)] | *743 [(7,4,3)] | *843 [(8,4,3)] | *∞43 [(∞,4,3)] |
| Figures | | | | | | | | |
| Config. | 4.8.6 | 6.8.6 | 8.8.6 | 10.8.6 | 12.8.6 | 14.8.6 | 16.8.6 | ∞.8.6 |
| Duals | | | | | | | | |
| Config. | V4.8.6 | V6.8.6 | V8.8.6 | V10.8.6 | V12.8.6 | V14.8.6 | V16.8.6 | V6.8.∞ |