# Trioctagonal tiling

Trioctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (3.8)$^2$ |
| Schläfli symbol | r{8,3} or ${\begin{Bmatrix}8\\3\end{Bmatrix}}$ |
| Wythoff symbol | 2 | 8 3| 3 3 | 4 |
| Coxeter diagram | or |
| Symmetry group | [8,3], (*832) [(4,3,3)], (*433) |
| Dual | Order-8-3 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

## Symmetry

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

## Related polyhedra and tilings

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular octagonal 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 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 |
The trioctagonal 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.∞)$^2$ | | | |
Dimensional family of quasiregular polyhedra and tilings: (8.n)$^2$ vte :
| Symmetry *8n2 [n,8] | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Hyperbolic... | Paracompact | Noncompact |
| Symmetry *8n2 [n,8] | *832 [3,8] | *842 [4,8] | *852 [5,8] | *862 [6,8] | *872 [7,8] | *882 [8,8]... | *∞82 [∞,8] | [iπ/λ,8] |
| Coxeter | | | | | | | | |
| Quasiregular figures configuration | 3.8.3.8 | 4.8.4.8 | 8.5.8.5 | 8.6.8.6 | 8.7.8.7 | 8.8.8.8 | 8.∞.8.∞ | 8.∞.8.∞ |