# Tetraoctagonal tiling

Tetraoctagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (4.8)$^2$ |
| Schläfli symbol | r{8,4} or ${\begin{Bmatrix}8\\4\end{Bmatrix}}$ rr{8,8} rr(4,4,4) t$_{0,1,2,3}$(∞,4,∞,4) |
| Wythoff symbol | 2 | 8 4 |
| Coxeter diagram | or or |
| Symmetry group | [8,4], (*842) [8,8], (*882) [(4,4,4)], (*444) [(∞,4,∞,4)], (*4242) |
| Dual | Order-8-4 quasiregular rhombic tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

## Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1$^+$], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1$^+$,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1$^+$,8,4,1$^+$], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Four uniform constructions of 4.8.4.8 :
| Name | Tetra-octagonal tiling | Rhombi-octaoctagonal tiling | | |
| Image | | | | |
| Symmetry | [8,4] (*842) | [8,8] = [8,4,1$^+$] (*882) = | [(4,4,4)] = [1$^+$,8,4] (*444) = | [(∞,4,∞,4)] = [1$^+$,8,4,1$^+$] (*4242) = or |
| Schläfli | r{8,4} | rr{8,8} =r{8,4}$^1$/$_2$ | r(4,4,4) =r{4,8}$^1$/$_2$ | t$_{0,1,2,3}$(∞,4,∞,4) =r{8,4}$^1$/$_4$ |
| Coxeter | | = | = | = or |

## Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

## Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)$^2$ vte :
| Symmetry *4n2 [n,4] | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact | Noncompact |
| Symmetry *4n2 [n,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | [ni,4] |
| Figures | | | | | | | | |
| Config. | (4.3)$^2$ | (4.4)$^2$ | (4.5)$^2$ | (4.6)$^2$ | (4.7)$^2$ | (4.8)$^2$ | (4.∞)$^2$ | (4.ni)$^2$ |
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.∞ |
Uniform octagonal/square tilings vte :
| [8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) |||||||
| {8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} |
| Uniform duals |||||||
| V8$^4$ | V4.16.16 | V(4.8)$^2$ | V8.8.8 | V4$^8$ | V4.4.4.8 | V4.8.16 |
| Alternations |||||||
| [1$^+$,8,4] (*444) | [8$^+$,4] (8*2) | [8,1$^+$,4] (*4222) | [8,4$^+$] (4*4) | [8,4,1$^+$] (*882) | [(8,4,2$^+$)] (2*42) | [8,4]$^+$ (842) |
| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} |
| Alternation duals |||||||
| V(4.4)$^4$ | V3.(3.8)$^2$ | V(4.4.4)$^2$ | V(3.4)$^3$ | V8$^8$ | V4.4$^4$ | V3.3.4.3.8 |
Uniform octaoctagonal tilings vte :
| Symmetry: [8,8], (*882) |||||||
| {8,8} | t{8,8} | r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} |
| Uniform duals |||||||
| V8$^8$ | V8.16.16 | V8.8.8.8 | V8.16.16 | V8$^8$ | V4.8.4.8 | V4.16.16 |
| Alternations |||||||
| [1$^+$,8,8] (*884) | [8$^+$,8] (8*4) | [8,1$^+$,8] (*4242) | [8,8$^+$] (8*4) | [8,8,1$^+$] (*884) | [(8,8,2$^+$)] (2*44) | [8,8]$^+$ (882) |
| h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} |
| Alternation duals |||||||
| V(4.8)$^8$ | V3.4.3.8.3.8 | V(4.4)$^4$ | V3.4.3.8.3.8 | V(4.8)$^8$ | V4$^6$ | V3.3.8.3.8 |
Uniform (4,4,4) tilings vte :
| Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | Symmetry: [(4,4,4)], (*444) | [(4,4,4)]$^+$ (444) | [(1$^+$,4,4,4)] (*4242) | [(4$^+$,4,4)] (4*22) |
| t$_0$(4,4,4) h{8,4} | t$_{0,1}$(4,4,4) h$_2${8,4} | t$_1$(4,4,4) {4,8}$^1$/$_2$ | t$_{1,2}$(4,4,4) h$_2${8,4} | t$_2$(4,4,4) h{8,4} | t$_{0,2}$(4,4,4) r{4,8}$^1$/$_2$ | t$_{0,1,2}$(4,4,4) t{4,8}$^1$/$_2$ | s(4,4,4) s{4,8}$^1$/$_2$ | h(4,4,4) h{4,8}$^1$/$_2$ | hr(4,4,4) hr{4,8}$^1$/$_2$ |
| Uniform duals ||||||||||
| V(4.4)$^4$ | V4.8.4.8 | V(4.4)$^4$ | V4.8.4.8 | V(4.4)$^4$ | V4.8.4.8 | V8.8.8 | V3.4.3.4.3.4 | V8$^8$ | V(4,4)$^3$ |