# Order-4 octagonal tiling

Order-4 octagonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic regular tiling |
| Vertex configuration | 8$^4$ |
| Schläfli symbol | {8,4} r{8,8} |
| Wythoff symbol | 4 | 8 2 |
| Coxeter diagram | or |
| Symmetry group | [8,4], (*842) [8,8], (*882) |
| Dual | Order-8 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

## Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1$^+$], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4$^*$], leaves remaining mirrors *4444 symmetry.
Four uniform constructions of 8.8.8.8 :
| Uniform Coloring | | | | |
| Symmetry | [8,4] (*842) | [8,8] (*882) = | [(8,4,8)] = [8,8,1$^+$] (*884) = = | [1$^+$,8,8,1$^+$] (*4444) = |
| Symbol | {8,4} | r{8,8} | r(8,4,8) = r{8,8}1⁄2 | r{8,4}1⁄8 = r{8,8}1⁄4 |
| Coxeter diagram | | | = = | = = = |

## Symmetry

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*2$^8$) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8$^*$,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
| *444 | *4222 | *832 |
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.

## Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4} vte :
| Spherical | Spherical | Euclidean | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| 2$^4$ | 3$^4$ | 4$^4$ | 5$^4$ | 6$^4$ | 7$^4$ | 8$^4$ | ...∞$^4$ |
Regular tilings: {n,8} vte :
| Spherical | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings | Hyperbolic tilings |
| {2,8} | {3,8} | {4,8} | {5,8} | {6,8} | {7,8} | {8,8} | ... | {∞,8} |
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
| {3,4} | {4,4} | {5,4} | {6,4} | {7,4} | {8,4} | ... | {∞,4} |
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 |