Octagonal tiling
# Octagonal tiling

Octagonal tiling :

| Poincaré disk model of the hyperbolic plane ||
| - | - |
| Type | Hyperbolic regular tiling |
| Vertex configuration | $8^3$ |
| Schläfli symbol | {8,3} t{4,8} |
| Wythoff symbol | 3 | 8 2 2 8 | 4 4 4 4 | |
| Coxeter diagram | |
| Symmetry group | [8,3], (*832) [8,4], (*842) [(4,4,4)], (*444) |
| Dual | Order-8 triangular tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

## Uniform colorings

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

| Regular | Truncations | Truncations |
| - | - | - |
| {8,3} | t{4,8} | $t{4^{[3]}}$ = = |
| Dual tiling |||
| {3,8} = | = | = = |

## Regular maps

The regular map ${8,3}_{2,0}$ can be seen as a 6-coloring of the {8,3} hyperbolic tiling. Within the regular map, octagons of the same color are considered the same face shown in multiple locations. The 2,0 subscripts show the same color will repeat by moving 2 steps in a straight direction following opposite edges. This regular map also has a representation as a double covering of a cube, represented by Schläfli symbol {8/2,3}, with 6 octagonal faces, double wrapped {8/2}, with 24 edges, and 16 vertices. It was described by Branko Grünbaum in his 2003 paper Are Your Polyhedra the Same as My Polyhedra?

## Related polyhedra and tilings

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

*n32 symmetry mutation of regular tilings: {n,3} vte :

| Spherical | Spherical | Spherical | Spherical | Euclidean | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| - | - | - | - | - | - | - | - | - | - | - | - |
| {2,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} | {12i,3} | {9i,3} | {6i,3} | {3i,3} |

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: $8^n$ vte :

| Space | Spherical | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracompact |
| - | - | - | - | - | - | - | - | - |
| Tiling | | | | | | | | |
| Config. | 8.8 | $8^3$ | $8^4$ | $8^5$ | $8^6$ | $8^7$ | $8^8$ | $...8^∞$ |

From a Wythoff construction there are ten 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 10 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 | $V3^5.4$ |

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 (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 | $V8^8$ | $V(4,4)^3$ |
