# Order-8 triangular tiling

Order-8 triangular tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic regular tiling |
| Vertex configuration | 3$^8$ |
| Schläfli symbol | {3,8} (3,4,3) |
| Wythoff symbol | 8 | 3 2 4 | 3 3 |
| Coxeter diagram | |
| Symmetry group | [8,3], (*832) [(4,3,3)], (*433) [(4,4,4)], (*444) |
| Dual | Octagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

## Uniform colorings

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

## Symmetry

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1$^+$,4,1$^+$,4,1$^+$,4)] (222222) is the commutator subgroup of [(4,4,4)].
A larger subgroup is constructed [(4,4,4$^*$)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).
The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.
Small index subgroups of [(4,4,4)] (*444) :
| Index | 1 | 2 | 2 | 2 | 4 | 4 |
| Diagram | | | | | | |
| Coxeter | [(4,4,4)] | [(1$^+$,4,4,4)] = | [(4,1$^+$,4,4)] = | [(4,4,1$^+$,4)] = | [(1$^+$,4,1$^+$,4,4)] | [(4$^+$,4$^+$,4)] |
| Orbifold | *444 | *4242 | *4242 | *4242 | 2*222 | 222× |
| Diagram | | | | | | |
| Coxeter | | [(4,4$^+$,4)] | [(4,4,4$^+$)] | [(4$^+$,4,4)] | [(4,1$^+$,4,1$^+$,4)] | [(1$^+$,4,4,1$^+$,4)] = |
| Orbifold | | 4*22 | 4*22 | 4*22 | 2*222 | 2*222 |
| Direct subgroups |||||||
| Index | 2 | 4 | 4 | 4 | 8 | 8 |
| Diagram | | | | | | |
| Coxeter | [(4,4,4)]$^+$ | [(4,4$^+$,4)]$^+$ = | [(4,4,4$^+$)]$^+$ = | [(4$^+$,4,4)]$^+$ = | [(4,1$^+$,4,1$^+$,4)]$^+$ = | [(4,1$^+$,4,1$^+$,4)]$^+$ = |
| Orbifold | 444 | 4242 | 4242 | 4242 | 222222 | 222222 |
| Radical subgroups |||||||
| Index | 8 | 8 | 8 | 16 | 16 | 16 |
| Diagram | | | | | | |
| Coxeter | [(4,4*,4)] | [(4,4,4*)] | [(4*,4,4)] | [(4,4*,4)]$^+$ | [(4,4,4*)]$^+$ | [(4*,4,4)]$^+$ |
| Orbifold | *22222222 | *22222222 | *22222222 | 22222222 | 22222222 | 22222222 |

## Related polyhedra and tilings

*n32 symmetry mutation of regular tilings: {3,n} vte :
| Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyper. | Compact hyper. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| 3.3 | 3$^3$ | 3$^4$ | 3$^5$ | 3$^6$ | 3$^7$ | 3$^8$ | 3$^∞$ | 3$^{12i}$ | 3$^{9i}$ | 3$^{6i}$ | 3$^{3i}$ |
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.
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 |
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} |
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 |
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)$^3$ |