# Truncated tetraapeirogonal tiling

Truncated tetraapeirogonal tiling :
| Poincaré disk model of the hyperbolic plane ||
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.∞ |
| Schläfli symbol | tr{∞,4} or $t{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$ |
| Wythoff symbol | 2 ∞ 4 | |
| Coxeter diagram | or |
| Symmetry group | [∞,4], (*∞42) |
| Dual | Order 4-infinite kisrhombille |
| Properties | Vertex-transitive |
In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

## Related polyhedra and tilings

Paracompact uniform tilings in [∞,4] family vte :
| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} |
| Dual figures |||||||
| V∞$^4$ | V4.∞.∞ | V(4.∞)$^2$ | V8.8.∞ | V4$^∞$ | V4$^3$.∞ | V4.8.∞ |
| Alternations |||||||
| [1$^+$,∞,4] (*44∞) | [∞$^+$,4] (∞*2) | [∞,1$^+$,4] (*2∞2∞) | [∞,4$^+$] (4*∞) | [∞,4,1$^+$] (*∞∞2) | [(∞,4,2$^+$)] (2*2∞) | [∞,4]$^+$ (∞42) |
| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} |
| Alternation duals |||||||
| V(∞.4)$^4$ | V3.(3.∞)$^2$ | V(4.∞.4)$^2$ | V3.∞.(3.4)$^2$ | V∞$^∞$ | V∞.4$^4$ | V3.3.4.3.∞ |
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n vte :
| Symmetry *n42 [n,4] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *n42 [n,4] | *242 [2,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] |
| Omnitruncated figure | 4.8.4 | 4.8.6 | 4.8.8 | 4.8.10 | 4.8.12 | 4.8.14 | 4.8.16 | 4.8.∞ |
| Omnitruncated duals | V4.8.4 | V4.8.6 | V4.8.8 | V4.8.10 | V4.8.12 | V4.8.14 | V4.8.16 | V4.8.∞ |
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n vte :
| Symmetry *nn2 [n,n] | Spherical | Spherical | Euclidean | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Compact hyperbolic | Paracomp. |
| Symmetry *nn2 [n,n] | *222 [2,2] | *332 [3,3] | *442 [4,4] | *552 [5,5] | *662 [6,6] | *772 [7,7] | *882 [8,8]... | *∞∞2 [∞,∞] |
| Figure | | | | | | | | |
| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ |
| Dual | | | | | | | | |
| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |

## Symmetry

The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. 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. The subgroup index-8 group, [1$^+$,∞,1$^+$,4,1$^+$] (∞2∞2) is the commutator subgroup of [∞,4].
A larger subgroup is constructed as [∞,4*], index 8, as [∞,4$^+$], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞$^4$), and another [∞*,4], index ∞ as [∞$^+$,4], (∞*2) with gyration points removed as (*2$^∞$). And their direct subgroups [∞,4*]$^+$, [∞*,4]$^+$, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2$^∞$).
Small index subgroups of [∞,4], (*∞42) :
| Index | 1 | 2 | 2 | 2 | 4 | 4 |
| Diagram | | | | | | |
| Coxeter | [∞,4] | [1$^+$,∞,4] = | [∞,4,1$^+$] = | [∞,1$^+$,4] = | [1$^+$,∞,4,1$^+$] = | [∞$^+$,4$^+$] |
| Orbifold | *∞42 | *∞44 | *∞∞2 | *∞222 | *∞2∞2 | ∞2× |
| Semidirect subgroups |||||||
| Diagram | | | | | | |
| Coxeter | | [∞,4$^+$] | [∞$^+$,4] | [(∞,4,2$^+$)] | [1$^+$,∞,1$^+$,4] = = = = | [∞,1$^+$,4,1$^+$] = = = = |
| Orbifold | | 4*∞ | ∞*2 | 2*∞2 | ∞*22 | 2*∞∞ |
| Direct subgroups ||||||||
| Index | 2 | 4 | 4 | 4 | 8 | 8 |
| Diagram | | | | | | |
| Coxeter | [∞,4]$^+$ = | [∞,4$^+$]$^+$ = | [∞$^+$,4]$^+$ = | [∞,1$^+$,4]$^+$ = | [∞$^+$,4$^+$]$^+$ = [1$^+$,∞,1$^+$,4,1$^+$] = = = | [∞$^+$,4$^+$]$^+$ = [1$^+$,∞,1$^+$,4,1$^+$] = = = |
| Orbifold | ∞42 | ∞44 | ∞∞2 | ∞222 | ∞2∞2 | ∞2∞2 |
| Radical subgroups |||||||
| Index | | 8 | ∞ | 16 | ∞ |
| Diagram | | | | | |
| Coxeter | | [∞,4*] = | [∞*,4] | [∞,4*]$^+$ = | [∞*,4]$^+$ |
| Orbifold | | *∞∞∞∞ | *2$^∞$ | ∞∞∞∞ | 2$^∞$ |