# Truncated order-8 hexagonal tiling

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Truncated order-8 hexagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.12.12
Schläfli symbol t{6,8}
Wythoff symbol 6
Coxeter diagram
Symmetry group [8,6], (*862)
Dual Order-6 octakis octagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

## Uniform colorings

This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 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 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86V6.16.16V(6.8)2V8.12.12V68V4.6.4.8V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6}s{8,6}hr{8,6}s{6,8}h{6,8}hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6V3.3.8.3.8.3V(3.4.4.4)2V3.4.3.4.3.6V(3.8)8V3.45V3.3.6.3.8

### Symmetry

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 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. The symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is the commutator subgroup of [(6,6,4)].

A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).

Small index subgroups of [(6,6,4)] (*664)
Fundamental
domains

Subgroup index 124
Coxeter [(6,6,4)]
[(1+,6,6,4)]
[(6,6,1+,4)]
[(6,1+,6,4)]
[(1+,6,6,1+,4)]
[(6+,6+,4)]
Orbifold *664 *6362 *4343 2*3333332×
Coxeter[(6,6+,4)]
[(6+,6,4)]
[(6,6,4+)]
[(6,1+,6,1+,4)]
[(1+,6,1+,6,4)]
Orbifold6*324*333*3232
Direct subgroups
Subgroup index248
Coxeter[(6,6,4)]+
[(1+,6,6+,4)]
[(6+,6,1+,4)]
[(6,1+,6,4+)]
[(6+,6+,4+)] = [(1+,6,1+,6,1+,4)]
=
Orbifold66463624343332332

## Related Research Articles

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.

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.

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h2{4,6}

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

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}.

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.

## References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN   978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN   0-486-40919-8. LCCN   99035678.