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.
Truncated order-4 hexagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | 4.12.12 |
Schläfli symbol | t{6,4} tr{6,6} or |
Wythoff symbol | 2 4 | 6 2 6 6 | |
Coxeter diagram | or |
Symmetry group | [6,4], (*642) [6,6], (*662) |
Dual | Order-6 tetrakis square tiling |
Properties | Vertex-transitive |
There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1^{+}], gives [6,6], (*662).
Name | Tetrahexagonal | Truncated hexahexagonal |
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Image | ||
Symmetry | [6,4] (*642) |
[6,6] = [6,4,1^{+}] (*662) = |
Symbol | t{6,4} | tr{6,6} |
Coxeter diagram |
The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group. |
The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 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 subgroup index-8 group, [1^{+},6,1^{+},6,1^{+}] (3333) is the commutator subgroup of [6,6].
Larger subgroup constructed as [6,6^{*}], removing the gyration points of (6*3), index 12 becomes (*333333).
The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, 8 hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.
*n42 symmetry mutation of truncated tilings: 4.2n.2n | |||||||||||
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Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | |||||||
*242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | ||||
Truncated figures |
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Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | |||
n-kis figures |
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Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |
Uniform tetrahexagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
= = = |
= |
= = = |
= |
= = = |
= |
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{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
Uniform duals | |||||||||||
V6^{4} | V4.12.12 | V(4.6)^{2} | V6.8.8 | V4^{6} | V4.4.4.6 | V4.8.12 | |||||
Alternations | |||||||||||
[1^{+},6,4] (*443) |
[6^{+},4] (6*2) |
[6,1^{+},4] (*3222) |
[6,4^{+}] (4*3) |
[6,4,1^{+}] (*662) |
[(6,4,2^{+})] (2*32) |
[6,4]^{+} (642) | |||||
= |
= |
= |
= |
= |
= |
||||||
h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |
Uniform hexahexagonal tilings | ||||||
---|---|---|---|---|---|---|
Symmetry: [6,6], (*662) | ||||||
= = |
= = |
= = |
= = |
= = |
= = |
= = |
{6,6} = h{4,6} |
t{6,6} = h_{2}{4,6} |
r{6,6} {6,4} |
t{6,6} = h_{2}{4,6} |
{6,6} = h{4,6} |
rr{6,6} r{6,4} |
tr{6,6} t{6,4} |
Uniform duals | ||||||
V6^{6} | V6.12.12 | V6.6.6.6 | V6.12.12 | V6^{6} | V4.6.4.6 | V4.12.12 |
Alternations | ||||||
[1^{+},6,6] (*663) |
[6^{+},6] (6*3) |
[6,1^{+},6] (*3232) |
[6,6^{+}] (6*3) |
[6,6,1^{+}] (*663) |
[(6,6,2^{+})] (2*33) |
[6,6]^{+} (662) |
= | = | = | ||||
h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
Small index subgroups of [6,6] (*662) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Index | 1 | 2 | 4 | ||||||||
Diagram | |||||||||||
Coxeter | [6,6] |
[1^{+},6,6] = |
[6,6,1^{+}] = |
[6,1^{+},6] = |
[1^{+},6,6,1^{+}] = |
[6^{+},6^{+}] | |||||
Orbifold | *662 | *663 | *3232 | *3333 | 33× | ||||||
Direct subgroups | |||||||||||
Diagram | |||||||||||
Coxeter | [6,6^{+}] |
[6^{+},6] |
[(6,6,2^{+})] |
[6,1^{+},6,1^{+}] = = = = |
[1^{+},6,1^{+},6] = = = = | ||||||
Orbifold | 6*3 | 2*33 | 3*33 | ||||||||
Direct subgroups | |||||||||||
Index | 2 | 4 | 8 | ||||||||
Diagram | |||||||||||
Coxeter | [6,6]^{+} |
[6,6^{+}]^{+} = |
[6^{+},6]^{+} = |
[6,1^{+},6]^{+} = |
[6^{+},6^{+}]^{+} = [1^{+},6,1^{+},6]^{+} = = = | ||||||
Orbifold | 662 | 663 | 3232 | 3333 | |||||||
Radical subgroups | |||||||||||
Index | 12 | 24 | |||||||||
Diagram | |||||||||||
Coxeter | [6,6*] |
[6*,6] |
[6,6*]^{+} |
[6*,6]^{+} | |||||||
Orbifold | *333333 | 333333 |
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