Truncated order-4 hexagonal 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.

Truncated order-4 hexagonal tiling
Truncated order-4 hexagonal tiling

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 CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png or CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
Symmetry group [6,4], (*642)
[6,6], (*662)
Dual Order-6 tetrakis square tiling
Properties Vertex-transitive

Constructions

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).

Two uniform constructions of 4.6.4.6
Name Tetrahexagonal Truncated hexahexagonal
Image Uniform tiling 64-t01 Uniform tiling 66-t012
Symmetry [6,4]
(*642)
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[6,6] = [6,4,1+]
(*662)
CDel node c1.pngCDel split1-66.pngCDel nodeab c2.png = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node h0.png
Symbol t{6,4} tr{6,6}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png

Dual tiling

Order-6 tetrakis square tiling Hyperbolic domains 662
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.

Related polyhedra and tiling

Symmetry

Truncated order-4 hexagonal tiling with mirrors
Truncated order-4 hexagonal tiling with *662 mirror lines

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.

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.

See also

External links

Order-4 hexagonal tiling honeycomb

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
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
Spherical square prism Uniform tiling 432-t12 Uniform tiling 44-t01 H2 tiling 245-3 H2 tiling 246-3 H2 tiling 247-3 H2 tiling 248-3 H2 tiling 24i-3
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
Spherical square bipyramid Spherical tetrakis hexahedron 1-uniform 2 dual Order-5 tetrakis square tiling Order-6 tetrakis square tiling Hyperbolic domains 772 Order-8 tetrakis square tiling H2checkers 2ii
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)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-66.pngCDel nodes 11.png
= CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
= CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 246-1 H2 tiling 246-3 H2 tiling 246-2 H2 tiling 246-6 H2 tiling 246-4 H2 tiling 246-5 H2 tiling 246-7
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 246b H2chess 246f H2chess 246a H2chess 246e H2chess 246c H2chess 246d H2checkers 246
V64 V4.12.12 V(4.6)2 V6.8.8 V46 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)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-66.pngCDel branch hh.pngCDel label2.png
CDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-66.pngCDel nodes 10lu.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel 2xa2xb-cross.pngCDel branch hh.pngCDel label2.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 443-t0 Uniform tiling 64-h02 Uniform tiling 64-h1 Uniform tiling 443-snub2 Uniform tiling 66-t0 Uniform tiling 3.4.4.4.4 Uniform tiling 64-snub
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)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel nodes 10ru.pngCDel split2-66.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png = CDel nodes 10ru.pngCDel split2-66.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png = CDel nodes.pngCDel split2-66.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-66.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-66.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png = CDel nodes 11.pngCDel split2-66.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png =CDel nodes 11.pngCDel split2-66.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H2 tiling 266-1 H2 tiling 266-3 H2 tiling 266-2 H2 tiling 266-6 H2 tiling 266-4 H2 tiling 266-5 H2 tiling 266-7
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png
H2chess 266b H2chess 266f H2chess 266a H2chess 266e H2chess 266c H2chess 266d H2checkers 266
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 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)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel branch 10ru.pngCDel split2-66.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h1.pngCDel 6.pngCDel node.png = CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes.png CDel node.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h1.png = CDel node.pngCDel split1-66.pngCDel branch 01ld.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png
Uniform tiling 66-h0 Uniform tiling verf 34343434 Uniform tiling 66-h0 Uniform tiling 64-h1 Uniform tiling 66-snub
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 662 symmetry 000 662 symmetry a00 662 symmetry 00a 662 symmetry 0a0 662 symmetry z0z 662 symmetry xxx
Coxeter [6,6]
CDel node c1.pngCDel 6.pngCDel node c3.pngCDel 6.pngCDel node c2.png
[1+,6,6]
CDel node h0.pngCDel 6.pngCDel node c3.pngCDel 6.pngCDel node c2.png = CDel branch c3.pngCDel split2-66.pngCDel node c2.png
[6,6,1+]
CDel node c1.pngCDel 6.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel split1-66.pngCDel branch c3.png
[6,1+,6]
CDel node c1.pngCDel 6.pngCDel node h0.pngCDel 6.pngCDel node c2.png = CDel branch c1.pngCDel 2a2b-cross.pngCDel branch c2.png
[1+,6,6,1+]
CDel node h0.pngCDel 6.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel branch c3.pngCDel 3a3b-cross.pngCDel branch c3.png
[6+,6+]
CDel node h2.pngCDel 6.pngCDel node h4.pngCDel 6.pngCDel node h2.png
Orbifold *662 *663 *3232 *3333 33×
Direct subgroups
Diagram 662 symmetry 0bb 662 symmetry bb0 662 symmetry b0b 662 symmetry 0ab 662 symmetry ab0
Coxeter [6,6+]
CDel node c1.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h2.png
[6+,6]
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node c2.png
[(6,6,2+)]
CDel node c3.pngCDel split1-66.pngCDel branch h2h2.pngCDel label2.png
[6,1+,6,1+]
CDel node c1.pngCDel 6.pngCDel node h0.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel split1-66.pngCDel branch h2h2.png
= CDel node c1.pngCDel 6.pngCDel node h0.pngCDel 6.pngCDel node h2.png = CDel branch c1.pngCDel 2a2b-cross.pngCDel branch h2h2.png
[1+,6,1+,6]
CDel node h0.pngCDel 6.pngCDel node h0.pngCDel 6.pngCDel node c2.png = CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node c2.png = CDel branch h2h2.pngCDel split2-66.pngCDel node c2.png
= CDel node h2.pngCDel 6.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel branch h2h2.pngCDel 2a2b-cross.pngCDel branch c2.png
Orbifold 6*3 2*33 3*33
Direct subgroups
Index 2 4 8
Diagram 662 symmetry aaa 662 symmetry abb 662 symmetry bba 662 symmetry bab 662 symmetry abc
Coxeter [6,6]+
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h2.png
[6,6+]+
CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h2.png = CDel branch h2h2.pngCDel split2-66.pngCDel node h2.png
[6+,6]+
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel node h2.pngCDel split1-66.pngCDel branch h2h2.png
[6,1+,6]+
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h2h2.pngCDel label2.png = CDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.png
[6+,6+]+ = [1+,6,1+,6]+
CDel node h4.pngCDel split1-66.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel 6.pngCDel node h0.pngCDel 6.pngCDel node h0.png = CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel branch h2h2.pngCDel 3a3b-cross.pngCDel branch h2h2.png
Orbifold 662 663 3232 3333
Radical subgroups
Index 12 24
Diagram 662 symmetry 0zz 662 symmetry zz0 662 symmetry azz 662 symmetry zza
Coxeter [6,6*]
CDel node c1.pngCDel 6.pngCDel node g.pngCDel 6g.pngCDel 3sg.pngCDel node g.png
[6*,6]
CDel node g.pngCDel 6g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node c2.png
[6,6*]+
CDel node h0.pngCDel 6.pngCDel node g.pngCDel 8g.pngCDel 3sg.pngCDel node g.png
[6*,6]+
CDel node g.pngCDel 6g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node h0.png
Orbifold *333333 333333

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