Truncated order-7 triangular tiling

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Truncated order-7 triangular tiling
Truncated order-7 triangular tiling.svg
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 7.6.6
Schläfli symbol t{3,7}
Wythoff symbol 3
Coxeter diagram CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group [7,3], (*732)
Dual Heptakis heptagonal tiling
Properties Vertex-transitive

In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, [1] is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.

Contents

Hyperbolic soccerball (football)

This tiling is called a hyperbolic soccerball (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space.

Comparison of truncated icosahedron and soccer ball.png
A truncated icosahedron
as a polyhedron and a ball
Uniform tiling 63-t12.png
The Euclidean hexagonal tiling
colored as truncated
triangular tiling
Hyperbolicsoccerball.jpg
A paper construction
of a hyperbolic soccerball

Dual tiling

The dual tiling is called a heptakis heptagonal tiling, named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point.

Heptakis heptagonal tiling.svg

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. CompactParac.Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*32
[,3]
[12i,3][9i,3][6i,3]
Truncated
figures
Hexagonal dihedron.svg Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.svg Truncated order-7 triangular tiling.svg H2-8-3-trunc-primal.svg H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png H2 tiling 23j-6.png
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 .6.6 12i.6.69i.6.66i.6.6
n-kis
figures
Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.svg Heptakis heptagonal tiling.svg H2-8-3-kis-dual.svg H2checkers 33i.png
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6V8.6.6V.6.6V12i.6.6V9i.6.6V6i.6.6

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Heptagonal tiling.svg Truncated heptagonal tiling.svg Triheptagonal tiling.svg Truncated order-7 triangular tiling.svg Order-7 triangular tiling.svg Rhombitriheptagonal tiling.svg Truncated triheptagonal tiling.svg Snub triheptagonal tiling.svg
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Order-7 triangular tiling.svg Order-7 triakis triangular tiling.svg 7-3 rhombille tiling.svg Heptakis heptagonal tiling.svg Heptagonal tiling.svg Deltoidal triheptagonal tiling.svg 3-7 kisrhombille.svg 7-3 floret pentagonal tiling.svg
V73 V3.14.14V3.7.3.7V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

This tiling features prominently in HyperRogue.

See also

Related Research Articles

Heptagonal tiling

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.

Order-7 triangular tiling

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.

Triheptagonal tiling

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Truncated triheptagonal tiling

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.

Truncated heptagonal tiling

In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of t{7,3}. The tiling has a vertex configuration of 3.14.14.

Rhombitriheptagonal tiling

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

Snub triheptagonal tiling

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.

Truncated trioctagonal tiling

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}.

Rhombitetrahexagonal tiling

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.

Truncated order-4 pentagonal tiling

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.

Snub trioctagonal tiling

In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.

Truncated octagonal tiling

In geometry, the Truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.

Truncated order-8 triangular tiling

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

Rhombitrioctagonal tiling

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

Truncated order-5 pentagonal tiling

In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,5}, constructed from one pentagons and two decagons around every vertex.

Order-4 heptagonal tiling

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Truncated order-4 heptagonal tiling

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.

Order-7 heptagonal tiling

In geometry, the order-7 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,7}, constructed from seven heptagons around every vertex. As such, it is self-dual.

Truncated order-7 heptagonal tiling

In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.

Snub heptaheptagonal tiling

In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.

References

  1. HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL