Truncated pentahexagonal tiling

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Truncated pentahexagonal tiling
H2 tiling 256-7.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.12
Schläfli symbol tr{6,5} or
Wythoff symbol 2 6 5 |
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 5.pngCDel node 1.png
Symmetry group [6,5], (*652)
Dual Order 5-6 kisrhombille
Properties Vertex-transitive

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.

Geometry branch of mathematics that measures the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Decagon shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon.

Dodecagon shape with twelve sides

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

Contents

Dual tiling

H2checkers 256.png Hyperbolic domains 652.png
The dual tiling is called an order-5-6 kisrhombille tiling, made as a complete bisection of the order-5 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,5] (*652) symmetry.

Symmetry

There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

Small index subgroups of [6,5], (*652)
Index 126
Diagram 652 symmetry 000.png 652 symmetry a00.png 652 symmetry 0bb.png 652 symmetry 0zz.png
Coxeter
(orbifold)
[6,5] = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 5.pngCDel node c2.png
(*652)
[1+,6,5] = CDel node h0.pngCDel 6.pngCDel node c2.pngCDel 5.pngCDel node c2.png = CDel branch c2.pngCDel split2-55.pngCDel node c2.png
(*553)
[6,5+] = CDel node c1.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
(5*3)
[6,5*] = CDel node c1.pngCDel 6.pngCDel node g.pngCDel 5.pngCDel 3sg.pngCDel node g.png
(*33333)
Direct subgroups
Index2412
Diagram 652 symmetry aaa.png 652 symmetry abb.png 652 symmetry azz.png
Coxeter
(orbifold)
[6,5]+ = CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png
(652)
[6,5+]+ = CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 5.pngCDel node h2.png = CDel branch h2h2.pngCDel split2-55.pngCDel node h2.png
(553)
[6,5*]+ = CDel node h2.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png
(33333)

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling.

Wythoff construction method for constructing a uniform polyhedron or plane tiling

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.

In hyperbolic geometry, a uniformhyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

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 [6,5] symmetry, and 3 with subsymmetry.

See also

Related Research Articles

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 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}.

Truncated tetrahexagonal tiling

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}.

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-6 square tiling

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

Truncated tetrapentagonal tiling

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}.

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}.

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-6 hexagonal tiling

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}

Truncated tetraheptagonal tiling

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

Truncated order-4 octagonal tiling

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.

Rhombitetraoctagonal tiling

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

Truncated tetraoctagonal tiling

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}.

Truncated order-8 octagonal tiling

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}.

Order-6 pentagonal tiling

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

Truncated order-5 hexagonal tiling

In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}.

Truncated order-6 pentagonal tiling

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.

Truncated hexaoctagonal tiling

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}.

Truncated order-6 octagonal tiling

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

Truncated order-8 hexagonal tiling

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

References

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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, and the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus.

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