Tetrahexagonal tiling

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Tetrahexagonal tiling
H2 tiling 246-2.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.6)2
Schläfli symbol r{6,4} or
rr{6,6}
r(4,4,3)
t0,1,2,3(,3,,3)
Wythoff symbol 2 | 6 4
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel split1-64.pngCDel nodes.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png or CDel node.pngCDel split1-66.pngCDel nodes 11.png
CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
Symmetry group [6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(,3,,3)], (*3232)
Dual Order-6-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

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.

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Schläfli symbol notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

Contents

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).

Kaleidoscope cylinder with mirrors containing loose, colored objects such as beads or pebbles and bits of glass

A kaleidoscope is an optical instrument with two or more reflecting surfaces tilted to each other in an angle, so that one or more objects on one end of the mirrors are seen as a regular symmetrical pattern when viewed from the other end, due to repeated reflection. The reflectors are usually enclosed in a tube, often containing on one end a cell with loose, colored pieces of glass or other transparent materials to be reflected into the viewed pattern. Rotation of the cell causes motion of the materials, resulting in an ever-changing view being presented.

Four uniform constructions of 4.6.4.6
Uniform
Coloring
H2 tiling 246-2.png H2 tiling 266-5.png H2 tiling 344-5.png 3222-uniform tiling-verf4646.png
Fundamental
Domains
642 symmetry 000.png 642 symmetry 00a.png 642 symmetry a00.png 642 symmetry a0b.png
Schläfli r{6,4}r{4,6}12r{6,4}12r{6,4}14
Symmetry [6,4]
(*642)
CDel node c3.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node c2.png
[6,6] = [6,4,1+]
(*662)
CDel node c3.pngCDel split1-66.pngCDel nodeab c1.png
[(4,4,3)] = [1+,6,4]
(*443)
CDel branch c1.pngCDel split2-44.pngCDel node c2.png
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
CDel labelinfin.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel labelinfin.png or CDel nodeab c1.pngCDel 3a3b-cross.pngCDel nodeab c1.png
Symbolr{6,4}rr{6,6}r(4,3,4)t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-66.pngCDel nodes 11.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel branch 11.pngCDel split2-44.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png =
CDel labelinfin.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel labelinfin.png or CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png

Symmetry

The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.

In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

Hyperbolic domains 3232.png Ord64 qreg rhombic til.png H2chess 246a.png Order-6 hexagonal tiling and dual.png

See also

Square tiling tiling of the plane by squares

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex.

Related Research Articles

Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.

Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

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}.

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 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 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}.

Order-6 hexagonal tiling

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.

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-4 octagonal tiling

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

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.

Tetraoctagonal tiling

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

Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Order-8 octagonal tiling

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.

Rhombitetraapeirogonal tiling

In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.

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}.

Order-6 octagonal tiling

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

Order-8 hexagonal tiling

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

Hexaoctagonal tiling

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

Quarter order-6 square tiling

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642.

References

John Horton Conway British mathematician

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.

International Standard Book Number Unique numeric book identifier

The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

Eric Wolfgang Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He currently works for Wolfram Research, Inc.

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.