Order-6 octagonal tiling

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Order-6 octagonal tiling
H2 tiling 268-1.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 86
Schläfli symbol {8,6}
Wythoff symbol 6 | 8 2
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
Symmetry group [8,6], (*862)
Dual Order-8 hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

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

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 Non-Euclidean 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

Symmetry

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *33333333 with 8 order-3 mirror intersections. In Coxeter notation can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [8,6] symmetry.

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.

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.

Coxeter notation system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation, with modifiers to indicate certain subgroups

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

Uniform constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [8,6,1+], gives [(8,8,3)], (*883). Removing two mirrors as [8,6*], leaves remaining mirrors (*444444).

Four uniform constructions of 8.8.8.8
Uniform
Coloring
H2 tiling 268-1.png H2 tiling 288-2.png H2 tiling 688-5.png
Symmetry [8,6]
(*862)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 6.pngCDel node c3.png
[8,6,1+] = [(8,8,3)]
(*883)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel branch c2.png
[8,1+,6]
(*4232)
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch c2.png
[8,6*]
(*444444)
CDel node c1.pngCDel 8.pngCDel node g.pngCDel 6sg.pngCDel node g.png
Symbol{8,6}{8,6}12r(8,6,8)
Coxeter
diagram
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-88.pngCDel branch.pngCDel node 1.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node.png = CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch.pngCDel node 1.pngCDel 8.pngCDel node g.pngCDel 6sg.pngCDel node g.png

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

Octagon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

n82 symmetry mutations of regular tilings: 8n
SpaceSphericalCompact hyperbolicParacompact
Tiling H2 tiling 238-1.png H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png H2 tiling 288-4.png H2 tiling 28i-4.png
Config. 8.8 83 84 85 86 87 88 ...8

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.

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}.

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}.

Order-6 square tiling

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

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-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.

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}.

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}.

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}.

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}.

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.

Order-4 apeirogonal tiling

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

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}.

Order-5 apeirogonal tiling

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

Order-6 apeirogonal tiling

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

References

John Horton Conway British mathematician

John Horton Conway 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.