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Order-4 octagonal tiling, the Glossary

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane.^[1]

23 relations: Checkerboard, Coxeter notation, Coxeter–Dynkin diagram, Euclidean tilings by convex regular polygons, Geometry, Hyperbolic geometry, John Horton Conway, Kaleidoscope, List of Euclidean uniform tilings, List of regular polytopes, Octagon, Octagonal tiling, Octahedron, Orbifold notation, Order-4 pentagonal tiling, Order-8 square tiling, Order-8 triangular tiling, Quasiregular polyhedron, Rhombitetraoctagonal tiling, Schläfli symbol, Square tiling, Truncated order-8 octagonal tiling, Truncated tetraoctagonal tiling.
Isohedral tilings
Octagonal tilings
Order-4 tilings
Regular tilings

Checkerboard

A checkerboard (American English) or chequerboard (British English; see spelling differences) is a game board of checkered pattern on which checkers (also known as English draughts) is played.

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In geometry, Coxeter notation (also Coxeter symbol) 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.

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Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

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Euclidean tilings by convex regular polygons

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. Order-4 octagonal tiling and Euclidean tilings by convex regular polygons are regular tilings.

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Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry.

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John Horton Conway

John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

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Kaleidoscope

A kaleidoscope is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a regular symmetrical pattern when viewed from the other end, due to repeated reflection.

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List of Euclidean uniform tilings

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.

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List of regular polytopes

This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.

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Octagon

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

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Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. Order-4 octagonal tiling and octagonal tiling are hyperbolic tilings, Isogonal tilings, Isohedral tilings, octagonal tilings and regular tilings.

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Octahedron

In geometry, an octahedron (octahedra or octahedrons) is a polyhedron with eight faces.

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Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

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Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. Order-4 octagonal tiling and order-4 pentagonal tiling are hyperbolic tilings, Isogonal tilings, Isohedral tilings, order-4 tilings and regular tilings.

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Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. Order-4 octagonal tiling and order-8 square tiling are hyperbolic tilings, Isogonal tilings, Isohedral tilings and regular tilings.

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Order-8 triangular tiling

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. Order-4 octagonal tiling and order-8 triangular tiling are hyperbolic tilings, Isogonal tilings, Isohedral tilings and regular tilings.

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Quasiregular polyhedron

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex.

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Rhombitetraoctagonal tiling

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. Order-4 octagonal tiling and rhombitetraoctagonal tiling are hyperbolic tilings and Isogonal tilings.

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Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.

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Square tiling

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. Order-4 octagonal tiling and square tiling are Isogonal tilings, Isohedral tilings and regular tilings.

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Truncated order-8 octagonal tiling

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. Order-4 octagonal tiling and truncated order-8 octagonal tiling are hyperbolic tilings, Isogonal tilings and octagonal tilings.

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Truncated tetraoctagonal tiling

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. Order-4 octagonal tiling and truncated tetraoctagonal tiling are hyperbolic tilings and Isogonal tilings.

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References

[1] https://en.wikipedia.org/wiki/Order-4_octagonal_tiling

Also known as 22222222 symmetry, 2^8 symmetry, Octaoctagonal tiling.