List of isotoxal polyhedra and tilings, the Glossary

Table of Contents

1. 74 relations: Convex polytope, Coxeter–Dynkin diagram, Cube, Cuboctahedron, Cubohemioctahedron, Dihedron, Ditrigonal dodecadodecahedron, Dodecadodecahedron, Dodecahedron, Dual polyhedron, Edge-transitive graph, Geometry, Great ditrigonal icosidodecahedron, Great dodecahedron, Great dodecahemicosahedron, Great dodecahemidodecacron, Great dodecahemidodecahedron, Great icosahedron, Great icosidodecahedron, Great icosihemidodecacron, Great icosihemidodecahedron, Great rhombic triacontahedron, Great stellated dodecahedron, Great triambic icosahedron, Hemipolyhedron, Heptagonal tiling, Hexagonal tiling, Hosohedron, Icosahedron, Icosidodecahedron, Isotoxal figure, Kepler–Poinsot polyhedron, Medial rhombic triacontahedron, Octagonal tiling, Octahedron, Octahemioctahedron, Order-4 hexagonal tiling, Order-4 octagonal tiling, Order-4 pentagonal tiling, Order-5 pentagonal tiling, Order-5 square tiling, Order-6 square tiling, Order-7 triangular tiling, Order-8 square tiling, Order-8 triangular tiling, Platonic solid, Polyhedra (book), Polyhedron, Quasiregular polyhedron, Regular polyhedron, ... Expand index (24 more) »

2. Isotoxal tilings

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.

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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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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces.

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Cuboctahedron

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

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Cubohemioctahedron

In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15.

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Dihedron

A dihedron is a type of polyhedron, made of two polygon faces which share the same set of n edges. List of isotoxal polyhedra and tilings and dihedron are polyhedra.

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Ditrigonal dodecadodecahedron

In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41.

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Dodecadodecahedron

In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.

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Dodecahedron

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces.

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Dual polyhedron

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. List of isotoxal polyhedra and tilings and dual polyhedron are polyhedra.

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Edge-transitive graph

In the mathematical field of graph theory, an edge-transitive graph is a graph such that, given any two edges and of, there is an automorphism of that maps to.

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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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Great ditrigonal icosidodecahedron

In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47.

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Great dodecahedron

In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedrons.

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Great dodecahemicosahedron

In geometry, the great dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65.

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Great dodecahemidodecacron

In geometry, the great dodecahemidodecacron is the dual of the great dodecahemidodecahedron, and is one of nine dual hemipolyhedra.

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Great dodecahemidodecahedron

In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70.

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Great icosahedron

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of.

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Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.

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Great icosihemidodecacron

In geometry, the great icosihemidodecacron is the dual of the great icosihemidodecahedron, and is one of nine dual hemipolyhedra.

See List of isotoxal polyhedra and tilings and Great icosihemidodecacron

Great icosihemidodecahedron

In geometry, the great icosihemidodecahedron (or great icosahemidodecahedron) is a nonconvex uniform polyhedron, indexed as U71.

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Great rhombic triacontahedron

In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron.

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Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol.

See List of isotoxal polyhedra and tilings and Great stellated dodecahedron

Great triambic icosahedron

In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. List of isotoxal polyhedra and tilings and great triambic icosahedron are polyhedra.

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Hemipolyhedron

In geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center.

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Heptagonal tiling

In geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane.

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Hexagonal tiling

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex.

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Hosohedron

In spherical geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. List of isotoxal polyhedra and tilings and hosohedron are polyhedra.

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Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

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Icosidodecahedron

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces.

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Isotoxal figure

In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. List of isotoxal polyhedra and tilings and isotoxal figure are polyhedra.

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Kepler–Poinsot polyhedron

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

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In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron.

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Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane.

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Octahedron

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

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Octahemioctahedron

In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as.

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Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane.

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Order-4 octagonal tiling

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

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Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane.

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Order-5 pentagonal tiling

In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane.

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Order-5 square tiling

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane.

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Order-6 square tiling

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane.

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Order-7 triangular tiling

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

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Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane.

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Order-8 triangular tiling

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane.

See List of isotoxal polyhedra and tilings and Order-8 triangular tiling

Platonic solid

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space.

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Polyhedra (book)

Polyhedra is a book on polyhedra, by Peter R. Cromwell. List of isotoxal polyhedra and tilings and polyhedra (book) are polyhedra.

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Polyhedron

In geometry, a polyhedron (polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. List of isotoxal polyhedra and tilings and polyhedron are polyhedra.

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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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Regular polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.

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Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.

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Rhombic triacontahedron

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces.

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Rhombille tiling

In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. List of isotoxal polyhedra and tilings and rhombille tiling are isotoxal tilings.

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Small ditrigonal icosidodecahedron

In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30.

See List of isotoxal polyhedra and tilings and Small ditrigonal icosidodecahedron

Small dodecahemicosacron

In geometry, the small dodecahemicosacron is the dual of the small dodecahemicosahedron, and is one of nine dual hemipolyhedra.

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Small dodecahemidodecacron

In geometry, the small dodecahemidodecacron is the dual of the small dodecahemidodecahedron, and is one of nine dual hemipolyhedra.

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Small dodecahemidodecahedron

In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as.

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Small icosihemidodecacron

In geometry, the small icosihemidodecacron is the dual of the small icosihemidodecahedron, and is one of nine dual hemipolyhedra.

See List of isotoxal polyhedra and tilings and Small icosihemidodecacron

Small icosihemidodecahedron

In geometry, the small icosihemidodecahedron (or small icosahemidodecahedron) is a uniform star polyhedron, indexed as. List of isotoxal polyhedra and tilings and small icosihemidodecahedron are polyhedra.

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Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol.

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Small triambic icosahedron

In geometry, the small triambic icosahedron is a star polyhedron composed of 20 intersecting non-regular hexagon faces.

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Square tiling

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. List of isotoxal polyhedra and tilings and square tiling are polyhedra.

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Tessellation

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. List of isotoxal polyhedra and tilings and tessellation are polyhedra.

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Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

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Tetrahemihexahedron

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4.

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Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. List of isotoxal polyhedra and tilings and tetrahexagonal tiling are isotoxal tilings.

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Tetrapentagonal tiling

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. List of isotoxal polyhedra and tilings and tetrapentagonal tiling are isotoxal tilings.

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Triangular tiling

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons.

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Triheptagonal tiling

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. List of isotoxal polyhedra and tilings and triheptagonal tiling are isotoxal tilings.

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Trihexagonal tiling

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. List of isotoxal polyhedra and tilings and trihexagonal tiling are isotoxal tilings.

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Trioctagonal tiling

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. List of isotoxal polyhedra and tilings and trioctagonal tiling are isotoxal tilings.

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Vertex configuration

In geometry, a vertex configuration by Walter Steurer, Sofia Deloudi, (2009) pp. List of isotoxal polyhedra and tilings and vertex configuration are polyhedra.

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Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. List of isotoxal polyhedra and tilings and vertex figure are polyhedra.

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Wythoff symbol

In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. List of isotoxal polyhedra and tilings and Wythoff symbol are polyhedra.

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See also

Isotoxal tilings

• Hexaoctagonal tiling
• List of isotoxal polyhedra and tilings
• Pentahexagonal tiling
• Regular tilings
• Rhombille tiling
• Tetraheptagonal tiling
• Tetrahexagonal tiling
• Tetraoctagonal tiling
• Tetrapentagonal tiling
• Triheptagonal tiling
• Trihexagonal tiling
• Trioctagonal tiling

References

[1] https://en.wikipedia.org/wiki/List_of_isotoxal_polyhedra_and_tilings

, Rhombic dodecahedron, Rhombic triacontahedron, Rhombille tiling, Small ditrigonal icosidodecahedron, Small dodecahemicosacron, Small dodecahemidodecacron, Small dodecahemidodecahedron, Small icosihemidodecacron, Small icosihemidodecahedron, Small stellated dodecahedron, Small triambic icosahedron, Square tiling, Tessellation, Tetrahedron, Tetrahemihexahedron, Tetrahexagonal tiling, Tetrapentagonal tiling, Triangular tiling, Triheptagonal tiling, Trihexagonal tiling, Trioctagonal tiling, Vertex configuration, Vertex figure, Wythoff symbol.