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Polytope compound

Index Polytope compound

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. [1]

73 relations: Archimedean solid, Catalan solid, Chirality (mathematics), Compound of cube and octahedron, Compound of dodecahedron and icosahedron, Compound of five cubes, Compound of five octahedra, Compound of five tetrahedra, Compound of great icosahedron and great stellated dodecahedron, Compound of small stellated dodecahedron and great dodecahedron, Compound of ten tetrahedra, Compound of three octahedra, Compound of twenty octahedra, Compound of two tetrahedra, Convex hull, Convex polytope, Cube, Cuboctahedron, Dihedral symmetry in three dimensions, Dodecahedron, Dual polyhedron, Faceting, Grand 600-cell, Grand stellated 120-cell, Great 120-cell, Great complex icosidodecahedron, Great dodecahedron, Great grand stellated 120-cell, Great icosahedron, Great snub dodecicosidodecahedron, Group action, Group theory, Harold Scott MacDonald Coxeter, Hexagonal tiling, Hexagram, Hypercubic honeycomb, Icosahedral symmetry, Icosahedron, Icosidodecahedron, Infinite-order apeirogonal tiling, Isogonal figure, Isohedral figure, Isotoxal figure, Johannes Kepler, List of finite spherical symmetry groups, Octahedral symmetry, Octahedron, Pentagonal icositetrahedron, Pentagram, Prism (geometry), ..., Regular Polytopes (book), Rhombic dodecahedron, Rhombic triacontahedron, Rhombitrihexagonal tiling, Schläfli symbol, Small complex icosidodecahedron, Small stellated dodecahedron, Snub cube, Square tiling, Stellated octahedron, Stellation, Subgroup, Tesseract, Tetrahedron, Triakis tetrahedron, Triangular tiling, Truncated tetrahedron, Uniform polyhedron, 120-cell, 16-cell, 24-cell, 5-cell, 600-cell. Expand index (23 more) »

Archimedean solid

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes.

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Catalan solid

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid.

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Chirality (mathematics)

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone.

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Compound of cube and octahedron

This polyhedron can be seen as either a polyhedral stellation or a compound.

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Compound of dodecahedron and icosahedron

In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.

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Compound of five cubes

The compound of five cubes is one of the five regular polyhedral compounds.

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Compound of five octahedra

The compound of five octahedra is one of the five regular polyhedron compounds.

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Compound of five tetrahedra

The compound of five tetrahedra is one of the five regular polyhedral compounds.

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Compound of great icosahedron and great stellated dodecahedron

This polyhedron can be seen as either a polyhedral stellation or a compound.

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Compound of small stellated dodecahedron and great dodecahedron

The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is interior to its dual, the small stellated dodecahedron.

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Compound of ten tetrahedra

The compound of ten tetrahedra is one of the five regular polyhedral compounds.

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Compound of three octahedra

In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other.

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Compound of twenty octahedra

This uniform polyhedron compound is a symmetric arrangement of 20 octahedra (considered as triangular antiprisms).

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Compound of two tetrahedra

In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

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Convex hull

In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.

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Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

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Dihedral symmetry in three dimensions

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn (n ≥ 2).

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Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, 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.

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Faceting

Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.

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Grand 600-cell

In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol.

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Grand stellated 120-cell

In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol.

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Great 120-cell

In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol.

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Great complex icosidodecahedron

In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron.

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Great dodecahedron

In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of.

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Great grand stellated 120-cell

In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol, one of 10 regular Schläfli-Hess 4-polytopes.

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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 snub dodecicosidodecahedron

In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Hexagonal tiling

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

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Hexagram

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol, 2, or.

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Hypercubic honeycomb

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn (or B~n-1) for n>.

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Icosahedral symmetry

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.

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Icosahedron

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

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Icosidodecahedron

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

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Infinite-order apeirogonal tiling

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane.

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Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

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Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same.

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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.

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Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

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List of finite spherical symmetry groups

Finite spherical symmetry groups are also called point groups in three dimensions.

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Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.

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Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

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Pentagonal icositetrahedron

In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube.

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Pentagram

A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a five-pointed star drawn with five straight strokes.

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Prism (geometry)

In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

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Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter.

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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

In geometry, 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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Rhombitrihexagonal tiling

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane.

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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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Small complex icosidodecahedron

In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron.

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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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Snub cube

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles.

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Square tiling

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane.

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Stellated octahedron

The stellated octahedron is the only stellation of the octahedron.

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Stellation

In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure.

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Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

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Tetrahedron

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

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Triakis tetrahedron

In geometry, a triakis tetrahedron (or kistetrahedron) is an Archimedean dual solid, or a Catalan solid.

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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.

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Truncated tetrahedron

In geometry, the truncated tetrahedron is an Archimedean solid.

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Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).

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120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

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16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

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24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

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600-cell

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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4-polytope compound, 4-polytope compounds, Compound polyhedron, Dual compound, Polyhedral compound, Polyhedron compound, Polypolyhedra, Regular 4-polytope compound, Regular 4-polytope compounds, Regular compound polyhedron, Regular polyhedral compound.

References

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

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