Compound of twenty tetrahemihexahedra

Compound of twenty tetrahemihexahedra

Type	Uniform compound Uniform polyhedron compound A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...
Index	UC₁₉
Polyhedra	20 tetrahemihexahedra Tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
Faces	20+60 triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .... s, 60 squares Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
Edges	240
Vertices	60
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	chiral Chirality (mathematics) In geometry, a figure is chiral 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. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object... icosahedral Icosahedral symmetry A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation... (I)
Subgroup Subgroup In group theory, 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 . More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H... restricting to one constituent	3-fold rotational Cyclic symmetries This article deals with the four infinite series of point groups in three dimensions with n-fold rotational symmetry about one axis , and no other rotational symmetry :Chiral:*Cn of order n - n-fold rotational symmetry... (C₃)

This uniform polyhedron compound

Uniform polyhedron compound

A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...

is a symmetric arrangement of 20 tetrahemihexahedra

Tetrahemihexahedron

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....

. It is chiral

Chirality (mathematics)

In geometry, a figure is chiral 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. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

with icosahedral symmetry

Icosahedral symmetry

A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...

(I).

John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra

Uniform polyhedron compound

, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by "adding symmetry to a group in which the basic polyhedron is uniform". Each tetrahemihexahedron in this compound is embedded with symmetry group C₃, which does not act transitively over the tetrahemihexahedron's six vertices. However, the compound as a whole can achieve uniformity because two tetrahemihexahedra coincide at each vertex.

Related polyhedra

This compound shares its edge arrangement with the great dirhombicosidodecahedron

Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...

, the great disnub dirhombidodecahedron

Great disnub dirhombidodecahedron

In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge...

, and the compound of 20 octahedra

Compound of twenty octahedra

This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...

.

The edges and 20 of the triangular faces occur in one enantiomer

Enantiomer

In chemistry, an enantiomer is one of two stereoisomers that are mirror images of each other that are non-superposable , much as one's left and right hands are the same except for opposite orientation. It can be clearly understood if you try to place your hands one over the other without...

of the great snub dodecicosidodecahedron

Great snub dodecicosidodecahedron

In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...

, with the other 60 triangular faces occurring in the other enantiomer.

Convex hull Convex hull In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....	Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...	Great dirhombicosidodecahedron Great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...
Great disnub dirhombidodecahedron Great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge...	Compound of twenty octahedra Compound of twenty octahedra This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...	Compound of twenty tetrahemihexahedra

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