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Compound of five tetrahedra

 

 
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Compound of five tetrahedra
 
 
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This compound polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 is also a stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
 of the regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. It was first described by Edmund Hess
Edmund Hess

Edmund Hess was a Germany mathematician who discovered several regular polytopes.See also* Schl?fli-Hess polychoron* Hess polytope...
 in 1876.

As a compound
It can be constructed by arranging five tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 in icosahedral symmetry
Icosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
 (I), as colored in the upper right model. It is one of five regular compounds which can be constructed from identical Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s.

It shares the same vertex arrangement
Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....
 as a regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
.

There are two enantiomorphous
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 forms (the same figure but having opposite chirality) of this compound polyhedron.






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Encyclopedia


This compound polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 is also a stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
 of the regular icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. It was first described by Edmund Hess
Edmund Hess

Edmund Hess was a Germany mathematician who discovered several regular polytopes.See also* Schl?fli-Hess polychoron* Hess polytope...
 in 1876.

As a compound


It can be constructed by arranging five tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 in icosahedral symmetry
Icosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
 (I), as colored in the upper right model. It is one of five regular compounds which can be constructed from identical Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s.

It shares the same vertex arrangement
Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....
 as a regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
.

There are two enantiomorphous
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 forms (the same figure but having opposite chirality) of this compound polyhedron. Both forms together create the reflection symmetric compound of ten tetrahedra
Compound of ten tetrahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....
.

Compoundoffivetetrahedra
:Transparent Models (Animation)

As a stellation


It can also be obtained by stellating
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
 the icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
, and is given as Wenninger model index 24
List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
.

The stellation facets for construction are:
Second Compound Stellation of Icosahedron Facets

An unusual dual property

This compound is unusual, in that the dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 figure is the enantiomorph of the original. This property seems to have led to a widespread idea that the dual of any chiral
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 figure has the opposite chirality. The idea is generally quite false: a chiral dual nearly always has the same chirality as its twin. For example if a polyhedron has a right hand twist, then its dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 will also have a right hand twist.

In the case of the compound of five tetrahedra, if the faces are twisted to the right then the vertices are twisted to the left. When we dualise
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
, the faces dualise to right-twisted vertices and the verices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare.

See also

Compound of ten tetrahedra
Compound of ten tetrahedra

This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....


External links

  • VRML
    VRML

    VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
     model:
  • by Sándor Kabai, The Wolfram Demonstrations Project.