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Polyhedral compounds {{polyhedron-stub
This uniform polyhedron compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...
is a composition of 2 icosahedra
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
. It has octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
''Oh''. As a holosnub, it is represented by Schläfli symbol β and Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
.
The triangles in this compound decompose into two orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
planes, while the other 24 lie in unique planes.
It shares the same 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.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
as a nonuniform truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
, having irregular hexagons alternating with long and short edges.
The icosahedron, as a uniform ''snub tetrahedron'', is similar to these snub-pair compounds: compound of two snub cubes and compound of two snub dodecahedra.
Together with its convex hull, it represents the icosahedron-first projection of the nonuniform snub tetrahedral antiprism.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the permutations of : (±1, 0, ±τ) where τ = (1+)/2 is thegolden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
(sometimes written φ).
Compound of two dodecahedra
The dual compound has two dodecahedra as pyritohedra in dual positions: :
References
*.See also
* Compound of two snub cubesExternal links
*VRML
VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...
modelPolyhedral compounds {{polyhedron-stub