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This uniform polyhedron compound is a composition of 2
Polyhedral compounds {{polyhedron-stub
This uniform polyhedron compound 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 symmetri ...
. It has octahedral symmetry ''Oh''. As a holosnub, it is represented by Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
β and Coxeter diagram .
The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral planes, while the other 24 lie in unique planes.
It shares the same vertex arrangement 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 hexagon, hexagons and 6 Squa ...
, 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
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram .
The vertex arrangement of this compound is shared by a convex nonuniform ...
and compound of two snub dodecahedra
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron.
The vertex arrangement of this compound is shared by a convex nonuniform truncated icosidodecahedron, with rectangular faces, alongside irregular he ...
.
Together with its convex hull, it represents the icosahedron-first projection of the nonuniform snub tetrahedral antiprism.
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of this compound are all the permutations of
: (±1, 0, ±τ)
where τ = (1+)/2 is the golden ratio (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 cubes
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram .
The vertex arrangement of this compound is shared by a convex nonuniform ...
External links
* VRML modelPolyhedral compounds {{polyhedron-stub