Compound of ten tetrahedra

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Compound of ten tetrahedra
Compound of ten tetrahedra.png
TypeRegular compound
Bowers style acronymE
Components10 tetrahedra
Faces40 triangles as 20 golden hexagrams
Vertex figureGolden hexagram, edge length 1
Measures (edge length 1)
Dihedral angle
Central density10
Number of external pieces180
Level of complexity10
Related polytopes
ArmyDoe, edge length
DualCompound of ten tetrahedra
ConjugateCompound of ten tetrahedra
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Schläfli type{3,3}
SymmetryH3, order 120

The icosicosahedron, e, or compound of ten tetrahedra is a weakly-regular polyhedron compound. It consists of 40 triangles which form 20 coplanar pairs, combining into golden hexagrams. The vertices also coincide in pairs, leading to 20 vertices where 6 triangles join. It can be seen as a compound of the two chiral forms of the chiricosahedron. It can also be seen as a rhombihedron, the compound of five cubes, with each cube replaced by a stella octangula.

This compound is sometimes considered to be regular, but it is not flag-transitive, despite the fact it is vertex, edge, and face-transitive. It is however regular if you consider conjugacies along with its other symmetries.

Its quotient prismatic equivalent is the tetrahedral decayottoorthowedge, which is twelve-dimensional.

Gallery[edit | edit source]


Vertex coordinates[edit | edit source]

Coordinates for the vertices of an icosicosahedron of edge length 1 are given by:

  • ,

plus all even permutations of:

  • .

Related polyhedra[edit | edit source]

It has connections to all weakly regular polyhedra and polyhedron compounds. It can be decomposed into 10 tetrahedra, 5 stella octangulas, or 2 chiricosahedra. It and each chiricosahedron has a dodecahedron convex hull and an icosahedron convex core while each stella octangula has a cube convex hull and an octahedron convex core, which form a rhombihedron and small icosicosahedron respectively.

External links[edit | edit source]

  • Klitzing, Richard. "e".