Compound of ten tetrahedra

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Compound of ten tetrahedra
Rank3
TypeRegular compound
SpaceSpherical
Notation
Bowers style acronymE
Elements
Components10 tetrahedra
Faces40 triangles as 20 golden hexagrams
Edges60
Vertices20
Vertex figureGolden hexagram, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt6}{4} \approx 0.61237}$
Inradius${\displaystyle \frac{\sqrt6}{12} \approx 0.20412}$
Volume${\displaystyle \frac{5\sqrt2}{6} \approx 1.17851}$
Dihedral angle${\displaystyle \arccos\left(\frac13\right) \approx 70.52877^\circ}$
Central density10
Number of external pieces180
Level of complexity10
Related polytopes
ArmyDoe, edge length ${\displaystyle \frac{\sqrt{10}-\sqrt2}{4}}$
RegimentE
DualCompound of ten tetrahedra
ConjugateCompound of ten tetrahedra
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Schläfli type{3,3}
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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.

Vertex coordinates

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

• ${\displaystyle \left(\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4},\,\pm\frac{\sqrt2}{4}\right)}$,

plus all even permutations of:

• ${\displaystyle \left(\pm\frac{\sqrt2+\sqrt{10}}{8},\,\pm\frac{\sqrt{10}-\sqrt2}{8},\,0\right)}$.

Related polyhedra

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.