Compound of ten tetrahedra

Compound of ten tetrahedra
(3D model) (OFF file)
Rank	3
Type	Regular compound
Space	Spherical
Notation
Bowers style acronym	E
Elements
Components	10 tetrahedra
Faces	40 triangles as 20 golden hexagrams
Edges	60
Vertices	20
Vertex figure	Golden hexagram, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{4} \approx 0.61237$
Inradius	$\frac{\sqrt6}{12} \approx 0.20412$
Volume	$\frac{5\sqrt2}{6} \approx 1.17851$
Dihedral angle	$\arccos\left(\frac13\right) \approx 70.52877^\circ$
Central density	10
Number of external pieces	180
Level of complexity	10
Related polytopes
Army	Doe, edge length $\frac{\sqrt{10}-\sqrt2}{4}$
Regiment	E
Dual	Compound of ten tetrahedra
Conjugate	Compound of ten tetrahedra
Convex core	Icosahedron
Abstract & topological properties
Flag count	240
Schläfli type	{3,3}
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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

Vertex coordinates

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

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

plus all even permutations of:

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

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.

External links

• Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#3).

• Klitzing, Richard. "e".

• Wikipedia Contributors. "Compound of ten tetrahedra".