Compound of five tetrahedra

Compound of five tetrahedra
(3D model) (OFF file)
Rank	3
Type	Weakly regular compound
Notation
Bowers style acronym	Ki
Elements
Components	5 tetrahedra
Faces	20 triangles
Edges	30
Vertices	20
Vertex figure	Equilateral triangle, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {6}}{4}}\approx 0.61237$
Inradius	${\frac {\sqrt {6}}{12}}\approx 0.20412$
Volume	${\frac {5{\sqrt {2}}}{12}}\approx 0.58925$
Dihedral angle	$\arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }$
Central density	5
Number of external pieces	60
Level of complexity	10
Related polytopes
Army	Doe, edge length ${\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}$
Regiment	Ki
Dual	Compound of five tetrahedra
Conjugate	Compound of five tetrahedra
Convex core	Icosahedron
Abstract & topological properties
Flag count	120
Schläfli type	{3,3}
Orientable	Yes
Properties
Symmetry	H3+, order 60
Flag orbits	2
Convex	No
Nature	Tame

The chiricosahedron, ki, or compound of five tetrahedra is a weakly-regular polyhedron compound. It consists of 20 triangles, 3 joining at each vertex. As the name suggests, it is chiral, and has faces in planes parallel to those of the convex icosahedron.

This compound is sometimes called 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 pentagyroprism, which is seven-dimensional.

Gallery

Vertex coordinates

Coordinates for the vertices of a chiricosahedron of edge length 1 are given by:

• $\left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}}\right)$,

plus all even permutations of:

• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,0\right)$.

Related polyhedra

The icosicosahedron is a compound of the two opposite chiral forms of the chiricosahedron.

External links

• Bowers, Jonathan. "Polyhedron Category C1: Compound Regulars" (#2).

• Klitzing, Richard. "ki".
• Wikipedia contributors. "Compound of five tetrahedra".