Compound of five tetrahedra

(Redirected from Chiricosahedron)
Compound of five tetrahedra
Rank3
TypeWeakly regular compound
Notation
Bowers style acronymKi
Elements
Components5 tetrahedra
Faces20 triangles
Edges30
Vertices20
Vertex figureEquilateral triangle, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{4}}\approx 0.61237}$
Inradius${\displaystyle {\frac {\sqrt {6}}{12}}\approx 0.20412}$
Volume${\displaystyle {\frac {5{\sqrt {2}}}{12}}\approx 0.58925}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density5
Number of external pieces60
Level of complexity10
Related polytopes
ArmyDoe, edge length ${\displaystyle {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}}$
RegimentKi
DualCompound of five tetrahedra
ConjugateCompound of five tetrahedra
Convex coreIcosahedron
Abstract & topological properties
Flag count120
Schläfli type{3,3}
OrientableYes
Properties
SymmetryH3+, order 60
Flag orbits2
ConvexNo
NatureTame

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.

Vertex coordinates

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

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

plus all even permutations of:

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

Related polyhedra

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