# Truncated chiricosahedron

Truncated chiricosahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTaki
Elements
Components5 truncated tetrahedra
Faces20 triangles, 20 hexagons
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths 1. 3, 3
Measures (edge length 1)
Circumradius$\frac{\sqrt{22}}{4} \approx 1.17260$ Volume$\frac{115\sqrt2}{12} \approx 13.55288$ Dihedral angles3–6: $\arccos\left(-\frac13\right) \approx 109.47122^\circ$ 6–6: $\arccos\left(\frac13\right) \approx 70.52878^\circ$ Central density5
Number of external pieces140
Level of complexity26
Related polytopes
ArmyNon-uniform Snid
RegimentTaki
DualTriakis chiricosahedron
ConjugateTruncated chiricosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count360
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The truncated chiricosahedron, taki, or compound of five truncated tetrahedra is a uniform polyhedron compound. It consists of 20 triangles and 20 hexagons, with one triangle and two hexagons joining at each vertex. As the name suggests, it can be derived as the truncation of the chiricosahedron, the compound of five tetrahedra.

Its quotient prismatic equivalent is the truncated tetrahedral pentachoroorthowedge, which is seven-dimensional.

## Vertex coordinates

The vertices of a truncated chiricosahedron of edge length 1 can be given by all even permutations and all even sign changes of:

• $\left(\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right),$ • $\left(\frac{\sqrt{10}-\sqrt2}{8},\,\frac{\sqrt{10}-3\sqrt2}{8},\,\frac{\sqrt2+\sqrt{10}}{4}\right),$ • $\left(\frac{\sqrt2+\sqrt{10}}{8},\,\frac{\sqrt2-\sqrt{10}}{4},\,\frac{3\sqrt2+\sqrt{10}}{8}\right),$ • $\left(\frac{3\sqrt2+\sqrt{10}}{8},\,\frac{\sqrt{10}-3\sqrt2}{8},\,\frac{\sqrt2}{4}\right),$ • $\left(\frac{\sqrt{10}}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt{10}}{4}\right).$ ## Related polyhedra

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