Compound of five truncated tetrahedra

Compound of five truncated tetrahedra
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Taki
Elements
Components	5 truncated tetrahedra
Faces	20 triangles, 20 hexagons
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle, edge lengths 1, √3, √3
Measures (edge length 1)
Circumradius	${\frac {\sqrt {22}}{4}}\approx 1.17260$
Volume	${\frac {115{\sqrt {2}}}{12}}\approx 13.55288$
Dihedral angles	3–6: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	6–6: $\arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }$
Central density	5
Number of external pieces	140
Level of complexity	26
Related polytopes
Army	Non-uniform Snid, edge lengths ${\frac {{\sqrt {5}}-1}{2}}$ (equilateral triangles), ${\sqrt {\frac {5-2{\sqrt {5}}}{5}}}$ (remaining edges)
Regiment	Taki
Dual	Compound of five triakis tetrahedra
Conjugate	Compound of five truncated tetrahedra
Convex core	Icosahedron
Abstract & topological properties
Flag count	360
Orientable	Yes
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

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.

Gallery

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:

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left({\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,{\frac {{\sqrt {10}}-3{\sqrt {2}}}{8}},\,{\frac {{\sqrt {2}}+{\sqrt {10}}}{4}}\right),$
• $\left({\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,{\frac {{\sqrt {2}}-{\sqrt {10}}}{4}},\,{\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),$
• $\left({\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,{\frac {{\sqrt {10}}-3{\sqrt {2}}}{8}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left({\frac {\sqrt {10}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {10}}{4}}\right).$

Related polyhedra

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

External links

• Bowers, Jonathan. "Polyhedron Category C2: Compound Truncates" (#8).

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