Truncated icosahedral prism

Truncated icosahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tipe
Coxeter diagram	x o5x3x ()
Elements
Cells	12 pentagonal prisms, 20 hexagonal prisms, 2 truncated icosahedra
Faces	30+60 squares, 24 pentagons, 40 hexagons
Edges	60+60+120
Vertices	120
Vertex figure	Sphenoid, edge lengths (1+√5)/2, √3, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{31+9\sqrt5}{8}} ≈ 2.52796$
Hypervolume	$\frac{125+43\sqrt5}{4} ≈ 55.28773$
Dichoral angles	Pip–4–hip: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	Hip–4–hip: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18968°$
	Ti–5–pip: 90°
	Ti–6–hip: 90°
Height	1
Central density	1
Number of pieces	34
Level of complexity	12
Related polytopes
Army	Tipe
Regiment	Tipe
Dual	Pentakis dodecahedral tegum
Conjugate	Truncated great icosahedral prism
Abstract properties
Flag count	2880
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

The truncated icosahedral prism or tipe is a prismatic uniform polychoron that consists of 2 truncated icosahedra, 12 pentagonal prisms, and 20 hexagonal prisms. Each vertex joins 1 truncated icosahedron, 1 pentagonal prism, and 2 hexagonal prisms. It is a prism based on the truncated icosahedron. As such it is also a CRF segmentochoron (designated K-4.127 on Richard Klitzing's list).

Gallery

• 

  Segmentochoron display, ti atop ti

• 

  Net

Vertex coordinates

Coordinates for the vertices of a truncated icosahedral prism of edge length 1 are given by all even permutations of the first three coordinates of:

• $\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±\frac12\right),$
• $\left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac12\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±\frac12\right).$

Representations

A truncated icosahedral prism has the following Coxeter diagrams:

• x o5x3x (full symmetry)
• oo5xx3xx&#x (bases considered separately)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#902).

• Klitzing, Richard. "Tipe".

• Wikipedia Contributors. "Truncated icosahedral prism".