Icosidodecahedral prism

Icosidodecahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Iddip
Coxeter diagram	x o5x3o ()
Elements
Cells	20 triangular prisms, 12 pentagonal prisms, 2 icosidodecahedra
Faces	40 triangles, 60 squares, 24 pentagons
Edges	30+120
Vertices	60
Vertex figure	Rectangular pyramid, edge lengths 1, (1+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{7+2\sqrt5}}{2} ≈ 1.69353$
Hypervolume	$\frac{45+17\sqrt5}{6} ≈ 13.83552$
Dichoral angles	Trip–4–pip: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	Id–3–trip: 90°
	Id–5–pip: 90°
Height	1
Central density	1
Number of pieces	34
Level of complexity	8
Related polytopes
Army	Iddip
Regiment	Iddip
Dual	Rhombic triacontahedral tegum
Conjugate	Great icosidodecahedral prism
Abstract properties
Flag count	1920
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

The icosidodecahedral prism or iddip, is a prismatic uniform polychoron that consists of 2 icosidodecahedra, 12 pentagonal prisms, and 20 triangular prisms. Each vertex joins 1 icosidodecahedron, 2 pentagonal prisms, and 2 triangular prisms. It is a prism based on the icosidodecahedron. As such it is also a convex segmentochoron (designated K-4.90 on Richard Klitzing's list).

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, id atop id

• 

  Net

Vertex coordinates

The vertices of an icosidodecahedral prism of edge length 1 are given by all permutations and sign changes of the first three coordinates of:

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

along with all even permutations and all sign changes of the first three coordinates of:

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

Representations

An icosidodecahedral prism has the following Coxeter diagrams:

• x o5x3o (full symmetry)
• oo5xx3oo&#x (bases seen separately)
• xxxxx xoxfo5ofxox&#xt (H₂×A₁ axial, pentagonal prism-first)

Related polychora

The regiment of the icosidodecahedral prism also includes the small icosihemidodecahedral prism and small dodecahemidodecahedral prism.

External links

• Bowers, Jonathan. "Category 19: Prisms" (#911).

• Klitzing, Richard. "Iddip".

• Wikipedia Contributors. "Icosidodecahedral prism".