Icosahedral prism
Icosahedral prism  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Ipe 
Coxeter diagram  x o5o3x () 
Elements  
Cells  20 triangular prisms, 2 icosahedra 
Faces  40 triangles, 30 squares 
Edges  12+60 
Vertices  24 
Vertex figure  Pentagonal pyramid, edge lengths 1 (base), √2 (legs) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Trip–4–trip: 
Ike–3–trip: 90°  
Height  1 
Central density  1 
Number of external pieces  22 
Level of complexity  4 
Related polytopes  
Army  Ipe 
Regiment  Ipe 
Dual  Dodecahedral tegum 
Conjugate  Great icosahedral prism 
Abstract & topological properties  
Flag count  960 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  H_{3}×A_{1}, order 240 
Convex  Yes 
Nature  Tame 
The icosahedral prism or ipe is a prismatic uniform polychoron that consists of 2 icosahedra and 20 triangular prisms. Each vertex joins 1 icosahedron and 5 triangular prisms. It is a prism based on the icosahedron. As such it is also a convex segmentochoron (designated K4.36 in Richard Klitzing's list).
Gallery[edit  edit source]

Card with cell counts, verf, and crosssections

Segmentochoron display, ike atop ike

Net
Vertex coordinates[edit  edit source]
The vertices of an icosahedral prism of edge length 1 are given by all even permutations and all sign changes of the first three coordinates of:
Representations[edit  edit source]
An icosahedral prism has the following Coxeter diagrams:
 x o5o3x (full symmetry)
 x2s3s4o () (bases as pyritohedral symmetry)
 x2s3s3s () (as snub tetrahedral prism)
 oo5oo3xx&#x (bases seen separately)
 xxxx oxoo5ooxo&#xt (H_{2}×A_{1} axial, edgefirst)
Related polychora[edit  edit source]
An icosahedral prism can be cut into a central pentagonal antiprismatic prism augmented with 2 pentagonal pyramidal prisms.
The regiment of the icosahedral prism also contains the great dodecahedral prism.
External links[edit  edit source]
 Bowers, Jonathan. "Category 19: Prisms" (#892).
 Klitzing, Richard. "Ipe".
 Wikipedia contributors. "Icosahedral prism".