Dodecahedral prism

Dodecahedral prism
	(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Dope
Coxeter diagram	x x5o3o ()
Elements
Cells	12 pentagonal prisms, 2 dodecahedra
Faces	30 squares, 24 pentagons
Edges	20+60
Vertices	40
Vertex figure	Triangular pyramid, edge lengths (1+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Pip–4–pip:
	Doe–5–pip: 90°
Height	1
Central density	1
Number of external pieces	14
Level of complexity	4
Related polytopes
Army	Dope
Regiment	Dope
Dual	Icosahedral tegum
Conjugate	Great stellated dodecahedral prism
Abstract & topological properties
Flag count	960
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

The dodecahedral prism or dope is a prismatic uniform polychoron that consists of 2 dodecahedra and 12 pentagonal prisms. Each vertex joins 1 dodecahedron and 3 pentagonal prisms. It is a prism based on the dodecahedron. As such it is also a convex segmentochoron (designated K-4.74 on Richard Klitzing's list).

Gallery[edit | edit source]

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

$\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right)$ ,

along with all even permutations and all sign changes of:

$\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$ .

A dodecahedral prism has the following Coxeter diagrams: