# Dodecahedral prism

Dodecahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymDope
Coxeter diagramx x5o3o ()
Elements
Cells12 pentagonal prisms, 2 dodecahedra
Faces30 squares, 24 pentagons
Edges20+60
Vertices40
Vertex figureTriangular pyramid, edge lengths (1+5)/2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {11+3{\sqrt {5}}}{8}}}\approx 1.48779}$
Hypervolume${\displaystyle {\frac {15+7{\sqrt {5}}}{4}}\approx 7.66311}$
Dichoral anglesPip–4–pip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Doe–5–pip: 90°
Height1
Central density1
Number of external pieces14
Level of complexity4
Related polytopes
ArmyDope
RegimentDope
DualIcosahedral tegum
ConjugateGreat stellated dodecahedral prism
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

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).

## Vertex coordinates

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:

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

along with all even permutations and all sign changes of:

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

## Representations

A dodecahedral prism has the following Coxeter diagrams:

• x x5o3o () (full symmetry)
• xx5oo3oo&#x (bases considered separately)