# Dodecahedral prism

Dodecahedral prism Rank4
TypeUniform
SpaceSpherical
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$\sqrt{\frac{11+3\sqrt5}{8}} ≈ 1.48779$ Hypervolume$\frac{15+7\sqrt5}{4} ≈ 7.66311$ Dichoral anglesPip–4–pip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ 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:

• $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12\right),$ along with all even permutations and all sign changes of:

• $\left(±\frac{3+\sqrt5}{4},\,±\frac12,\,0,\,±\frac12\right).$ ## Representations

A dodecahedral prism has the following Coxeter diagrams:

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