Dodecahedral prism

Dodecahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
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	$\sqrt{\frac{11+3\sqrt5}{8}} ≈ 1.48779$
Hypervolume	$\frac{15+7\sqrt5}{4} ≈ 7.66311$
Dichoral angles	Pip–4–pip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	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

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, doe atop doe

• 

  Net

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)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#891).

• Klitzing, Richard. "Dope".

• Wikipedia Contributors. "Dodecahedral prism".