# Snub dodecahedral prism

Snub dodecahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSniddip
Coxeter diagramx2s5s3s ()
Elements
Cells20+60 triangular prisms, 12 pentagonal prisms, 2 snub dodecahedra
Faces40+120 triangles, 30+60+60 squares, 24 pentagons
Edges60+60+120+120
Vertices120
Vertex figureMirror-symmetric (topologically irregular) pentagonal pyramid, edge lengths 1, 1, 1, 1, (1+5)/2 (base), 2 (legs)
Measures (edge length 1)
Hypervolume≈ 37.61665
Dichoral anglesTrip–4–trip: ≈ 164.17537°
Trip–4–pip: ≈ 152.92992°
Snid–5–pip: 90°
Snid–3–trip: 90°
Height1
Central density1
Number of pieces94
Level of complexity20
Related polytopes
ArmySniddip
RegimentSniddip
DualPentagonal hexecontahedral tegum
ConjugatesGreat snub icosidodecahedral prism, great inverted snub icosidodecahedral prism, great inverted retrosnub icosidodecahedral prism
Abstract properties
Flag count4800
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH3+×A1, order 120
ConvexYes
NatureTame

The snub dodecahedral prism or sniddip is a prismatic uniform polychoron that consists of 2 snub dodecahedra, 12 pentagonal prisms, and 20+60 triangular prisms. Each vertex joins 1 snub dodecahedron, 1 pentagonal prism, and 4 triangular prisms. It is a prism based on the snub dodecahedron. As such it is also a convex segmentochoron (designated K-4.110 on Richard Klitzing's list).

## Vertex coordinates

The coordinates of a snub dodecahedral prism, centered at the origin and with unit edge length, are given by all even permutations with an odd number of sign changes of the first three coordinates of:

• ${\displaystyle \left(\frac{\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\xi\phi\sqrt{3-\xi^2}}2,\,\frac{\phi\sqrt{\xi(\xi+\phi)+1}}2,\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\phi\sqrt{3-\xi^2}}2,\,\frac{\xi\phi\sqrt{1-\xi+\frac{1+\phi}\xi}}2,\,\frac{\phi\sqrt{\xi(\xi+1)}}2,\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\xi^2\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\phi\sqrt{\xi+1-\phi}}2,\,\frac{\sqrt{\xi^2(1+2\phi)-\phi}}2,\,±\frac12\right),}$

as well as all even permutations with an even number of sign changes of the first three coordinates of:

• ${\displaystyle \left(\frac{\xi^2\phi\sqrt{3-\xi^2}}2,\,\frac{\xi\phi\sqrt{\phi(\xi-1-\frac1\xi)}}2,\,\frac{\phi^2\sqrt{\xi(\xi+\phi)+1}}{2\xi},\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\sqrt{\phi(\xi+2)+2}}2,\,\frac{\phi\sqrt{1-\xi+\frac{1+\phi}\xi}}2,\,\frac{\xi\sqrt{\xi(1+\phi)-\phi}}2,\,±\frac12\right),}$

where

• ${\displaystyle \phi = \frac{1+\sqrt5}2,}$
• ${\displaystyle \xi = \sqrt[3]{\frac{\phi+\sqrt{\phi-\frac5{27}}}2}+\sqrt[3]{\frac{\phi-\sqrt{\phi-\frac5{27}}}2}.}$