Snub dodecahedral prism

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).

Snub dodecahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sniddip
Coxeter diagram	x2s5s3s ()
Elements
Cells	20+60 triangular prisms, 12 pentagonal prisms, 2 snub dodecahedra
Faces	40+120 triangles, 30+60+60 squares, 24 pentagons
Edges	60+60+120+120
Vertices	120
Vertex figure	Mirror-symmetric (topologically irregular) pentagonal pyramid, edge lengths 1, 1, 1, 1, (1+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	≈ 2.21306
Hypervolume	≈ 37.61665
Dichoral angles	Trip–4–trip: ≈ 164.17537°
	Trip–4–pip: ≈ 152.92992°
	Snid–5–pip: 90°
	Snid–3–trip: 90°
Height	1
Central density	1
Number of external pieces	94
Level of complexity	20
Related polytopes
Army	Sniddip
Regiment	Sniddip
Dual	Pentagonal hexecontahedral tegum
Conjugates	Great snub icosidodecahedral prism, great inverted snub icosidodecahedral prism, great inverted retrosnub icosidodecahedral prism
Abstract & topological properties
Flag count	4800
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3+×A1, order 120
Convex	Yes
Nature	Tame

Gallery

• 

  Geogebra render

• 

  Net

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:

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

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

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

where

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

External links

• Bowers, Jonathan. "Category 19: Prisms" (#952).

• Klitzing, Richard. "Sniddip".

• Wikipedia Contributors. "Snub dodecahedral prism".