Snub cubic prism

Snub cubic prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSniccup
Coxeter diagramx2s4s3s ()
Elements
Cells8+24 triangular prisms, 6 cubes, 2 snub cubes
Faces16+48 triangles, 12+12+24+24 squares
Edges24+24+48+48
Vertices48
Vertex figureMirror-symmetric (topologically irregular) pentagonal pyramid, edge lengths 1, 1, 1, 1, 2 (base), 2 (legs)
Measures (edge length 1)
Hypervolume≈ 7.88948
Dichoral anglesTrip–4–trip: ≈ 153.23459°
Trip–4–cube: ≈ 142.98343°
Snic–4–cube: 90°
Snic–3–trip: 90°
Height1
Central density1
Number of pieces40
Level of complexity20
Related polytopes
ArmySniccup
RegimentSniccup
DualPentagonal icositetrahedral tegum
ConjugateSnub cubic prism
Abstract properties
Flag count1920
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB3+×A1, order 48
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The snub cubic prism or sniccup is a prismatic uniform polychoron that consists of 2 snub cubes, 6 cubes, and 8+24 triangular prisms. Each vertex joins 1 snub cube, 1 cube, and 4 triangular prisms. It is a prism based on the snub cube. As such it is also a convex segmentochoron (designated K-4.60 on Richard Klitzing's list).

Vertex coordinates

The vertices of a snub cubic prism of edge length 1 are given by all even permutations and even sign changes, as well as odd permutations and odd sign changes of the first three coordinates of:

• ${\displaystyle \left(c_1,\,c_2,\,c_3,\,±\frac12\right),}$

where

• ${\displaystyle c_1=\sqrt{\frac{1}{12}\left(4-\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{17-3\sqrt{33}}\right)},}$
• ${\displaystyle c_2=\sqrt{\frac{1}{12}\left(2+\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{17-3\sqrt{33}}\right)},}$
• ${\displaystyle c_3=\sqrt{\frac{1}{12}\left(4+\sqrt[3]{199+3\sqrt{33}}+\sqrt[3]{199-3\sqrt{33}}\right)}.}$