# Snub cube

Snub cube
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSnic
Coxeter diagrams4s3s ()
Elements
Faces8+24 triangles, 6 squares
Edges12+24+24
Vertices24
Vertex figureFloret pentagon, edge lengths 1, 1, 1, 1, 2
Measures (edge length 1)
Volume≈ 7.88948
Dihedral angles3–3: ≈ 153.23459°
4–3: ≈ 142.98343°
Central density1
Number of pieces38
Level of complexity5
Related polytopes
ArmySnic
RegimentSnic
DualPentagonal icositetrahedron
ConjugateSnub cube
Abstract properties
Flag count240
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3+, order 24
ConvexYes
NatureTame

The snub cube or snic, also called the snub cuboctahedron, is one of the 13 Archimedean solids. Its surface consists of 24 snub triangles, 8 more triangles, and 6 squares, with four triangles and one square meeting at each vertex. It can be obtained by alternation of the great rhombicuboctahedron, followed by adjustment of edge lengths to be all equal.

It is one of two chiral Archimedean solids, the other being the snub dodecahedron.

The triangles come in two types. One group of 8 has rotational triangular symmetry and joins only to other triangles. The other set of 24 has no local symmetry and connects to the square faces, and are the snub faces of the polyhedron.

This is one of nine uniform snub polyhedra generated with one set of digonal faces.

## Measures

The circumradius R ≈ 1.34371 of the snub cube with unit edge length is the largest real root of

${\displaystyle 32x^6-80x^4+44x^2-7.}$

Its volume V ≈ 7.88948 is given by the largest real root of

${\displaystyle 729x^6-45684x^4+19386x^2-12482}$.

Its dihedral angles can be given as acos(α) for the angle between two triangular faces, and acos(β) for the angle between a square face and a triangular face, where α ≈ –0.89286 equals the unique real root of

${\displaystyle 27x^3-9x^2-15x+13,}$

and β ≈ –0.79846 equals the unique negative real root of

${\displaystyle 27x^6-99x^4+129x^2-49.}$[1]

## Vertex coordinates

A snub cube of edge length 1, centered at the origin, has coordinates given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of

• (c1, c2, c3),

where

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

## Variations

The snub cube has a general variant that maintains the chiral cubic symmetry. It generally has 6 squares of one edge size, 8 triangles of a second size, and 24 scalene triangles using the first two edge sizes along with a third edge size, that joins two scalene triangle faces.

The most notable of these variants is the one obtained as the direct alternation of the uniform great rhombicuboctahedron. This variant uses 6 squares of size ${\displaystyle \sqrt{2+\sqrt2}}$, 8 triangles of size ${\displaystyle \sqrt3}$, and 24 scalene triangles, with one of each of these edges along with edges joining them of length ${\displaystyle \sqrt2}$, as faces.

## Related polyhedra

The disnub cuboctahedron is a uniform polyhedron compound that consists of the two opposite chiral forms of the snub cube.

It is also related to the cuboctahedron and small rhombicuboctahedron through a twisting operation. Twisting the faces of the small rhombicuboctahedron so the edge-squares become pairs of triangles results in the snub cube. Continuing the twisting until those triangles become edges results in a cuboctahedron.

o4o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4,3} x4o3o
Truncated cube tic t{4,3} x4x3o
Cuboctahedron co r{4,3} o4x3o
Truncated octahedron toe t{3,4} o4x3x
Octahedron oct {3,4} o4o3x
Small rhombicuboctahedron sirco rr{4,3} x4o3x
Great rhombicuboctahedron girco tr{4,3} x4x3x
Snub cube snic sr{4,3} s4s3s

## References

1. Wolfram Research, Inc. (2023). "Wolfram|Alpha Knowledgebase". Champaign, IL. "PolyhedronData["SnubCube", {"Circumradius", "Volume", "DihedralAngles"}]".