In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.

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Snub cube
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 38, E = 60, V = 24 (χ = 2)
Faces by sides	(8+24){3}+6{4}
Conway notation	sC
Schläfli symbols	sr{4,3} or ${\displaystyle s{\begin{Bmatrix}4\\3\end{Bmatrix}}}$
	ht0,1,2{4,3}
Wythoff symbol	| 2 3 4
Coxeter diagram
Symmetry group	O, 1/2B3, [4,3]+, (432), order 24
Rotation group	O, [4,3]+, (432), order 24
Dihedral angle	3-3: 153°14′04″ (153.23°) 3-4: 142°59′00″ (142.98°)
References	U12, C24, W17
Properties	Semiregular convex chiral
Colored faces	3.3.3.3.4 (Vertex figure)
Pentagonal icositetrahedron (dual polyhedron)	Net

It is a chiral polyhedron; that is, it has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron.

Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol ${\displaystyle s\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}$, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol ${\displaystyle t\scriptstyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}$.

For a snub cube with edge length ${\displaystyle a}$, its surface area and volume are:^[1]

${\displaystyle {\begin{aligned}A&=a^{2}\cdot \left(6+8{\sqrt {3}}\right)&&\approx 19.856\,406\,460\,6a\\V&=a^{3}\cdot {\frac {8t+6}{3{\sqrt {2(t^{2}-3)}}}}&&\approx 7.889\,477\,399\,98a\end{aligned}}}$

where t is the tribonacci constant

${\displaystyle t={\frac {1+{\sqrt[{3}]{19-3{\sqrt {33}}}}+{\sqrt[{3}]{19+3{\sqrt {33}}}}}{3}}\approx 1.839\,286\,755\,21.}$

If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths

${\displaystyle {\frac {1}{\sqrt {t+1}}}{\approx 0.593\,465}\quad {\text{and}}\quad {\frac {\sqrt {t+1}}{2}}\approx 0.842\,509.}$.

Cartesian coordinates for the vertices of a snub cube are all the even permutations of

(±1, ±1/t, ±t)

with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where t ≈ 1.83929 is the tribonacci constant. Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs, gives a different snub cube, the mirror image. Taking all of them together yields the compound of two snub cubes.

This snub cube has edges of length ${\displaystyle \alpha ={\sqrt {2+4t-2t^{2}}}}$, a number which satisfies the equation

${\displaystyle \alpha ^{6}-4\alpha ^{4}+16\alpha ^{2}-32=0,\,}$

and can be written as

${\displaystyle {\begin{aligned}\alpha &={\sqrt {{\frac {4}{3}}-{\frac {16}{3\beta }}+{\frac {2\beta }{3}}}}\approx 1.609\,72\\\beta &={\sqrt[{3}]{26+6{\sqrt {33}}}}\end{aligned}}}$

To get a snub cube with unit edge length, divide all the coordinates above by the value α given above.

The snub cube has two special orthogonal projections, centered, on two types of faces: triangles, and squares, correspond to the A₂ and B₂ Coxeter planes.

The snub cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Great circle arcs (geodesics) on the sphere are projected as circular arcs on the plane.

The snub cube can be generated by taking the six faces of the cube, pulling them outward so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with equilateral triangles.

The snub cube can also be derived from the truncated cuboctahedron by the process of alternation. 24 vertices of the truncated cuboctahedron form a polyhedron topologically equivalent to the snub cube; the other 24 form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform.

An "improved" snub cube, with a slightly smaller square face and slightly larger triangular faces compared to Archimedes' uniform snub cube, is useful as a spherical design.^[2]

The snub cube is one of a family of uniform polyhedra related to the cube and regular octahedron.

More information Uniform octahedral polyhedra, Symmetry: [4,3], (*432) ...

Uniform octahedral polyhedra
Symmetry: [4,3], (*432)							[4,3]+ (432)	[1+,4,3] = [3,3] (*332)		[3+,4] (3*2)
{4,3}	t{4,3}	r{4,3} r{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

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This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n = 6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

More information Symmetry n32, Spherical ...

n32 symmetry mutations of snub tilings: 3.3.3.3.n v t e
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
	232	332	432	532	632	732	832	∞32
Snub figures
Config.	3.3.3.3.2	3.3.3.3.3	3.3.3.3.4	3.3.3.3.5	3.3.3.3.6	3.3.3.3.7	3.3.3.3.8	3.3.3.3.∞
Gyro figures
Config.	V3.3.3.3.2	V3.3.3.3.3	V3.3.3.3.4	V3.3.3.3.5	V3.3.3.3.6	V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.∞

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The snub cube is second in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

More information Symmetry 4n2, Spherical ...

4n2 symmetry mutations of snub tilings: 3.3.4.3.n v t e
Symmetry 4n2	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	242	342	442	542	642	742	842	∞42
Snub figures
Config.	3.3.4.3.2	3.3.4.3.3	3.3.4.3.4	3.3.4.3.5	3.3.4.3.6	3.3.4.3.7	3.3.4.3.8	3.3.4.3.∞
Gyro figures
Config.	V3.3.4.3.2	V3.3.4.3.3	V3.3.4.3.4	V3.3.4.3.5	V3.3.4.3.6	V3.3.4.3.7	V3.3.4.3.8	V3.3.4.3.∞

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In graph theory, a snub cubical graph is the graph of vertices and edges of the snub cube, one of the Archimedean solids. It has 24 vertices and 60 edges, and is an Archimedean graph.^[3]

• Compound of two snub cubes
• Snub (geometry)
• Snub dodecahedron
• Snub square tiling
• Truncated cube