Snub cube

Snub cube

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 $s\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$
ht0,1,2{4,3}
Wythoff symbol | 2 3 4
Coxeter diagram
Symmetry group O, ½BC3, [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

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.

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 $s\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}$.

Dimensions

For a snub cube with edge length 1, its surface area is $\scriptstyle{6+8\sqrt{3}}$ and its volume is $\sqrt{\tfrac{613t+203}{9(35t-62)}}$, where t is the tribonacci constant $\tfrac{1}{3}\scriptstyle{\left(1+\sqrt[3]{19-3\sqrt{33}}+\sqrt[3]{19+3\sqrt{33}}\right) \approx 1.83929}$.

If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths $\tfrac{1}{\sqrt{t+1}} \scriptstyle{\approx 0.593465}$ and $\tfrac{1}{2}\scriptstyle{\sqrt{t+1} \approx 0.842509}$.

Cartesian coordinates

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

(±1, ±ξ, ±1/ξ)

with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where ξ is the real solution to

$\xi^3+\xi^2+\xi=1, \,$

which can be written

$\xi = \frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}} - \sqrt[3]{-17+3\sqrt{33}} - 1\right)$

or approximately 0.543689. ξ is the reciprocal of 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.

This snub cube has edges of length α, a number which satisfies the equation

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

and can be written as

$\alpha = \sqrt{\frac{4}{3}-\frac{32}{6\sqrt[3]{2}\beta}+\frac{6\sqrt[3]{2}\beta}{9}}\approx1.60972$
$\beta = \sqrt[3]{13+3\sqrt{33}}$

For a snub cube with unit edge length, use all the even permutations of

$(\pm C_1,\pm C_2,\pm C_3)$

having an even number of plus signs, along with all the odd permutations having an odd number of plus signs.

$C_1=\sqrt{\frac{4 - c_1 + c_2}{12}}\approx 0.337754$
$C_2=\sqrt{\frac{2 + c_1 - c_2}{12}}\approx 0.621226$
$C_3=\sqrt{\frac{4 + c_3 + c_4}{12}}\approx 1.14261$
$c_1=\sqrt[3]{3\sqrt{33}+17}$
$c_2=\sqrt[3]{3\sqrt{33}-17}$
$c_3=\sqrt[3]{199+3\sqrt{33}}$
$c_4=\sqrt[3]{199-3\sqrt{33}}$

Orthogonal projections

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.

Orthogonal projections
Centered by Face
Triangle
Face
Square
Edge
Image
Projective
symmetry
[3] [4]+ [2]
Dual
image

Spherical tiling

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. Straight lines on the sphere are projected as circular arcs on the plane.

Geometric relations

A snub cube can be constructed from a transformed rhombicuboctahedron by rotating the 6 blue square faces until the 12 white square become pairs of equilateral triangles.

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.

 Cube Rhombicuboctahedron (Expanded cube) Snub cube

It can also be constructed as an alternation of a nonuniform omnitruncated cube, deleting every other vertex and creating new triangles at the deleted vertices. A properly proportioned (nonuniform) great rhombicuboctahedron will create equilateral triangles at the deleted vertices. Depending on which set of vertices are alternated, the resulting snub cube can have a clockwise or counterclockwise twist.

A "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.[1]

Related polyhedra and tilings

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

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

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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure

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.∞
Coxeter
Schläfli

sr{2,3}

sr{3,3}

sr{4,3}

sr{5,3}

sr{6,3}

sr{7,3}

sr{8,3}

sr{∞,3}
Snub
dual
figure

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.∞
Coxeter

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

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
242
[2,4]+
342
[3,4]+
442
[4,4]+
542
[5,4]+
642
[6,4]+
742
[7,4]+
842
[8,4]+...
∞42
[∞,4]+
Snub
figure

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.∞
Coxeter
Schläfli

sr{2,4}

sr{3,4}

sr{4,4}

sr{5,4}

sr{6,4}

sr{7,4}

sr{8,4}

sr{∞,4}
Snub
dual
figure

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.∞
Coxeter

Snub cubical graph

Snub cubical graph
4-fold symmetry
Vertices 24
Edges 60
Automorphisms 24
Properties Hamiltonian, regular

In the mathematical field of 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.[2]