Snub cube

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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 s\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}
Wythoff symbol | 2 3 4
Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
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
Snub hexahedron.png
Colored faces
Snub cube vertfig.png
(Vertex figure)
Pentagonal icositetrahedron
(dual polyhedron)
Snub cube flat.svg

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}.


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)}} \approx 7.88929467771, 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[edit]

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 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 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{16}{3\beta}+\frac{2\beta}{3}}\approx1.60972
\beta = \sqrt[3]{26+6\sqrt{33}}

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

Orthogonal projections[edit]

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
Image Snub cube A2.png Snub cube B2.png Snub cube e1.png
[3] [4]+ [2]
Dual snub cube A2.png Dual snub cube B2.png Dual snub cube e1.png

Spherical tiling[edit]

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.

Spherical snub cube.png Snub cube stereographic projection.png
Orthographic projection Stereographic projection

Geometric relations[edit]

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.

Small rhombicuboctahedron.png
(Expanded cube)
Snub hexahedron.png
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.

Snubcubes in grCO.svg

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[edit]

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

This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure ( and Coxeter–Dynkin diagram CDel node h.pngCDel n.pngCDel node h.pngCDel 3.pngCDel node h.png. 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.

n32 symmetry mutations of snub tilings:
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.png Uniform tiling 73-snub.png Uniform tiling 83-snub.png Uniform tiling i32-snub.png
Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.png Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Ord7 3 floret penta til.png Order-3-infinite floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.3.3.∞

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

4n2 symmetry mutations of snub tilings:
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Spherical square antiprism.png Spherical snub cube.png Uniform tiling 44-snub.png Uniform tiling 54-snub.png Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
Spherical tetragonal trapezohedron.png Spherical pentagonal icositetrahedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg Order-5-4 floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.4.3.∞

Snub cubical graph[edit]

Snub cubical graph
Snub cubic graph.png
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]

Orthogonal projection
Snub cube A2.png

See also[edit]


  1. ^ "Spherical Designs" by R.H. Hardin and N.J.A. Sloane
  2. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  • Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette 89 (514): 76–81. 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

External links[edit]