Snub cube explained

Snub cube
Faces:	38
Edges:	60
Vertices:	24
Symmetry:	Rotational octahedral symmetry O
Angle:	triangle-to-triangle: 153.23° triangle-to-square: 142.98°
Vertex Figure:	Polyhedron snub 6-8 left vertfig.svg
Net:	Polyhedron snub 6-8 left net.svg

s\scriptstyle\begin{Bmatrix}4\ 3\end{Bmatrix}

, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol

t\scriptstyle\begin{Bmatrix}4\ 3\end{Bmatrix}

Construction

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 may also be constructed from a rhombicuboctahedron. It started by twisting its square face (in blue), allowing its triangles (in red) to be automatically twisted in opposite directions, forming other square faces (in white) to be skewed quadrilaterals that can be filled in two 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.

Cartesian coordinates

Cartesian coordinates for the vertices of a snub cube are all the even permutations of $\left(\pm 1, \pm \frac, \pm t \right),$ 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 them together yields the compound of two snub cubes.

This snub cube has edges of length

\alpha=\sqrt{2+4t-2t^2}

, a number which satisfies the equation

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

and can be written as

\begin \alpha &= \sqrt\approx1.609\,72 \\ \beta &= \sqrt[3]. \end

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

Properties

For a snub cube with edge length

, its surface area and volume are:

\begin A &= \left(6+8\sqrt\right)a^2 &\approx 19.856a^2 \\ V &= \fraca^3 &\approx 7.889a^3.\end

. The polygonal faces that meet for every vertex are four equilateral triangles and one square, and the vertex figure of a snub cube is

3⁴ ⋅ 4

. The dual polyhedron of a snub cube is pentagonal icositetrahedron, a Catalan solid.

Graph

The skeleton of a snub cube can be represented as a graph with 24 vertices and 60 edges, an Archimedean graph.

References

Jayatilake . Udaya . Calculations on face and vertex regular polyhedra . Mathematical Gazette . March 2005 . 89 . 514 . 76–81. 10.1017/S0025557200176818 . 125675814 .
(Section 3-9)

External links

- - - - The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Editable printable net of a Snub Cube with interactive 3D view