Snub cube

Snub cube
(Click here for rotating model)
Type	Archimedean solid
Elements	F = 38, E = 60, V = 24 (χ = 2)
Faces by sides	(8+24){3}+6{4}
Schläfli symbol	<math>s\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}</math>
Wythoff symbol	| 2 3 4
Coxeter-Dynkin
Symmetry	O
References	U12, C24, W17
Properties	Semiregular convex chiral
Colored faces	3.3.3.3.4 (Vertex figure)
Pentagonal icositetrahedron (dual polyhedron)	Net

The snub cube, or snub cuboctahedron, is an Archimedean solid. The snub cube has 38 faces, of which 6 are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

Contents 1 Cartesian coordinates 2 Geometric relations 3 See also 4 References 5 External links

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

ξ³+ξ²+ξ=1,

which can be written

<math>\xi = \frac{1}{3}\left(\sqrt[3]{17+\sqrt{297}} - \sqrt[3]{-17+\sqrt{297}} - 1\right)</math>

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

α⁶-4α⁴+16α²-32=0,

and can be written as

<math>\alpha = \sqrt{\frac{4}{3}-\frac{8\sqrt[3]{4}}{3\beta}+\frac{4\beta}{3}}\approx1.60972</math>

<math>\beta = \sqrt[3]{13+3\sqrt{33}}</math>

For a snub cube with unit edge length, use the following coordinates instead:

<math>(\pm C_1,\pm C_2,\pm C_3)</math>

<math>C_1=\sqrt{\frac{1}{6}-\frac{1}{6c_1}+\frac{c_1}{12}}\approx0.621226</math>

<math>C_2=\sqrt{\frac{1}{3}-\frac{1}{6c_2}+\frac{c_2}{12}}\approx0.337754</math>

<math>C_3=\sqrt{\frac{1}{3}+\frac{1}{12c_3}+\frac{c_4}{12}}\approx1.14261</math>

<math>c_1=\sqrt[3]{3\sqrt{33}+17}</math>

<math>c_2=\sqrt[3]{3\sqrt{33}-17}</math>

<math>c_3=\sqrt[3]{199+3\sqrt{33}}</math>

<math>c_4=\sqrt[3]{199-3\sqrt{33}}</math>

Geometric relations

The snub cube can be generated by taking the six faces of the cube, pulling them outward so they no longer touch. Then give them all 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)

It can also be constructed as an alternation of a nonuniform great rhombicuboctahedron, 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.

See also

• Truncated cube
• Snub dodecahedron
• Snub square tiling
• Snub hexagonal tiling

References

• 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)

External links

• Eric W. Weisstein, Snub cube (Archimedean solid) at MathWorld.
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra