Editing Snub cube (section)

== Construction ==
The snub cube can be generated by taking the six faces of the cube, [[Expansion (geometry)|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 triangle]]s.{{r|holme}}

[[File:A5-A7.gif|thumb|left|150px|Process of snub cube's construction by rhombicuboctahedron]]
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.{{sfnp|Conway|Burgiel|Goodman-Struss|2008|p=287&ndash;288}}

The snub cube can also be derived from the [[truncated cuboctahedron]] by the process of [[alternation (geometry)|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.
{{multiple image
 | align = center  | total_width = 450
 | image1 = Polyhedron great rhombi 6-8 subsolid snub left maxmatch.png
 | image2 = Polyhedron great rhombi 6-8 max.png
 | image3 = Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png
 | footer = Uniform alternation of a truncated cuboctahedron
}}

=== Cartesian coordinates ===
[[Cartesian coordinates]] for the [[vertex (geometry)|vertices]] of a snub cube are all the [[even permutation]]s of
<math display="block"> \left(\pm 1, \pm \frac{1}{t}, \pm t \right), </math>
with an even number of plus signs, along with all the [[odd permutation]]s with an odd number of plus signs, where <math> t \approx 1.83929 </math> is the [[Generalizations of Fibonacci numbers#Tribonacci numbers|tribonacci constant]].{{r|collins}} 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 <math>\alpha = \sqrt{2+4t-2t^2}</math>, a number which satisfies the equation
<math display="block">\alpha^6-4\alpha^4+16\alpha^2-32=0, </math>
and can be written as
<math display="block">\begin{align} \alpha &= \sqrt{\frac{4}{3}-\frac{16}{3\beta}+\frac{2\beta}{3}}\approx1.609\,72 \\ \beta &= \sqrt[3]{26+6\sqrt{33}}. \end{align}</math>
To get a snub cube with unit edge length, divide all the coordinates above by the value ''α'' given above.