Editing Snub cube (section)

=== 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.