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Snub Cube

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An Archimedean Solid also called the Snub Cuboctahedron whose Vertices are the 24 points on the surface of a Sphere for which the smallest distance between any two is as great as possible. It has two Enantiomers, and its Dual Polyhedron is the Pentagonal Icositetrahedron. It has Schläfli Symbol s $\left\{{3\atop 4}\right\}$. It is also Uniform Polyhedron $U_{12}$ and has Wythoff Symbol $\vert\,2\,3\,4$. Its faces are $32\{3\}+6\{4\}$.

The Inradius, Midradius, and Circumradius for $a=1$ are

$\displaystyle r$ $\textstyle =$ $\displaystyle 1.157661791\ldots$  
$\displaystyle \rho$ $\textstyle =$ $\displaystyle 1.247223168\ldots$  
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{x^2-8x+4\over x^2-5x+4}=1.3437133737446\ldots,$  


x\equiv (19+3\sqrt{33}\,)^{1/3},

and the exact expressions for $r$ and $\rho$ can be computed using
$\displaystyle r$ $\textstyle =$ $\displaystyle {R^2-{\textstyle{1\over 4}}a^2\over R}$  
$\displaystyle \rho$ $\textstyle =$ $\displaystyle \sqrt{R^2-{\textstyle{1\over 4}}a^2}.$  

See also Snub Dodecahedron


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 139, 1987.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.

© 1996-9 Eric W. Weisstein