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Great Rhombicuboctahedron (Archimedean)


An Archimedean Solid sometimes (improperly) called the Truncated Cuboctahedron and also called the Rhombitruncated Cuboctahedron. Its Dual is the Disdyakis Dodecahedron, also called the Hexakis Octahedron. It has Schläfli Symbol t $\left\{{3\atop 4}\right\}$. It is also Uniform Polyhedron $U_{11}$ and has Wythoff Symbol $2\,3\,4\,\vert$. Its faces are $8\{6\}+12\{4\}+6\{8\}$. The Small Cubicuboctahedron is a Faceted version. The Inradius, Midradius, and Circumradius for unit edge length are

$\displaystyle r$ $\textstyle =$ $\displaystyle {\textstyle{3\over 97}}(14+\sqrt{2}\,)\sqrt{13+6\sqrt{2}}\approx 2.20974$  
$\displaystyle \rho$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{12+6\sqrt{2}}\approx 2.26303$  
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{13+6\sqrt{2}}\approx 2.31761.$  

Additional quantities are
$\displaystyle t$ $\textstyle =$ $\displaystyle \tan({\textstyle{1\over 8}}\pi)=\sqrt{2}\,-1$  
$\displaystyle l$ $\textstyle =$ $\displaystyle 2t=2(\sqrt{2}\,-1)$  
$\displaystyle h$ $\textstyle =$ $\displaystyle 1+l\sin({\textstyle{1\over 4}}\pi) = 3-\sqrt{2}.$  

See also Small Rhombicuboctahedron, Great Truncated Cuboctahedron


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 138, 1987.

© 1996-9 Eric W. Weisstein