Great Rhombicuboctahedron (Archimedean)

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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

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

References

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