Snub Dodecahedron

$\begin{figure}\begin{figure}\begin{center}\BoxedEPSF{snub_dodec_net.epsf scaled 500}\end{center}\end{figure}\par\end{figure}$

An Archimedean Solid, also called the Snub Icosidodecahedron, whose Dual Polyhedron is the Pentagonal Hexecontahedron. It has Schläfli Symbol s $\left\{{3\atop 5}\right\}$ . It is also Uniform Polyhedron $U_{29}$ and has Wythoff Symbol $\vert\,2\,3\,5$ . Its faces are $80\{3\}+12\{5\}$ . For , it has Inradius, Midradius, and Circumradius

$\displaystyle r$	$\textstyle =$	$\displaystyle 2.039873155\ldots$
$\displaystyle \rho$	$\textstyle =$	$\displaystyle 2.097053835\ldots$
$\displaystyle R$	$\textstyle =$	$\displaystyle {1\over 2}\sqrt{8\cdot 2^{2/3}-16x+2^{1/3} x^2\over 8\cdot 2^{2/3}-10x+2^{1/3}x^2}$
	$\textstyle =$	$\displaystyle 2.15583737511564\dots,$

where

$\begin{displaymath} x\equiv \left({49+27\sqrt{5}+3\sqrt{6}\sqrt{93+49\sqrt{5}}\,}\right)^{1/3}, \end{displaymath}$

and the exact expressions for

and $\rho$ can be computed using

$\displaystyle r$	$\textstyle =$	$\displaystyle {R^2-{\textstyle{1\over 4}}a^2\over R}$
$\displaystyle \rho$	$\textstyle =$	$\displaystyle \sqrt{R^2-{\textstyle{1\over 4}}a^2}.$

References

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.