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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 $a=1$, it has Inradius, Midradius, and Circumradius

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


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

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


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