Pentagonal Cupola

$\begin{figure}\begin{center}\BoxedEPSF{J05_net.epsf scaled 700}\end{center}\end{figure}$

Johnson Solid $J_{5}$ . The bottom 10 Vertices are

$\begin{displaymath} \left({\pm{(1+\sqrt{5})\sqrt{5+\sqrt{5}}\over 4\sqrt{2}}, \p... ...r 2}, 0}\right), (0, \pm{\textstyle{1\over 2}}(1+\sqrt{5}), 0) \end{displaymath}$

and the top five vertices are

$\left({{\sqrt{5+\sqrt{5}}\over\sqrt{10}}, 0, {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\... ...\textstyle{1\over 4}}(1+\sqrt{5}), {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\,}\right),$

$\left({{-(\sqrt{5}+1)\sqrt{5+\sqrt{5}}\over 4\sqrt{10}}, \pm{1\over 2}, {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\,}\right).$

$\left({{\sqrt{5+\sqrt{5}}\over\sqrt{10}}, 0, {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\... ...\textstyle{1\over 4}}(1+\sqrt{5}), {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\,}\right),$
$\left({{-(\sqrt{5}+1)\sqrt{5+\sqrt{5}}\over 4\sqrt{10}}, \pm{1\over 2}, {\sqrt{5-\sqrt{5}}\over\sqrt{10}}\,}\right).$