Small Stellated Dodecahedron

$\begin{figure}\begin{center}\BoxedEPSF{small_stellated_dodec_net.epsf}\end{center}\end{figure}$

One of the Kepler-Poinsot Solids whose Dual Polyhedron is the Great Dodecahedron. Its Schläfli Symbol is $\{{\textstyle{5\over 2}}, 5\}$ . It is also Uniform Polyhedron $U_{34}$ and has Wythoff Symbol $5\,\vert\,2\,{\textstyle{5\over 2}}$ . It was originally called the Urchin by Kepler. It is composed of 12 Pentagrammic faces. Its faces are $12\{{\textstyle{5\over 2}}\}$ . The easiest way to construct it is to build twelve pentagonal Pyramids

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

and attach them to the faces of a Dodecahedron. The Circumradius of the small stellated dodecahedron with is

$\begin{displaymath} R={\textstyle{1\over 2}}5^{1/4}\phi^{-1/2} ={\textstyle{1\over 4}}5^{1/4}\sqrt{2(\sqrt{5}-1)}\,. \end{displaymath}$

References

Fischer, G. (Ed.). Plate 103 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 102, 1986.

Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 219, 1997.