info prev up next book cdrom email home

Cube 5-Compound

\begin{figure}\begin{center}\BoxedEPSF{5Cubes_net.epsf scaled 650}\end{center}\end{figure}

A Polyhedron Compound consisting of the arrangement of five Cubes in the Vertices of a Dodecahedron. In the above figure, let $a$ be the length of a Cube Edge. Then

$\displaystyle x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a(3-\sqrt{5}\,)$  
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \tan^{-1}\left({3-\sqrt{5}\over 2}\right)\approx 20^\circ 54'$  
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \tan^{-1}\left({\sqrt{5}-1\over 2}\right)\approx 31^\circ 43'$  
$\displaystyle \psi$ $\textstyle =$ $\displaystyle 90^\circ-\phi\approx 58^\circ 17'$  
$\displaystyle \alpha$ $\textstyle =$ $\displaystyle 90^\circ-\theta\approx 69^\circ 6'.$  

The compound is most easily constructed using pieces like the ones in the above line diagram. The cube 5-compound has the 30 facial planes of the Rhombic Triacontahedron (Ball and Coxeter 1987).

See also Cube, Cube 2-Compound, Cube 3-Compound, Cube 4-Compound, Dodecahedron, Polyhedron Compound, Rhombic Triacontahedron


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 135 and 137, 1987.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 135-136, 1989.

© 1996-9 Eric W. Weisstein