Cube 5-Compound

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A Polyhedron Compound consisting of the arrangement of five Cubes in the Vertices of a Dodecahedron. In the above figure, let 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).

References

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.