The compound of five dodecahedra illustrated above has the symmetry of the icosahedron and can be constructed by taking a dodecahedron with top and bottom vertices aligned along the z-axis and one vertex oriented in the direction of the -axis, rotating about the y-axis by an angle

(1)

and then rotating this solid by angles for , 1, ..., 4. The right illustration above shows a paper sculpture of the dodecahedron 5-compound.

It is implemented in Mathematica as PolyhedronData["DodecahedronFiveCompound"].

Nets for the first dodecahedron 5-compound are shown above, where the lengths are given by

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

The compound has surface area

(11)

Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 145-147, 1983.

CITE THIS AS:

Weisstein, Eric W. "Dodecahedron 5-Compound." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Dodecahedron5-Compound.html