Cube 10-Compound

A number of attractive cube 10-compounds can be constructed. The first can be obtained by beginning with an initial cube and rotating it by an angle about the axis, then adding a second cube obtained by rotating the first by an angle about the axis, where is the golden ratio. The angle places corresponding faces of first two cubes in a symmetrical position relative to one another, and makes each of these faces cut the other in an isosceles right triangle. The remaining eight cubes of the compound are then generating by adding four more pairs of cubes rotated by angles about the axis (the same rotations used to construct the cube 5-compound) for , 2, 3, 4.

The second and third compounds can be constructed from the vertices of the second and third dodecahedron 2-compounds, respectively.

These compounds are implemented in the Wolfram Language as PolyhedronData["CubeTenCompound", n] for , 2, 3.

These cube 10-compounds are illustrated above together with their octahedron 10-compound duals and common midspheres.

For the first compound, the common solid has the connectivity of a pentakis dodecahedron. All other interiors and convex hulls are unnamed solids illustrated above.

A net for constructing the first compound is illustrated above. The edge lengths are given by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

where indicated a polynomial root.

The surface area of the hull of the compound is

			(14)
			(15)