TOPICS

Dodecahedron 2-Compound

A number of attractive 2-compounds of the regular dodecahedron can be constructed. The first (left figures) has the symmetry of the cube and arises by combining two regular dodecahedra rotated with respect to each other about a common axis (Holden 1991, p. 37). The second (middle figures) takes a regular dodecahedron with a axis along the -axis and adds a second dodecahedron rotated by around the axis. The third (right figure) can be obtained by rotating one regular dodecahedron by an angle of radians () with respect to another about a axis.

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

These dodecahedron 2-compounds are illustrated above together with their icosahedron 2-compound duals and common midspheres.

For the first compound, the common solid has the connectivity of the tetrakis hexahedron. All other common solids and convex hulls of these compounds are unnamed polyhedra illustrated above.

The (first) compound can be constructed from the net above. For dodecahedra with unit edge lengths, the side lengths are

 (1) (2) (3) (4)

where is the golden ratio. The surface area of the first compound hull is

 (5)

Icosahedron 2-Compound, Polyhedron Compound, Regular Dodecahedron

Explore with Wolfram|Alpha

More things to try:

References

Holden, A. Shapes, Space, and Symmetry. New York: Dover, p. 37, 1991.

Cite this as:

Weisstein, Eric W. "Dodecahedron 2-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dodecahedron2-Compound.html