Introduction to rhombic polyhedra

Izidor Hafner
Tomislav Zitko
Faculty of Electrical Engineering, University of Ljubljana
Trzaska 25 , 1000 Ljubljana , Slovenia

We have investigated rhombic polyhedra where ratio of diagonals was golden ratio in [1]. In this introduction we are dealling with rhombic polyhedra where the ratio of diagonals is square root of 2. The most important polyhedron of this type is rhombic dodecahedron discoverd by J. Kepler in 1611. There are only two another convex polyhedra of this type - prolate and oblate rhombohedron.

The next three ilustrations give the relation of rhombic dodecahedron to cube, octahedron and tetrahedron.

The rhombic dodecahedra fill the space.

Join four dodecahedra along axes of threefold symmetry to obtain the following combination. To get similar figure use 12 oblate rhombohedra.

To make a double dodecahedron use 6 dodecahedra and 8 oblate rhombohedra.

Take 6 halves of rhombic dodecahedra and 8 trianguar pyramids (1/4 of tetrahedra) to get cuboctahedron.

Put 12 dodecahedra on the dodecahedron. Put 6 halves of dodecahedron on poles of fourfold axes, level pyramids around threefold axes to get a truncated octahedron.

To make models use the following nets


[1] I. Hafner, T. Zitko, Introduction to golden rhombic polyhedra