We already know two Catalan's polyhedra whose faces are identical rhombuses: the rhombic dodecahedron and the rhombic triacontahedron, duals of the cuboctahedron and the icosidodecahedron. On such polyhedra we can remove a "n-zone" (ring of n rhombuses sharing the same edge's direction) and thus obtain other interesting polyhedra, even if their symmetry groups are reduced.
Removing a 6-zone on the rhombic dodecahedron leads to a flat rhombohedron.
The rhombic triacontahedron leads successively to a nice rhombic icosahedron by removing a 10-zone, then to a second rhombic dodecahedron by removing a 8-zone, and finally, by removing a 6-zone, to two rhombohedra, one flat, the other pointy.
The faces of the polyhedra of this sequence are all "golden rhombuses".
Remark: This process is reversible (we can insert zones to a polyhedron) and thus permits to create other polyhedra, especially if different faces are allowed. We must chose a convex kernel and a set of directions allowing the insertion of zones.
A simple example: starting with a regular octahedron and the three directions of its diagonals, we get, by inserting three 8-zones (in all 18 squares), the small rhombicuboctahedron.
http://www.georgehart.com/virtual-polyhedra/vp.html by George W. Hart
(pages "zonohedra" and "zonish polyhedra")
||convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects||February 2004 |