On  rhombic polyhedra with icosahedral symmetry

by Izidor Hafner, Faculty of electrical engineering, University of Ljubljana
email: izidor.hafner@fe.uni-lj.si
Fakulteta za elektrotehniko
Trzaska 25
1000 Ljubljana

Abstract (AMS. Subj. Class. 52B10, 52B12)
There is only one convex rhombic polyhedron with icosahedral symmetry.  It is triacontahedron discovered by Kepler in 1611. On the other  hand  there are infinite  nonconvex rhombic polyhedra with such symmetry. We describe some of them.

1. Introduction

Kepler classified solids with congruent faces into regular (Platonic) and half-regular (rhombic) solids. He knew two examples of  his half-regular polyhedra. The first is bounded by twelve rhombi whose diagonals are in the ratio 1:. The second rhombic polyhedron is bounded by thirty rhombi whose diagonals are in the golden ratio. They are called the rhombic dodecahedron and the rhombic triacontahedron, respectively. In the De Nive Sexangula, a work written in 1611 as a New Year's gift for a councellor at Rudolph's court in Prague, Kepler provides a hint as to how he discovered these two polyhedra. These rhombic polyhedra have some resemblance to the Platonic solids. Like them, they are spherical in shape, have congruent faces,  and a high degree of symmetry.
We shall limit ourselves to consider only the solids whose faces are congruent golden rhombi. The protate and the oblate rhomohedra have been known since old Greece. There are two more convex polyhedra whose faces are all congruent golden rhombi. One was discovered by Evgraf Stepanovich Fedorov in 1885. It is an oblate solid with twenty faces - a rhombic icosahedron. The last was the second rhombic dodecahedron. It was discovered in 1960 by Stanko Bilinski when he made an exhaustive enumeration of rhombic solids. He called it the rhombic dodecahedron of the second kind to distibguish it from the one described by Kepler. The figure belove shows all the five solids. [P.R.Cromwell: Polyhedra, Chambridge U. Press, 1997, p. 156].

Bilinski showed that we could get the rhombic icosahedron by collapsing a belt of rhombi that ran around the triacontahedron. The direction of the collaps is indicated by green colour. The dodecahedron is obtained from the icosahedron by collapsing in direction indicated by violet. The oblate rhombohedron is obtained from the dodecahedron by collaps in direction of brown colour.

The next tables show the dihedral  and solid angles of all five rhombic solids, the full angle being 1.

Icosahedral symmetry

The rhombic triacontahedron  has axes of 2-fold, 3-fold and 5-fold rotational symmetry. A 5-fold axix passes through each pair of opposite vertices od degree 5; a 3-fold axis passes through each pair of opposite vertices of degree 3; a 2-fold axis passes through the centres of each pair of opposite faces. Thus there are six 5-fold axes, ten 3-fold axes and fifteen 2-fold axes. This system of rotational symmetry is called the icosahedral system. Namely 12 vertices of degree 5 are the vertices of regular icosahedron.