Patterns in nested Platonic solids

Abstract

In 1970 Dr. Georg Unger (1909–1999) of Dornach, Switzerland, suggested to me the Platonic solids as a topic worthy of contemplation. I was both startled and intrigued to hear a trained mathematician speak thus about such an elementary topic in mathematics. Soon thereafter I read George Adams’ Physical and Ethereal Space (Adams, 1965), with its imaginations of geometric forms created by lines and planes coming in from the infinitely distant plane, and I also came across L. Gordon Plummer’sMathematics of the Cosmic Mind (GordonPlummer, 1970); the theosophical symbolism seemed tome a bit contrived, but I found the drawings of nested Platonic solids to show some wonderful and surprising geometry. Beautiful geometric forms were swimming in my mind. I wanted to make some models to show to my students, but how could I make them physically, so they would actually hold together. Moreover, I wanted little material and clean lines that emphasize the geometric forms, so the viewer is stimulated to inwardly imagine the pure geometric forms; it is that inner activity that engages one and makes the subject meaningful. So, how to make them physically? After some trial and error, I eventually settled on metal tubes connected with bent wires and glue, with more tubes suspended inside on taut wires. When teaching projective geometry and group theory, the models captivate students’ attention and make the concepts very accessible, and in art galleries the viewers gaze with interest and wonder. Looking from different perspectives shows the viewer striking patterns. For that to work, however, precision is needed to make all of the tubes and wires line up exactly. The compound of five regular tetrahedra is very well known, models typically made in cardboard with solid faces. Figure 1 is a photo of this form cast in bronze. Figure 2 shows the same five tetrahedra, with brass tubes for edges. Figure 1 has solid faces. It adorns our home garden where the aging natural patina and the shifting daylight create varied and beautiful effects. However, due to the opaqueness of the solid faces, one cannot see a whole tetrahedron from any one perspective, and one cannot see at all the shape of the inner core. Figure 2 is actually a tensegrity figure. The 5 tetrahedra do not touch one another, but are suspended from one another with wires that form the edges of a dodecahedron on the outside. Moreover, aluminium tubes, suspended on diagonal strings, form an icosahedron