# Martin Levin

I received a Ph.D. in mathematics from Johns Hopkins. For most of my career I taught high school mathematics in Waldorf schools, where the pedagogy encourages the bridging of mathematics and art. I'm now retired.

The Platonic solids are, in a way, quite simple geometric forms, and yet, as one contemplates them and builds up and holds the forms in one’s imagination, they become quite captivating. One can view a cube, for instance, as a bounded solid, but it is more than that. The center point of the figure has a dual (in the sense of projective geometry), which is the plane at infinity. Opposite vertices have a common line that lies on the center point, while opposite faces have a common line that lies on the plane at infinity. One can imagine the form carved out by planes and lines coming in from the infinitely distant periphery. The models shown here are designed to suggest shapes that are not solid blocks, but rather created by lines and planes coming from the periphery.

For any two Platonic Solids, by lining up axes of order 3, and adjusting relative sizes, one finds many interesting relationships, a few of which are shown here.

A. Octahedron, Cube, and Dodecahedron

This could be called a tensegrity figure, since the cube and octahedron are suspended from one another by strings in tension. I have not seen this model anywhere else.

B. Dodecahedron

This highlights the dodecahedron shown in black string in model A.

C. Dodecahedron and Icosahedron: Superposition of models B and D.

D. Icosahedron

This highlights the icosahedron shown in white string in model E.

E. Octahedron, Cube, and Icosahedron

This is the dual of model A. The cube becomes the octahedron and vice versa in the dual, but the black string dodecahedron becomes the white string icosahedron in passing to the dual.

Mental exercise: while looking at model E, picture in your mind the black strings of model A added to it.

There is a polarity of space that pairs the vertices of the icosahedron with the faces of the dodecahedron and vice versa; it leaves invariant a sphere that lies between them. The ratio of the mid-radius (i.e. to the midpoint of an edge) of the dodecahedron to the mid-radius of the icosahedron is the golden mean squared.

The ratio of the mid-radius of the icosahedron to the mid-radius of the dodecahedron is the golden mean.