Martin Levin

Portland, Oregon, USA

I received a Ph.D. in mathematics from Johns Hopkins. For most of my career I taught high school math in Waldorf schools, where the pedagogy encourages the bridging of mathematics and art. I'm now retired.
The Platonic solids are 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. The center has a dual (in the sense of projective geometry), which is the plane at infinity. One can imagine the form carved out by planes and lines coming in from the infinitely distant periphery. The model shown here is designed to suggest shapes that are not solid blocks, but rather created by lines and planes coming from the periphery.

A Cube Inscribed On The Faces Of A Dodecahedron
A Cube Inscribed On The Faces Of A Dodecahedron
45 x 45 x 45 cm
brass and aluminum tubing, beading wire, and 3D printed steel joints

Each of the 12 edges of the cube lies on one face of the dodecahedron, and conversely. Hence, a cube edge meets each of the five edges of the dodecahedron face; it meets two of them at internal golden mean division points, two at external golden mean division points, and one at infinity. Moreover, each dodecahedron edge, since it lies on two dodecahedron faces, must intersect two cube edges, which makes it possible to suspend the dodecahedron inside the cube, as shown here. The ratio of the cube edge to the dodecahedron edge is φ^4, where φ is the golden mean.
There is another way of inscribing a cube on the faces of a dodecahedron, well known all the way back to Euclid, but this figure seems to be new. I have not seen it elsewhere.