Martin Levin
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.
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.