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 point 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 regular dodecahedron, with each of its 12
edges on one of the 12 faces of the dodecahedron, was used by
Euclid to prove the existence of the regular dodecahedron. This
model shows a second way of inscribing the cube on the
dodecahedron, with each edge on a face of the dodecahedron. This
is done with a larger cube and the dodecahedron faces extended.
This second way seems to have not been noticed before. Each edge
of the dodecahedron meets two edges of the cube, thereby
suspending the dodecahedron inside the cube. Each pentagonal face
of the dodecahedron lies on one edge of the cube; its 5 edges meet
the one edge of the cube, at the 2 internal and 2 external golden
mean points and the point at infinity.