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
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 model shown here is designed to suggest shapes that are not solid blocks, but rather created by lines and planes coming from the periphery.
Beginning with an icosahedron and then filling in its diagonals creates a dodecahedron inside. Repeating the process creates a second icosahedron with orientation identical to the first, but whose size has been shrunk by the cube of the golden mean. This model shows the duality of icosahedron and dodecahedron with the vertex of each lying directly above or below the center of the face of the other, while the edges cross each other at their midpoints.