Here is another brief model, since it is summer...
Problem: Build the dual to last month's
construction of an icosahedron inscribed in a cube.
Answer: An octahedron inscribed in a dodecahedron, so that every
octahedron vertex is at the midpoint of a dodecahedron edge. The
icosahedron's edge was centered in the cube's face, so dualizing puts the
octahedron's vertex centered in the dodecahedron's edge.
Construction: It is easy. With a 2b1 dodecahedron
(the smallest size that has a zomeball available at an edge midpoint),
the octahedron edge length is g3 (which you have to make as g1+g2,
of course).
(If someone sends me a picture, I'll add it to this page.)