Model of the solar system based on the five perfect solids.
What it shows:
Kepler attempted to describe the orbits of the planets in terms of the five regular polyhedrons. The polyhedrons, inscribed within one another define the distances of the planets from the Sun. They act as (invisible) supporting structures for the spheres on which the planets move. The order of the solids outwards from the Sun are the octahedron, icosahedron, dodecahedron, tetrahedron, and hexahedron.
How it works:
A contemporary illustration of Kepler's Universe appears in Sagan's Cosmos (see References). We have a 50cm cardboard and a 26cm balsa wood model. For an idea of the relative sphere sizes, the balsa model has:
octahedron circumscribed by 2cm diameter sphere
icosahedron circumscribed by 4cm sphere
dodecahedron circumscribed by 5.5cm sphere
tetrahedron circumscribed by 15cm sphere
hexahedron circumscribed by 26cm sphere
A proof of there only being five perfect solids is given in Appendix 2 of Cosmos. Pythagoras (6th century B.C.) knew of all the regular polyhedrons except the dodecahedron, which was discovered by Hippasus (5th century B.C.). A good historical example of a really neat idea that turns out to be a load of dingo's kidneys.
C. Sagan, Cosmos, (Random House, 1980) p.58 and Appendix 2.
J. V. Field, Kepler's Geometrical Cosmology (University of Chicago Press, 1988)