Johannes Kepler’s amazing book, Mysterium Cosmographicum, was published in 1596. Kepler’s central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer sphere. The inner sphere is tangent to the centre of each face and the outer sphere contains all the vertices of the polyhedron.
In Kepler’s model, each planet is on a sphere, the inner sphere of a polyhedron whose outer sphere contains the next planet. That is, until we come to the sixth sphere, representing Saturn, the outermost planet. The five regular solids thus rationalize the existence of six planets. The distances between the spheres can be calculated. With a particular ordering of the polyhedra, Kepler was able to achieve reasonable agreement with the observed spacings of the planets. He found the arrangement in best accord with the known orbits: the six planets from Mercury out to Saturn were separated by the solids in the sequence octahedron, icosahedron, dodecahedron, tetrahedron and cube. The Sun was at the centre of the six concentric spheres.
There is a Mathematica Demonstration that computes the distances between the six spheres (see reference [2] below). The images at the top of this post have been generated using that source: they show the five polyhedra, and the two planets involved in each case. The distances can be deduced from the geometry of the polyhedra. The calculation is particularly easy for the cube separating Jupiter and Saturn. If a cube has a sidelength of 2 units, then the distance from its centre to a vertex is √3 = 1.732. We see that, for Kepler’s model, the ratio of the distances for Jupiter and Saturn is (6.539 / 3.775) = 1.732.
Modern measurements give the mean distances for Jupiter and Saturn as 778 Mm and 1429 Mm (Megametres) respectively, with ratio 1.837, which is close, but not too close, to the value from Kepler’s model.
More accurate observations indicate further discrepancies in Kepler’s magnificent model. In any case, the (much) later discovery of Uranus and Neptune would have demolished it, as there are five and no more regular solids.
The graph on the left below shows the ratios between the six inner planets using modern values (blue) and the values from Kepler’s model (red). The general pattern is reasonable. Note especially the large relative gap between Mars and Jupiter, reflecting the presence of the asteroid belt, unknown to Kepler. On the right, the logarithm of planetary radius (in AU) is shown.
The polyhedral model is a magnificent failure. Its conception is ingenious and it did much to advance our understanding of the heliocentric system, liberating us from the bonds of Ptolemaic epicycles. But observations are paramount, and the model did not fit the data. We recall the aphorism of Thomas Huxley: “The great tragedy of science – the slaying of a beautiful hypothesis by an ugly fact”.
Kepler made sweeping advances over the following twenty years, publishing his first two laws, on elliptic orbits and constancy of areal speed in his Astronomia Nova (1609) and his third law, relating the temporal and spacial scales of the orbits in his masterpiece Harmonices Mundi (1619). But he never abandoned his polyhedral model, issuing a revised edition of Mysterium Cosmographicum in 1621.
Sources:

[1] Kepler, Johannes, 1596, 1621: Mysterium Cosmographicum. Online version (PDF) available at:
http://www.mindserpent.com/American_History/books/Kepler/1621_kepler_mysterium_cosmographicum.pdf

[2] Mathematica Demonstration that computes the distances between the six spheres:
http://demonstrations.wolfram.com/KeplersMysteriumCosmographicum/ .

[3] Wikipedia articles on Johannes Kepler and on his works, Mysterium Cosmographicum, Astronomia Nova and Harmonices Mundi.