Why are there six planets, and why are they spaced the way they are? Johannes Kepler's first book, the Mysterium Cosmographicum (The Cosmographic Mystery, 1596), answered with geometry: God had spaced the planets with the five perfect solids.
Kepler was 24 and teaching mathematics in Graz when, in July 1595, he was drawing the great conjunctions of Jupiter and Saturn on the blackboard. They make a triangle in the zodiac, and he noticed that the circle inside the triangle and the circle around it stood in roughly the ratio of the orbits of Jupiter and Saturn. Flat shapes did not work for the other planets, so he went to three dimensions.
There are exactly five Platonic solids, solids whose faces are all the same regular polygon: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. Each one fits snugly between two spheres, one through its corners and one touching its faces. Five solids leave room for six spheres, and in Kepler's day there were six known planets. To him that was not a coincidence.
Kepler wanted to have the model made as a silver drinking cup for his patron, Duke Frederick of Württemberg, with each planet's sphere a separate bowl and seven drinks flowing from hidden pipes. The cup was never made, but the large folding engraving of the nested solids in his book became one of the most famous images in the history of science.
The geometry fixes the ratio between neighboring spheres. Inside a cube, the inner sphere is 1/√3 = 0.577 of the outer one; inside a tetrahedron it is 1/3; inside a dodecahedron or an icosahedron 0.795; inside an octahedron 0.577. With planets as thin spheres, those ratios are off by 6 to 21%. Kepler improved the fit by giving each sphere a thickness, enough to hold the planet's whole swing toward and away from the Sun, and by letting Mercury's sphere touch the edges of the octahedron instead of its faces. The model on the main page lets you try both.
It was close, but not close enough for Kepler, who took the remaining errors seriously. He wanted better observations, and the best in the world belonged to the Danish astronomer Tycho Brahe. Kepler joined Tycho in Prague in 1600. Working through Tycho's data on Mars, he found that the planets move on ellipses (1609) and, later, the exact relation between their distances and their periods (1619), the three laws of planetary motion. He never gave up on the solids, though. He reissued the Mysterium in 1621 with notes, still believing in it.
The discovery of Uranus in 1781 gave the solar system a seventh planet and left no solid to fit it. Today the spacing of the planets is explained by how the solar system formed, not by a hidden plan. But the wrong idea that set Kepler looking was what led him to the right ones.
The solids are computed from their corners, and every sphere's size follows from the solid outside it. The planets orbit with their real relative periods (Kepler's third law), compressed so that a year takes about three seconds. Real distances and eccentricities are today's values, not the ones Kepler had.