Kepler's Nested Solids
This is the model of the universe Johannes Kepler published in 1596, turning slowly in three dimensions. Six spheres, one for each planet known to him, are held apart by the five regular solids, each fitted snugly between two of them. The table in the corner sets Kepler's spacing beside the planets' real distances from the Sun: close, and wrong.
| The five ratios | √3, 3, 1.258, 1.258, √3 |
|---|---|
| Worst gap | Mars to Earth, 17.4% too narrow |
| Mercury's sphere | 0.070 of Saturn's |
| Published | 1596, Tübingen |
What you are looking at
From the outside in: the sphere of Saturn, a cube, the sphere of Jupiter, a tetrahedron, Mars, a dodecahedron, Earth, an icosahedron, Venus, an octahedron, and Mercury, with the Sun at the centre. Each sphere is drawn as two glowing rings, its equator and a meridian, and each solid as gold edges whose corners just touch the sphere outside it. The lines nearer to you are brighter, so the depth reads as the model turns.
The inner four spheres are small, because Kepler's spacing really does crowd them toward the Sun, so the circle in the corner shows them enlarged, the way Kepler's own engraving did. The table compares each gap, the outer sphere's radius divided by the inner one's, with the same ratio for the real orbits.
The math
A regular solid has a sphere through all its corners and a smaller sphere touching the middle of every face. The ratio of the two radii is fixed for each solid: the square root of 3, about 1.732, for the cube and the octahedron; exactly 3 for the tetrahedron; and about 1.258 for both the dodecahedron and the icosahedron, which are mirror images of each other in this sense. For a solid whose faces have p sides, with q faces at each corner, the ratio is tan(180/p degrees) times tan(180/q degrees).
Stack them in Kepler's order and Saturn's sphere is 1.732 times Jupiter's, Jupiter's 3 times Mars's, and so on down to Mercury, whose sphere comes out at 0.070 of Saturn's. The real orbits are wider at every step. Saturn's mean distance is 1.83 times Jupiter's, Jupiter's 3.41 times Mars's, Mars's 1.52 times the Earth's, the Earth's 1.38 times Venus's and Venus's 1.87 times Mercury's. Every one of Kepler's gaps is too narrow, from 5.5 percent between Saturn and Jupiter to 17.4 percent between Mars and the Earth.
How it is computed
Each solid is built from its standard corner coordinates, scaled so its corners sit on the sphere outside it, and every solid shares the cube's axes so the nest stays symmetrical. The picture is a perspective projection written for this page: the model turns about its vertical axis, tilts toward you, and is drawn through a simple pinhole camera, with the edges sorted into depth bands so the near ones glow brighter.
The real distances are the planets' mean distances from the Sun in NASA JPL's mean orbital elements, in astronomical units: Mercury 0.387, Venus 0.723, Earth 1, Mars 1.524, Jupiter 5.203 and Saturn 9.537. Kepler worked with Copernicus's distances and gave each sphere a thickness wide enough to hold the planet's nearest and farthest distance from the Sun, which made his fit look closer than the bare solids do here.
Questions
Why did Kepler think this was true?
He wanted a reason why there were six planets and why they were spaced as they were. There are exactly five regular solids, and five gaps between six spheres, so the fit seemed too neat to be chance. The idea came to him in Graz in 1595 while he was drawing the pattern of Jupiter and Saturn conjunctions for a class.
Was Kepler right?
No. The gaps are close but every one is too narrow, and the discovery of Uranus in 1781 broke the idea that there could only be six planets. But the questions it made him ask, about the planets' distances and speeds, led him to his three laws of planetary motion, which are right.
What are the Platonic solids?
The five solids whose faces are all the same regular polygon, with the same number meeting at every corner: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. Plato linked four of them to the elements and the fifth to the heavens, which is how they got the name.