In 1596 the young Kepler published the Mysterium Cosmographicum, in which he tried to explain two mysteries with a single geometric idea: why the planets then known were exactly six (Mercury, Venus, Earth, Mars, Jupiter, Saturn) and why their orbits had precisely those relative sizes.
His insight was that the five Platonic solids could be nested, in the right order, between the six planetary spheres: each solid inscribed in one sphere and circumscribed about the next. Starting from the outside, Saturn’s sphere encloses a cube, inside which Jupiter’s sphere is inscribed; between Jupiter and Mars a tetrahedron; between Mars and Earth a dodecahedron; between Earth and Venus an icosahedron; between Venus and Mercury an octahedron. Since the regular polyhedra are exactly five, exactly six intervals remain — and hence six planets: precisely the number observed.
The model translated the structure of the cosmos into geometric ratios. For each solid, the link between two consecutive orbits was given by the ratio of the radius of the inscribed sphere to that of the circumscribed sphere. For a cube of edge , for instance, the inscribed sphere has radius and the circumscribed one , so that
Each polyhedron thus supplies a fixed ratio, and the sequence of the five solids produced the predicted sizes of the orbits.
The appeal of the idea was enormous: purely geometric objects seemed to impose the architecture of the solar system. Yet the observational data did not fit well enough, and for a deep reason: the real orbits are not circles but ellipses, as Kepler himself would discover years later in the Astronomia Nova (1609), where he stated the first two laws of planetary motion (Boyer, Kline). The Mysterium Cosmographicum remains a celebrated example of how an elegant but mistaken mathematical idea can nonetheless push science forward.
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Topics: Synthetic geometry in space
Concepts: Platonic solids · Regular polyhedra
People: Kepler