# Kepler's Explanation for Spacing of Planets

## Conjecture

In the 16th century there were $6$ known planets.

It was also known that there were Five Platonic Solids.

Johannes Kepler reasoned that it was possible to fit the orbits of the planets around the sun into spheres such that:

- $(1): \quad$ Between each sphere was a (real or imaginary) platonic solid
- $(2): \quad$ The sphere of one planet was inscribed within the platonic solid between that and the next planet out
- $(3): \quad$ The sphere of one planet was circumscribed around the platonic solid between that and the next planet in.

Thus:

- Between the orbits of Mercury and Venus was an octahedron
- Between the orbits of Venus and Earth was an icosahedron
- Between the orbits of Earth and Mars was a dodecahedron
- Between the orbits of Mars and Jupiter was a tetrahedron
- Between the orbits of Jupiter and Saturn was a cube.

## Source of Name

This entry was named for Johannes Kepler.

## Historical Note

Johannes Kepler's conjecture on the spacing of the planets was made in his $1596$ work *Mysterium Cosmographicum*.

*The earth's orbit is the measure of all things; circumscribe around it a dodecahedron, and the circle containing this will be Mars: circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter: circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the earth an icosahedron, and the circle contained in it will be Venus; inscribe within Venus an octahedron, and the circle contained within it will be Mercury. You now have the reason for the number of planets.*

His model, however ingenious, did not agree well with observations, particularly in light of subsequent, more accurate, observational data.

However, it made his reputation as an imaginative thinker and keen astronomer, and importantly this was one of the first theses based on a heliocentric universe.

He brought him to the attention of the scientific community, including Galileo Galilei, and soon he was offered a position working for Tycho Brahe.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.10$: Kepler ($\text {1571}$ – $\text {1630}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Kepler