You've probably heard of 17th-century astronomer Johannes Kepler,

one of the key scientific figures of his time. He is most famous for

having determined that planetary orbits are actually elliptical rather

than circular, as part of his general laws of planetary motion. He

also made some significant contributions to the developing field of

optics. But before he made his famous contributions, he published an

earlier book called the Mysterium Cosmographicum, or "The Sacred

Mystery of the Cosmos", proposing the rather absurd theory that all

planetary orbits were determined by the five platonic solids.

The platonic solids, as you may remember from earlier podcasts,

are the five three-dimensional convex solids in which every face and

every angle are congruent. There are five of these: the 4-faced

tetrahedron, the 6-faced cube, the 8-faced octahedron, the 12-faced

dodecahedron, and the 20-faced icosahedron. Kepler noticed the odd

coincidence that there were exactly five of these solids, and also

exactly six planets known at the time. Determined that this must be

part of God's plan for the universe rather than a simple coincidence,

he decided to relate the two numbers.

How did these numbers relate to each other? Kepler began by

expanding the orbits of each planet into a sphere. (When he wrote

this book, he had not yet determined that planetary orbits are

elliptical.) Then he figured out that he could tightly fit a platonic

solid between each pair of planetary spheres, at least to the degree

of accuracy with which the orbits were known at that time. To make

this work, he had to choose a rather strange order for the solids:

the innermost one was the octahedron, followed by the icosahedron, the

dodecahedron, the tetrahedron, and the cube. To get from each

planet's orbit to the next orbit, you would draw a platonic solid

circumscribed around that orbital sphere, or the minimal-sized version

of that solid that could completely contain the orbit. Then you would

draw a circumscribed sphere outside that solid, and that sphere would

coincide with the next planet's orbit.

It's pretty easy to laugh at this theory today. To start with, of

course, there are the additional planets Uranus and Neptune discovered

after Kepler's time, which cannot be accounted for since, as we proved

in an earlier podcast, there can be no more than five platonic

solids. But as a more basic question, why the heck should the orbits

of planets be affected by the fact that we could inscribe giant

polyhedrons between them? And even if we play devil's advocate for

the moment and say that the theory is reasonable, why are the solids

used in the arbitrary order that Kepler had to choose? The polyhedrons

are not sorted by vertex angle, number of faces, or face type.

Strangely, Kepler never completely abandoned his theory even after his

later discoveries, and pubished a revised version of the Mysterium in

1621, presumably reconciling its earlier conclusions with his more

recent, and more accurate, discoveries.

But we shouldn't be too hard on poor Kepler. Back then, the

boundaries between physics, math, and religion were all kind of fuzzy,

and scientists like him believed they were uncovering God's plan for

the universe. And if you accept that everything around us was created

manually by an anthropomorphic God, why shouldn't He insert mysterious

mathematical connections that would otherwise not be needed, just to

satisfy His desire for an orderly universe? The other important

factor to keep in mind is that his willingness to entertain

off-the-wall theories, ideas that might be considered ridiculous by

his colleagues, is what made Kepler such a great mathematician and

scientist. To many of his contemporaries, the theory that planetary

orbits were determined by the platonic solids was probably a lot more

reasonable the idea that they were ellipses rather than circles!

And this has been your math mutation for today.

Kepler on Wikipedia

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