Tuesday, December 27, 2011

26: That Krazy Kepler

    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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