Kepler's Model of the Solar System

Disclaimer:  I know that this model is not terribly accurate, but I still think it is a pretty neat idea and so I am going to tell you about it.  If you get annoyed by people getting excited about outdated attempts at science, you probably should not read this.

Before we begin talking about Kepler, we are going to have to take a trip back to ancient Greece and talk about geometry.  You may remember discussing regular polygons in your high school geometry class: those two-dimensional figures which had all of their sides the same length and all of their angles equal (the math word for same size is "congruent").  There are infinitely many such figures, as you can make one for any number of sides greater than or equal to three.  What if we move up a dimension?

In three dimensions, a regular solid is a figure which has congruent regular polygons as its faces (all the same shape and the same size) and all of the vertices the same (meaning the same number of polygons meeting at any vertex) and convex (meaning they all poke out rather than in).  You might be tempted to think that since there are infinitely many regular polygons to choose from, there are also infinitely many regular solids to be made, but in fact there are only five.

To see why, we must start with the fact that the sum of the angles meeting at a vertex must be less than 360 degrees, because 360 degrees would make a flat plane (so that you don't get a point and thus don't really have a vertex) and any more than that would have to have some part poking in the wrong way in order to make everything fit, so that the vertex is not convex.  Moreover, in order to form a vertex you must have at least three faces meeting, because if you only have two they are lying flat on top of each other and you no longer have a three-dimensional object.

Armed with this information, we can start trying to construct some regular solids.

First suppose that all of our faces are equilateral triangles.  Every angle of an equilateral triangle measures sixty degrees.  If you have three meeting, the sum is 180 degrees, which is less than 360, so we can get a solid this way.  This gives a regular tetrahedron (solid with four faces).

If we have four equilateral triangles meet, the angles sum to 240 degrees, which is still less than 360, so we get another regular solid.  This time it is the octahedron (solid with eight faces).

 With five, the angles will sum to 300 degrees, which is still less than 360, so we get yet another
regular solid: the icosahedron (solid with twenty faces).

If we try to have a vertex with six equilateral triangles, the angles sum to 360 degrees, which doesn't work, so we now move on to regular solids with square faces.  Each angle of a square measures ninety degrees, so if we put three together we get 270 degrees.  This is less than 360, so it yields a regular solid: the cube.

If we put four squares together, the angles at the vertex sum to 360, which again means we have run out of solids.  We now move on to solids with regular pentagons for faces.  Each angle of a regular pentagon measures 108 degrees, so three of them together makes 324 degrees.  Once again, this is less than 360 so we get a regular solid.  This time it is the dodecahedron (solid with 12 faces).

Moving up to four pentagons, the angles sum to 432 degrees, which is greater than 360, so we won't get any more regular solids from the pentagon.  Let's move on to hexagons.  A regular hexagon has angles measuring 120 degrees each.  Three of them together will measure 360 degrees, so this doesn't work.  Since a regular polygon with more sides will have even larger angles, none of them will allow us to construct another regular solid.  We have found them all.

A quick review: there are exactly five regular solids.  They are the regular tetrahedron, octahedron, icosahedron, cube, and dodecahedron, having four, eight, twenty, six, and twelve faces, respectively.

These solids are often referred to as the platonic solids after the Greek philosopher Plato.  This is because Plato believed that the classical elements were composed of these solid figures.  According to Plato, water was composed or icosahedra (this is the plural of icosahedron) because this is the regular solid closest to the sphere, and water flows smoothly as though its base particles are rolling smoothly over each other.  Earth was made from cubes, since these shapes are solid and strong and would thus make things which are difficult to break, such as rocks.  Air was made from octahedra because these shapes are light and will float easily.  Fire was made from tetrahedra because they are the sharpest and fire hurts if you get to close.  At this point we've exhausted the classical elements found on earth, but we haven't done anything with the dodecahedron.  Plato postulated that everything in the heavens was made from a mysterious fifth element or "quintessence" whose base particles were shaped like dodecahedra.  This fifth element is more perfect than anything found on earth, which is why celestial bodies glow and are shaped (as far as Plato could tell) like perfect spheres.

During the Renaissance in Europe, the old Greek classics were rediscovered and held in high esteem as relics of a past golden age of science and learning in Europe.  The Catholic Church adopted Plato's classification of matter and fit it into Christian theology, using it as evidence of Earth's uniqueness in the universe, an explanation of human imperfection, and so on.

In the fifteenth and sixteenth centuries, with Europeans sailing all over the world, it became increasingly evident that the existing geocentric model of the solar system was not good enough.  Over time, the deviation between the predicted and actual positions of the planets grew, which presented problems for sailors who needed accurate sky charts for navigation.  In response, people began positing heliocentric models of the solar system, which were found to be simpler and more accurate for several reasons which I won't go into here.

Generally when you hear the phrase "heliocentric model of the solar system" the two names that come to mind are Copernicus and Galileo, because their model comes closest to the one we use today.  But there were many people developing models, most of which have fallen out of fashion.  Among them were the German Johannes Kepler.

Kepler set out to answer two questions: why are there six planets in our solar system instead of some other number (six was the amount they were aware of at that time, since nothing beyond Saturn can be seen by the naked eye), and what determines the size of their orbits.  He began by assuming a heliocentric model with circular orbits for the planets.  He then found that if you let the radius of Saturn's orbit be the radius of a sphere and inscribe a cube in that sphere then another sphere in that cube, the radius of the inner sphere is the radius of Jupiter's orbit.  If you then inscribe a tetrahedron and another sphere, you will get the radius of Mars's orbit.  Continuing down with a dodecahedron, sphere, icosahedron, sphere, octahedron, and sphere, you will get the radius of the orbit for Earth, Venus, and Mercury to within reasonable experimental error based on his data (before you get too excited, we now know that some of them are off by as much as ten percent, which is quite a lot on the scale of planetary orbits).  If that's a lot to take in, the picture looks something like this:

If you're concerned that this is a bit random, I should mention he did come up with detailed reasons for this configuration as opposed to some other ordering of the solids, based upon Plato's classification of matter.  For example, Earth's place inside the dodecahedron is significant because the dodecahedron is supposed to be the shape of the fifth element from which the heavens are composed, and Earth is the only planet in the solar system with life and the one on which we are found.  Thus, our special place in God's creation is marked off by the one shape that is used in building the perfection of the rest of the rest of the universe.

So, with one model Kepler had answered both his questions: there are six planets because there are five platonic solids, leaving room for exactly six spheres in this nested system, and the sizes of their orbits are determined by the nesting.  His results have not withstood modern scrutiny, but he did succeed in building a single model that elegantly answered his questions and offered further information about the system he was studying (through the perceived properties of the platonic solids and their order in the nesting), which is still one of the main goals of science today.