The problem of doubling the volume of a cube, which the ancient Greeks were famous for experimenting with, has more to do with epistemology than with actually having to produce a doubled cube, which is just the mechanical part of the problem. As you probably will notice, the mechanical problems seem to jointly point to more metaphysical questions, which in fact supersede the mechanics. Gottfried Wilhelm Leibniz, in his many fights on this question, was always trying to carve out these paradoxes from the blockheads of the Cartesians and other empiricists.
The numerical values of certain magnitudes that make up areas and volumes, have specific relationships to those areas and volumes, in the sense that the magnitudes that make up an area are not quite the same as those which make up a volume, even though they're both magnitudes of finite length. This will be made clearer after a brief examination of certain basics in constructive geometry for those unfamiliar with geometric concepts.
The question of doubling an area of a square presents itself as finding what is called one mean between two extremes. For example, between two squares there is one quantity that will maintain a geometric proportion. Like 2 between 1 and 4, and 6 between 4 and 9 (see figure).
So, what will be the mean between a square of area one and a square of area two? Plato, in his Meno dialogue, presents this as a key geometrical problem amidst a discussion about the nature of knowledge, which comes up in the discovery of the relationship of the side of a square to it's diagonal (see figure).
The importance of it is that once you have discovered the magnitude that will double the area of the square, that magnitude cannot be measured numerically by either extremes. That is, no matter how many parts you divide the side of a square into, you'll never be able to add any number of those parts to the side of the square itself to produce the length of the diagonal. Try it out!
Yet they are both lengths. Shouldn't two finite lengths have a common measure, especially when you know the geometrical relationship between them? In Book I, Kepler described these magnitudes as "expressible only in square." Such magnitudes can be expressed generally by the relationships of right triangles in circular action (see animation).
Now take this idea to the doubling of a cube. According to cubes, you may create two means between two extremes. For example, 2 and 4 between 1 and 8; 12 and 18 between 8 and 27 (see figure).
So, a proportion is created: 1:2::2:4::4:8, and so forth. So what possibly could be our two extremes between a cube of volume one and a cube of volume two? As you may soon realize, the circle will not be able to help us.
Yes, you may construct another right triangle, thus generating the possibility of another mean (see animation).
So in this animation the proportion is: OB:OQ::OQ:OP::OP:OA. Thus, if OB is one and OA is two, we will have our two means between one and two. You'll even see that point B traces out a non-uniform, non-symmetrical curve.
But how would you knowably construct such a curve as opposed to being a faker and plotting a bunch of moments at point B and then connecting the dots? For now, that problem can rest on the mechanic's head.
So now we see that this extra mean has produced a new type of non-uniformity which has made our objects of sense behave quite differently. This is because we are now confronted with a double incommensurability. That is, in dealing with the square, we had one mean that was incommensurable to both extremes. Now, with the cube, we have two means that are not only incommensurable to the two extremes, but to each other as well.
Plato's friend, Archytas, discovered one way to double the cube. He created two means through the intersection of a torus, cylinder, and cone (see figure).
Let's look at the motions that are created, specifically between the two semi-circles; the one lying flat representing the cylinder and the one upright representing the torus (see animation).
Take notice of the way in which the diameter of the upright circle is intersecting the circumference of the circle lying flat. This intersection, called point Q, is also the point at which a perpendicular is erected onto the circumference of the rotating circle, at point P. If you actually cut out two semicircles and perform this construction, you'll see that point P is tracing out a very non-uniform curve.
As Bruce Director has pointed out (see Riemann for Anti-Dummies Part 67) there are multiply-connected actions occurring here. In the case of one mean between two extremes, the uniform motion P along the circular arc projects a non-uniform motion of Q along the diameter, and vice versa, if Q is uniform, then P non-uniform. Now, in the case of two means, not only is point P moving along a circular arc, according to the rotating circle, but is also traveling along that non-uniform, non-symmetrical curve. Also, point Q is moving along a circular arc, according to the circle that is lying flat, but also is moving in a straight line according to the diameter of the rotating circle. This is what Gauss and Riemann, after Kepler, would generally describe as, an elliptical function.
Elliptical? What is it about the ellipse that expresses non-uniformity? The properties of the ellipse will be discussed to a greater extent later in Book V, but let's take a quick look at the incommensurabilities between the circle and ellipse. Take the sine of a circle. There is no common measure between the sine and the circular arc that subtends the sine. Not even, for example, can you say that as the angle grows three times itself, that the sine also triples or doubles or any relationship that can be expressible in proportion (see figure).
However, in a circle, equal angular rotations correspond to equal circular arcs. This is not true for the ellipse, where not only is the sine incommensurable with the elliptical arc, but equal angles do not correspond to equal elliptical arcs. In fact, the elliptical arcs are constantly changing, for equal angles taken upon the elliptical circumference (see figure).
Now you may begin to see why such problems like doubling a cube might be called elliptical functions. You may not see the ellipse directly in the Archytas construction, nor is an ellipse acting upon the construction from the outside, like some deus ex machina. However, the elliptical function is most readily experienced as a physical function. It can never be fully understood from a formal, mathematical point of view. An elliptical function is a creature of the physical universe which sometimes wears the mask of Geometry in order to be able to present itself to our sense perception.
Let's turn our attention back to the Archimedeans and the relationships of their volumes. The only definitive statement Kepler makes about volumes is in Book V, where he states that the tetrahedron, inscribed in the cube, is a third of the cube, and the octahedron inscribed in the cube is half of the tetrahedron and thus a sixth of the cube (see figure).
The reader is encouraged to have fun, and try to prove, geometrically, why this is the case.
Constructing the solids out of material that can hold water is one way to see the relationships of the volumes, however just looking at the levels of the water does not give you the actual proportions. In fact this is just what a few of my associates on the animations team and I worked through. A workshop was formed in which we built the wells of the solids out of acrylic plexiglass and poured their volumes into a giant glass cylinder. By "wells" I mean the pyramidal shapes whose base is made of the edges of the solid, and whose height is made of the radius of the sphere it is inscribed in. (See Figure)
There would be a well for each type of face that the Archimedean would have. So for example, the cuboctahedron would have two wells built, one for the triangle (seen in yellow), and one for the square (seen in orange) (see figure).
This is all, of course, based on the dimensions of the truncation, then the radius of the sphere which the Archimedean is inscribed in had to be calculated. The calculation of the radii can be equated to a similar method Kepler uses in the Mysterium Cosmographicum.
Again, once the water was poured, the levels were marked off (see figure).
However, just looking at the levels gives us a sense of the relationships to each other, but further investigation has to happen in order to find the actual proportion. Once the relationship of the cube, tetrahedron, and octahedron are geometrically found, as described above, then the other truncations become manageable.
Some truncations are more difficult to calculate than others, but the reader is encouraged to experience the fun of trying to find all of them. The volumes become more difficult to calculate for the dodecahedric family, for as Kepler says in Book V, they are "inexpressible indeed, but divine." The reader should also try to think about how to approach that problem.
Once the volumes were found, then the following table was made, based on all the relationships among each other:
The reader may not fully appreciate these proportions, unless one has worked through certain parts of Book III (such as, the divisions of the string and the musical scale constructed of harmonic, melodic, and other proportions).
Take note that among the proportions of the volumes, you don't see the 1:2 ratio. This will become clearer shortly. But first, let's take a fresh, physical look at the doubling of the square and cube.
There may be hundreds of ways to double a square, but our attention will for now will focus on one. If we divide the area of the square into four parts, such that lengths are created from the center of the area to the vertices, then we take those areas and flip them out, so to speak, then we create another square whose area is double, based on the angles of the construction (see figure).
Now let's see what happens when we analogously perform this experiment on the cube, in terms of volume. Do it yourself, first. Find the center of the volume of a cube, then the cube's volume will be divided into six parts. Take each part, which would be a sort of square-based pyramid shape, then flip them out onto the tops of each face of the cube (see figure).
Whatever it is, it's double the volume of the cube. That is, if you were to flip out all of the faces.
As was said before, Kepler was the first to discover this solid figure, called the rhombic dodecahedron, which is among the semi-regular, perfect congruences.
Based on the construction, you'll notice that, like the ellipse, the rhombus has a major axis and a minor axis. In the rhombic dodecahedron, for each rhombus, the edges of the cube make up the minor axes. But what solid figure's edges could make up the major axes? (see figure)
It appears that the rhombic dodecahedron is made up of both a cube and an octahedron. However the relationship of their edges is a specific one. For if the cube's edge is equal to one, then based on how the rhombic dodecahedron was constructed, what is the edge of the rhombus? (see figure)
It also appears that within the rhombic dodecahedron, we have the entire cubic family of truncations, since we have the two duals and their rightful proportions to each other as well. However, on our table of proportions, there was no 1:2 ratio. This is accounted for now by the cube and the rhombic dodecahedron. And as you know from musical intervals, the 1:2 ratio is expressed by the octave, which is the greatest interval you can create within the relationships of the tones. For if you create a relationship of a tone within the octave to one outside of the octave, that relationship can always be replicated within the octave, which will be looked at more thoroughly in Book III. And of course all the relationships are bounded by the octave. So, all of the proportions among the truncations, or congruences rather, of the cubic family are bounded within a space in which a cube is being doubled, which is inherently an elliptical function, as was seen in the Archytas construction.
One may ask, however, "Isn't the whole idea behind the construction of Archytas based on finding the two means between two extremes, one of the means being the edge of the cube whose volume is double? You have not produced that mean in creating the Rhombic Dodecahedron, so how can you imply that the elliptical function is inherently present?" The reply is as follows: "True, the edge of the figure which is double the volume of cube, the Rhombic Dodecahedron, has an edge length which is commensurable in square to the edge of the cube. However, when discussing volumes, what is primary is the volumes of the figures, not their edge length. The cube with edge length of ³√2 and the Rhombic Dodecahedron have different edge lengths to each other, but they both have the same relationship in volume to the original cube. This points to more general problems of what Kepler called "Solid Problems" in Book I. Don't lose sight of the physical content of the operation as opposed to the formal descriptions of these ontologically different objects. And keep in mind, that it is the relationships of the volumes that are bounded by the octave, and not the actual bodies themselves. Haven't you ever wondered why the word "Volume" is used to describe both intensity or quantity of sound and mass or quantity of body?"
Perhaps, this may shed light on Kepler's investigations into the harmonic motions of the planets, which happen to be traveling in elliptical orbits, to be seen in Book V.