[deleted]
A lot of the essays in this contest have to do with the issue of mathematics versus reality. While numbers can be used to count things, they are not those things, but I'm not sure why it's important to make that obvious distinction. I don't recall that it was ever a big issue in geometry.
In my case, what is being counted are units of space and time. The 3D expansion of space upon which it is based is an observed physical phenomenon. The operational interpretation of the mathematical expression s3/t = 23/1 describes how many measures of space per measure of time are generated from a given point, as the progression continues. Thus, after n linear measures of time pass, (n*2)3 volume measures are generated, giving us the mathematical progression, 8, 64, 216, 512, ... quantitatively, corresponding to the linear march of time, 1, 2, 3, 4, ....
Geometrically, the time units are constructed as the radius of the inner circle of figure 1 in my essay (considering only one of the two constructions there). The volume units are the volume of the 2x2x2 stack of 8 one-unit cubes, which grows to a 4x4x4 stack of 64 one-unit cubes, a 6x6x6 stack of 216 one-unit cubes, an 8x8x8 stack of 512 one-unit cubes, and so on, ad infinitum, as time marches on.
Admittedly, these discrete volume units in the progression correspond to nothing real, since space doesn't expand outward cubically, in only eight "directions," but outward radially, in an infinite number of directions. However, the inner and outer radial volumes are determined by the 3D stack of one-unit cubes and the ratio of their volumes is equal to R3, where R = 21/2. Hence, the progression of the ratio of the two volumes of the expanding spheres (technically balls) is, (n*21/2)3, or 81/2, 5121/2, 58321/2, 327681/2..., as time marches on.
It may not appear at first that these numbers correspond to anything real, but the fact that their squares are integers, and very familiar integers at that, indicates otherwise. Indeed, when we look at the ratio of the surface areas of the spheres, the formula for the progression of which is (n*R)2, the numbers are very curious: (1*21/2)2 = 2, (2*21/2)2 = 8, (3*22)2 = 18, (4*21/2)2 = 32.
These are the numbers of elements in the half-periods of the periodic table, something we might be tempted to dismiss as conincidental, smacking of numerology, if it weren't for the fact that Le Cornec has made what C.K. Whitney calls "a stunning demonstration:" Le Cornec discovered a previously un-noted pattern underlying all ionization potential data: the square roots of all ionization orders, when plotted as a function of atomic number Z, lie on a straight line. (See Whitney's "Relativistic Dynamics in Basic Chemistry")
That the IPs are something real is undeniable, but that their order appears closely related to a natural numerical progression is astonishing, in my view.