In an earlier post about fractal coordinate systems, I linked to a draft paper about the topic by Prof. Jacky Cresson: Fractal Coordinates Systems and Scale Relativity [PDF format]. I'd like to share with you a few interesting and relatively accessible passages from the paper. Begging your indulgence, I'll append my crazy idea for describing 3-D space using only two coordinates.
In the introduction of his paper, Cresson explains that fractal space-time offers an alterative to superstring theory in the struggle to develop a unified theory of physics. He first describes the schism that currently divides fundamental physics: "The origin of the fundamental incompatibility between quantum mechanics and Einstein's general relativity theory lies in the microscopic structure of space-time . . . . [S]pace-time is not a differentiable manifold at the atomic scale, in contradiction with the assumptions of general relativity," (emphasis removed).
(What does that bit about the non-differentiability of space-time mean? In brief and in rough, it means that space-time is not unbroken. Hence the "quantum" in "quantum mechanics"; space-time shatters at the atomic scale.)
Cresson then explains that superstring theory represents only one of two general approaches to merging quantum mechanics with general relativity. Superstring theory "enriches the structure of space-time, assuming the existence of very tiny new dimensions." (I take it that, mathematically speaking, superstrings render space-time differentiable. Ditto membranes.) Cresson and a few other theoreticians, in contrast, take an alternative approach called "scale relativity." which abandons the assumption of the differentiability of space-time in favor of "a fractal space-time, and extend[s] the relativity principle on [sic] this kind of spaces [sic]," (emphasis in the original).
(Cut Cresson some slack on the grammar. He apparently approaches English as a native French-speaker and, anyhow, this is a draft paper. Besides, his struggle with English results in an unconventional but cool use of "precise" as a verb, as in "fractal coordinates systems allow us to precise the following points . . . .")
That introduction sets up Cresson's particular project: developing a rigorous definition of fractal space-time. I'd say his approach looks promising, but my opinion in these matters probably isn't worth the paper it isn't printed on. He uses not only symbols but operations I've never encountered! Heck, I'm still trying to figure out if Cresson's paper addresses the particular, and peculiar, idea that originally got me interested in fractal coordinate systems: The possibility of defining a point in three-dimensional Euclidean space using only two coordinates.
Here's the idea, in brief: Assume a volume-filling, self-avoiding, fractal curve, like Peano curve or Hilbert curve. In one or more of its iterations, the curve will touch each point in the volume. It seems to me that you could thereby define a point in 3-D space using only two coordinates: The iteration of the fractal and the distance from a specified origin to the point's position on the curve.
That sort of fractal coordinate system would appear to use only two coordinates to define three dimensions. It seems logically possible, but it also seem to violate the very definition of Euclidean space. Strange? Yes. Wrong? Perhaps. Interesting? To me, at least, yes.