Fractal Coordinate Systems

How do you define the position of a point in space? You label it with set of numbers that define a unique location. In fancy math talk, you use a coordinate system to assign a tuple of numbers to each point in n-dimensional space.

Does that verbiage fail to clarify? Well, then, you may find it easier to reminisce about elementary geometry, where you learned how the Cartesian coordinate system assigns a unique set of (x, y, z) coordinates to each point in 3-dimensional space. Measures of height, width, and length thus suffice to describe, say, the dimensions of a box.

The Cartesian coordinate system isn't the only way to define points in space, however. Other popular alternatives include polar coordinate systems such as the spherical or cylindrical coordinate systems. I've worked only with those coordinate systems. But googling the topic has introduced me to other, more exotic ones. For example, you can define a space using parabolic coordinates, toroidal coordinates, or bicyclide coordinates. Those—and several alternatives--all qualify as orthogonal curvilinear coordinate systems. Apparently, there also exist skew cooridinate systems, so-called because they involve families of surfaces that intersect at other than right angles.

(Two asides: First, does vector space count as another alternative? I don't think so; rather, I think it more proper to say that vectors are described in a coordinate system even though they exist independently of it. Second aside: Granted that each tuple in a coordinate system has to define a unique point. But does each point have to have a unique tuple? Of course not. Consider, for example, that the cylindrical coordinates (10, 0, 5) defines the same point as (10, 2π, 5), (10, 4π, 5), and so forth. There can, in other words, be redundant tuples for each point.))

So much for background. Now on to the hypothesis that got me started: Given that some fractal curves completely fill the space they occupy, could we use a curve as the foundation of a coordinate system? After all, since each point in the space would correspond to a point on the fractal curve, a description of the curve might suffice to describe the space.

That led me to google "fractal coordinates," which led to an interesting preliminary work on that topic [PDF format]. It's far beyond my present comprehension, I confess. But I'm thrilled that my hunch might not have erred. If I were more conversant in these topics, I'd love to pursue fractal coordinates. I have another hunch, you see, that the basic structure of the universe is fractal. I'm thus willing to bet—from an almost entirely uninformed point of view, admittedly—that fractal coordinate systems could prove very useful in fundamental physics.