I thought that was where the whole concept of fractal dimension came into play. You can have a fractal line that sort of fills a plane. As the fractal dimension proceeds from 1 to 2, the crazy-curve-shape fills the plane with greater and greater density. Here's the thing though - you need more than one finite number to describe your position along a crazy-curve-shape of dimension > 1, because the curve is infinitely convoluted. Just one real number won't do, unless you plan returning infinity for any deviation whose linear distance from the base-point is finite.
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