Magnify an ordinary curve, like a parabola, and no new detail appears. As one magnifies it further and further, a line becomes a very good approximation. Most curves have this property, and likewise most surfaces can be very well approximated by a single plane at sufficient magnification. Fractals, on the other hand, have structure at all levels of magnification, thus they do not become simpler as one looks at them more closely
Most precisely, a fractal is a mathematical object with a fractional dimension. When one covers it with identical boxes and lets the boxes’ size approach zero, the ratio of the logarithm of the number of boxes to the logarithm of the length the side of a box approaches a fraction, whereas for a non-fractal it approaches a whole number.
We tend to think of objects as having integer-number dimensions. A curve is one-dimensional, as a point on it may be specified by a single parameter (arc length), likewise, because a point on it may be located using Cartesian coordinates, a plane has two dimensions. Putting these intuitive notions onto rigorous mathematical footing is a nontrivial task involving the Lebesgue covering dimension and other topological methods; dimension quantified in this way is known as “topological dimension”.
The process gets trickier when considering certain mathematical objects, the simplest among them the space-filling curves, which are curves defined in such a way as to pass through every point in a given region of (higher-dimensional) space, thus being in some intuitive sense two-dimensional, but nonetheless have a topological dimension equal to 1! (That this can be done implies that a square and its edge contain the same number of points, but that is a matter for another essay.) We may define other ways of measuring dimensionality which better reflect how much of space is filled by an object, the simplest among them-but by no means the only or even the best useful definition-being the box-counting dimension.
We may approximate the dimension of the object by dividing the space in which it is embedded into boxes-in two-dimensional space this is equivalent to drawing the object on graph paper-and counting the number of boxes it takes to cover it. Where L is the length of a box edge, D is the object’s dimension, and N is the number of boxes (a function of L), 1/(L^D)=N(L). Try this for edge lengths 1/2 and 1/3 for the unit square and unit cube, and you’ll see that D=2 and 3, respectively.
A little arithmetic shows that D(L)=log(N(L))/log(1/L). For complicated objects, such as the Louisiana coastline, D will vary dramatically as the measurement scale is dropped from meters to microns to molecules, but eventually approach some limit. There are mathematical objects even more complicated, with substructure at all measurement scales, but all, nonetheless, approach some limit. That limit, lim(L->0) log(N(L))/log(1/L) is known as the box-counting dimension.
For a space-filling curve, the box-counting dimension is 2. For a line, it is 1. Objects with dimensions in-between-fractional dimensions-are known as fractals. These are necessarily objects with similar levels of structure at all distance scales. They may be deterministic: begin with the curve
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_____/ ______
and replace all edges with scaled-down copies of itself, ad infinitum, and you have one canonical example. They may also be random; mathematical idealizations of random walks such as Brownian motion or Levy flights are fractals, self-similar at all scales and with geometric dimension greater than their classic topological dimension.
