Fractal Dimension

In a nutshell, FractalDimension measures how "fractal" something is; a fractal is a structure which looks the same if you observe it under a magnifying glass at different powers of magnification. (Think of the "powers of ten" exploration of the universe - atoms looking somewhat like solar systems which look somewhat like galaxies.) The simpler fractals can be constructed out of one-dimensional line segments; such one-dimensional fractals are extremely "folded up" or "indented", a bit like a very irregular coastline. In the mathematical idealization, these "one-dimensional" objects turn out to have an infinite length; some can in fact "cover" entire two-dimensional areas. Hence the FractalDimension, which is a formal mathematical treatment of the notion that such objects are "in between" one-dimensional line segments and two-dimensional areas.

It is simply how the fractal's "weight" scales with its length. For normal stuff, the scale will be in integer powers, e.g. a line's "weight" scales linearly with length, a plane scales as length^2, a cube scales as length^3. But for a fractal, the "weight" scales as a fractional (usually non-integer) power of its length, and that fractional value is the Fractal Dimension.

Enlarge an object by a scale factor s. If you see n 'objects' like the original, the (fractal) dimension, d, is (log n)/(log s), so that s^d = n, but one can find 'plausible' alternative definitions in some cases which give differing values.

Hmmm, interesting.... Where can I go for a more formal description?

For crying out loud! Go directly to Mandelbrot's definition(s) in his beautiful book, TheFractalGeometryOfNature?.

[Or try http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node31.html.]

(BrokenLink 2005-02-01)

The 'self-similarity under magnification' definition might not be very satisfactory. While some fractals are self-similar, the book also describes some that are not, the so-called non-scaling fractals.

In a perhaps less useful nutshell, the fractal dimension of a set is its HausdorffBesicovitchDimension?, or D. Mandelbrot defines a fractal as a set for which D strictly exceeds the topological dimension.

The D of (the outline of) a polygon is 1, which is also the Euclidean dimension of a straight line. (By polygon I mean one with a finite number of sides.) The D of the interior of a polygon is 2, the Euclidean dimension of a plane. The D of the interior of a polyhedron is 3. And so on. A fractal, unlike a polygon for example, can have a non-integer D.

If you approximate a circle by drawing a polygon inside it, you find that as you give the polygon more sides, making the sides shorter, the polygon's perimeter gets close to that of the circle in a stable manner. (It converges.) But if you approximate a coastline by drawing a polygon, you find that making the sides shorter does not seem to make the approximation stabilize.

Mandelbrot says that fractals are good models of real-world curves such as coastlines. The Seirpinski gasket, which can be visualized using a simple recursive program, has a D of (log 3 / log 2), or about 1.5849.

All the good stuff above is from Mandelbrot or helpful Wiki folk, the rest is mine. -- DavidVincent