4/26/2004: Buddhabrot

[paranoid]

Originally posted by Undertoad
In computer science we call that recursion, although on reading the Wikipedia entry on fractals I learned that they are not all recursive, and so I don't think I understand fractals exactly. (Paranoid, do you want to take a shot here?)

It seems that another *noid already took the chance, but luckily (or unluckily), he made a few minor errors in attempt to simplify complex ideas enough to be fit for consumption.

First, we need to understand that there are different dimensionalities (i.e. ways to determine the dimensionality of an object). The simpliest one is topological. A line is 1-dimensional, a surface is 2-dimensional, a body is 3-dimensional and so on, going further up is easy. Let's also add that a countable set (i.e. a collection of points that can be counted like 1, 2, 3, .... ; it can be infinite) is 0-dimensional and an empty set is -1-dimensional.

There are also other ways to define the dimensionality. The one we need is called Hausdorff-Besikovitch dimensionality (some mathematicians use Minkovsky dimensionality to define fractals instead, which usually, but not always, gives the same result, but we will not go that deep). It's extremely simple. Take a square (or cubic, etc.) grid, draw your object and count how many squares it crosses. Now scale the grid (change the size of the squares, their total number changes accordnigly). Calculate the same. Now look at the dependence between the size of the square edge (d) and the number of squares that contain some points of your object (N). Try it with any curve drawn on a piece of paper. Usually N is (not exactly, of course, but if you decrease the d, the limit will be the x, but let's not go into that) proportional to 1/d^x (d to the power of x). x is the Hausdorff-Besikovitch dimensionality (of course, this is not the strict definition).

If we are talking about simple curves on a plane (or in space, it doesn't matter), then x will be 1, if we are talking about 2-dimensional figures on a place (or in space, etc.), x will be 2. But it turns out, that there are such "curves" that have fractional Hausdorff-Besikovitch dimensionality (how does the number of squares change), even though their topological dimensionality (is it a curve or a surface or a body) is integer. These things are fractals. It's often said that something is a fractal only if its Hausdorff-Besikovitch dimensionality is <B>greater</B> than its topological dimensionality and if it is less, it's called a Cantor set, but it's pretty much the same thing.

The characteristics of a fractal are self-similarity (not necessarily), discontinuity and fractional dimensionality.

Discontinuity is basically what separates them from normal figures, that they have no "defined" shape. Self-simularity is a common thing (Mandelbrot defined fractals in some of his papers based on self-simularity and the name fractal may actually come from latin "fractio" ("fractus"?), which hints at self-simularity. But there are things which are not self-similar, but are very much like fractals, so we usually call them fractals too. You can easily imagine them if you first imagine a simple regular (algebraic) fractal, but then remove the requirement of regularity and understandability. Basically that would be some crazy set of points confined in a square. If you add enough points and if they are crazy enough, they will form a non-regular fractal. This isn't hard math, though. As for fractional dimensionality, this is usually true (and some people [wrongly] define fractals as objects with fractional dimensionality), but not always. For example, Peano curve has a H-B dimensionality of exactly 2 and topological dimensionality of 1. It is a fractal.

These are some funny things to ponder, but the real question that only masters could answer is whether Budda has a fractal nature?