What is the basic formula for fractals?

Answer 1

The self-similarity dimension is a simplification of the Hausdorff dimension which can be applied to exactly self-similar objects.

The following analysis of the Koch Snowflake suggests how self-similarity can be used to analyze fractal properties.

The total length of a number, N, of small steps, L, is the product NL. Applied to the boundary of the Koch snowflake this gives a boundless length as L approaches zero. But this distinction is not satisfactory, as different Koch snowflakes do have different sizes. A solution is to measure, not in meter, m, nor in square meter, m², but in some other power of a meter, mx. Now 4N(L/3)x = NLx, because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives x = (log 4)/(log 3) ≈ 1.26186. So the unit of measurement of the boundary of the Koch snowflake is approximately m1.26186.

More generally, suppose that a fractal consists of N identical parts that are similar to the entire fractal with the scale factor of L and that the intersection between part is of the Lebesgue measure 0. Then the Hausdorff dimension of the fractal is \frac{\log N}{\log L}\,\!. For example, the Hausdorff dimension of

* the Cantor set is \frac{\log 2}{\log 3}\approx 0.63\,\!,
* the Sierpinski gasket is \frac{\log 3}{\log 2}\approx 1.58\,\!,
* the Sierpinski carpet is \frac{\log 8}{\log 3}\approx 1.89\,\!,

and so on. Even more generally one may assume that each of N parts is similar to the fractal with a different scale factor Li, i = 1...N. Then the Hausdorff dimension can be calculated by solving the following equation in the variable s:

\sum_{i=1}^N L_i^s = 1.

Answer 2

All fractals require lots and lots of computation to draw. There is no formula like y = a*x^2 + b or y = 1/(x-2) where you can just plot (x,y) and the answer has the fractal property. Even the simplest ideas for drawing fractals are impractical to execute by hand, and require a good deal of time even on a fast computer.

Answer 3

construction of fractal sets is based on interplay between algebra and analysis here fractals are the limits of a sequence of compact subsets of Rn with iterates of a given word under an appropriate senigroup endomorphism. google for Dekking

Answer 4

One classic example for generating fractals is the Mandelbrot recursion fromula.

http://www.devx.com/amd/Article/21827