Fractals

Here are some Fractal Images that I reference in my podcast on “Fascinating Fractals”  — Jeanne Lazzarini

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Fractal Dimension:

First, consider the dimension of a line. (Notice the blue line). A line can be divided into n separate equal sized parts. For example:  Each of those parts is 1/n the size of the whole line and each part, if magnified n times, would look exactly the same as the original.  For a line, ln(number of divisions) = ln (n¹).  (Where the exponent is the dimension number).

Now, for a square — Repeating this process shows that a square can be divided into n² parts, so that ln(number of divisions) = ln(n²).     When scaling it by a factor of 2, its area increases by a factor of 2²= 4

A cube has dimension 3. When scaling it by a factor of 2, its volume increases by a factor of 2³= 8. Each of the n³ pieces would be 1/n the size of the whole figure.  For a cube, ln(number of divisions) = ln(n³). So, the larger cube consists of 8 copies of the smaller one!

ln(# of divisions) / ln (magnification factor)

For a Line:  D = ln (n¹) / ln(n)  = 1

For a Square:  D = ln(n²) / ln(n) = 2

For a Cube:   D = ln(n³) / ln(n) = 3

In each of these examples, the magnification factor was always n. But for fractals, the magnification factor will be a constant which varies for each fractal causing it to have a non-integer dimension.

Some simple iterative Fractal Images:

Each of these diagrams shows the first few steps of construction of the figure.

Imagine the iterations being carried on indefinitely.

The Koch Snowflake would have infinite perimeter.

Fractals in Nature

Here are just a few examples of how fractals appear in a wide range of natural situations.

Coastlines are Fractal

Mandelbrot’s early work discussed the difficulty of measuring the coastline of Britain.

Computer-generated fractal terrain

Fractals can also be used to create realistic “copies” of nature, for example, as landscapes and textures used in video games or computer-generated movies. The water, mountains and clouds in this image are made entirely by a computer, with the help of fractals!