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 (n^{1}). (Where the exponent is the dimension number).

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

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

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

For a Line: D = ln (n^{1}) / ln(n) = 1

For a Square: D = ln(n^{2}) / ln(n) = 2

For a Cube: D = ln(n^{3}) / 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!