# 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.