Fractals

Most people don’t think of mathematics as beautiful

but when you show them pictures of fractals…

…“Beautiful” is what they usually think.

The beautiful pictures we’ve seen are

fractals that come from looking at the

Mandelbrot set. For now, we’ll look at other

types of fractals. Although they aren’t

as beautiful, they are easier to understand.

Fractals are…..

• made by following mathematical rules

• recursively made

• self similar

For each black equilateral triangle, find the midpoint

Recursively made

of each side, connect them to make another, smaller,

equilateral triangle which you color white.

After 1 iteration we have:

After 2 iterations we have:

If you do this forever and ever, it will be a fractal.

Look at any black triangle from the 1st iteration.

Self similar

It will be a copy of the entire 1st iteration.

31 2 4 5

Creating a fractal

Do this forever and it is a fractal

Figures which aren’t fractals

Zoom in and (except at corners) it will be a line

Figures which aren’t fractals

Have a finite perimeter and finite area

Many fractals have a finite area and a perimeter

which has an infinite length!