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

31245 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!