What is a Fractal?

1. What is a Fractal? • A fractal is a mathematical object that is both self-similar and chaotic. • self-similar: As you magnify, you see the object over and over again in its parts. • chaotic: Fractals are infinitely complex. • Amazingly, these beautiful objects of breath-taking complexity are generated by relatively simple mathematical processes.

2. The first fractals were discovered by a french Mathematician named Gaston Julia who discovered them decades before the advent of computer graphics. Julia’s work was rediscovered by Benoit Mandelbrot. The most famous of all fractals is the Mandelbrot set.

6. ...And we can continue to zoom in. As we magnify the object, we see the same thing over and over again.....This is Self Similarity

7. These two pictures are interesting because they show the same portion of the mandelbrot set colored differently. • The choice of color scheme really influences what we see in the picture. • Is this mathematics or art?

8. The Fractal Geometry of Nature Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line... ...Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of patterns is for all purposes infinite. The existence of these patterns challenges us to study those forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel. ---Benoit Mandelbrot (1984)

9. Aristid Lindenmeyer Aristid Lindenmeyer invented L-Systems to model plant growth. See the “fractal” he is holding?

10. Fractal Landscapes • In the following slides we will see some landscapes that are progressively more complex. These landscapes are NOT drawings. They are created entirely by a computer using “fractal interpolation.” • This is the procedure that is used by special effects artists to create computer generated scenes for the big screen. (“Independence Day” was full of them.) Of course, mine are much cruder than theirs. They were produced in about an hour on my (ancient) pentium computer at home by a program called VISTAPRO.

11. This is the crudest version. “Nature made of triangles.”

12. In this picture, the grid of triangles is a bit finer

13. The triangles are finer still and the mountains in the distance begin to look a bit better.

14. The next step. The mountains in the background look like real mountains.

15. Now we add some texture to the triangles.

16. Here we add some crude clouds and improve the landscape.

17. Here we add some crude trees, and improve the landscape and the clouds.

18. This is the best resolution I got.

19. We can make some artistic choices of color to give us the same landscape in different seasons.. Summer Autumn Winter---we also got rid of the leaves Spring