## Chaos and Fractals

Some references...:

### Fractals

"A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale".

• Here's sierpinski.sc (take a look at the Sierpinski Triangle in Wikipedia), the file I was working on during the class, it contains functions you can evaluate and examples you can run.
• more Sierpinski, including a Java app that creates the curve.
• more Sierpinski , and the Menger Sponge as well.

### Chaos

"Chaos is apparently unpredictable behavior arising in a deterministic system because of great sensitivity to initial conditions."

From the Fractal FAQs

Q9: What is the logistic equation?
A9: It models animal populations. The equation is x -> c*x*(1-x), where x is the population (between 0 and 1) and c is a growth constant. Iteration of this equation yields the period doubling route to chaos. For c between 1 and 3, the population will settle to a fixed value. At 3, the period doubles to 2; one year the population is very high, causing a low population the next year, causing a high population the following year. At 3.45, the period doubles again to 4, meaning the population has a four year cycle. The period keeps doubling, faster and faster, at 3.54, 3.564, 3.569, and so forth. At 3.57, chaos occurs; the population never settles to a fixed period. For most c values between 3.57 and 4, the population is chaotic, but there are also periodic regions. For any fixed period, there is some c value that will yield that period. See "An Introduction to Chaotic Dynamical Systems" for more information.

A very nice Java applet that shows the logistics equation in action

• Here's logistic.sc, the file I was working on during the class, it contains functions you can evaluate and examples you can run.

See the output of the bifur1d program in the "Computational Beauty of Nature" examples

SuperCollider comes with several UGens that can generate Chaos, see:

• /usr/share/SuperCollider/Help/UGens/Chaos/