- Sorry: haven't graded your graphs unit homework yet.
- Your Fractal/Tree homework is due next time, Monday!
- It's a graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative) definition.
- We're going to be looking at more graphs in this unit, but they're
graphs of a very different nature.
- Let's start by having a glimpse at the beautiful gallery of
section 6.1, for these are the subjects of our unit.
- Deterministic
fractals: delve into an infinite, dynamic world, created by
a simple mathematical process:
- a simple rule
- repeated over and over and over.... but
- simple rules, intense content!
- Can you explain how these fractal images illustrate chaos?
- The pictures remind me of the "acres of dominoes" from Robert Graves's Warning to Children.... Here's the poem in manyeyes.
- Building deterministic fractals with sticks
- Fractal ferns (p. 443), and even names!
- Let's see the process, but a little slower.
- Now let's introduce a little bit of randomness:
- Fractal Islands, and coastlines.
- Sierpinski triangle (p. 436):
- First deterministically: fractals by subtraction
- Let's try #3, p. 450.
- After we're finished removing all that stuff, how much area is left in the triangle?
- The Chaos game - generating fractals using random movement! (p. 447)
- Follow the example of p. 447. Use a die and create your own pattern using 5 rolls.
- Let's try #23, p. 454
- Once again, simple rules lead to complex objects. This time, however, we don't proceed methodically, but rather haphazardly - and yet we still produce the complex images!
- Here's
a better version of that, that has more general options.
From randomness comes order; from simple rules comes complicated objects!
- Let's make our own fractal (deterministically), starting from a
square. (We'll use some graph paper.)
- The
collage making process -- see page 437.
- Let's test out the collage on p. 442 - another Sierpinski!
| Here's our mascot for this unit: Bessie the fractal cow: |
|
| We might have easily taken the Japanese artist Hokusai. |
So what is a fractal? A couple of things we'll generally see?
