- You have an assignment due: Problems pp. 421-, #1, 5, 8, 9, 12-15
- We have an exam on Friday!
- New assignment:
Mon 3/15 More Fractals - No new assignment in preparation for the next exam.
- The exam will cover 4.7, 5.2, 6.1-6.3 (and Fortunatus's purse!).
- It's a graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative) definition.
- 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!
- Here's a description of the mathematics behind the fractal.
- Building deterministic fractals with sticks
- Let's see the process, but a little slower (check out the beautiful bushes...).
- The Koch snowflack
- Infinitely long curves in a finite space
- Let's make a different one....
- It's not just sticks: we can also make fractals using areas. Let's consider
the Sierpinski triangles (p. 436):
- Sierpinski triangle graph paper (make a starting triangle with 32 segments on a side)
- First deterministically: fractals by subtraction
- 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)
- 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! Then all hell broke loose....
- Mindscape #14, p. 452.
| Here's a suitable mascot for this unit: Bessie the fractal cow: |
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Have you ever lined up a couple of mirrors and seen an infinite number of
copies of yourself disappearing off into the distance?
Well, try my new funhouse mirror generator for fun (but no profit): make your own funhouse mirror image, using my web interface and your own image, and I'll post them on our "website gallery". |
So what is a fractal? A couple of things we'll generally see:
