- I've put a homework up that will give you some idea of what the quiz will be like on Friday.
- Additional reading: Please read this chapter from Tony Crilly's "50 Mathematical Ideas You Really Need to Know". It gives the basics of fractals.
- One person has already submitted her logo and "report". Feel free to do the same. You will then simply have to get up on the final Friday and explain your logo.
- I hope that you've read chapter 8 -- today we're starting fractals. The example in chapter 8 is a little subtle, so we'll start with some simpler examples and built up to that one.
- The concept of fractals is the most modern mathematics that we'll
study. We've worked our way up from the Babylonians all the way to the
20th century.
- This will be our intuitive definition of a Fractal:
-
A world within a world!
This key notion is more formally called "self-similarity": "a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts)."
- Some examples you have seen before:
- Our Fibonacci spirals
- The tree diagram of the rabbit problem that gave rise to the Fibonacci numbers.
- Often these arose out of the idea that we carry out a process, then "do it again, do it again"....
- Some examples you might have seen before:
- Here's a suitable mascot for this unit: Bessie the fractal cow:
- 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 funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them to our website.
- World's Most Famous Fractal (Benoit Mandelbrot, Novel Mathematician, Dies at 85: 10/14/2010 -- a good image of him)
- Here's a suitable mascot for this unit: Bessie the fractal cow:
- So what is a fractal? A
couple of things we'll generally see:
- It's a process or graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative)
definition -- that is, we do something, and then we "do
it again!"
- Let's check out a beautiful gallery of natural
fractals.
It turns out that nature loves fractals, just like it loves Fibonacci numbers.
- Building deterministic fractals with sticks
- We'll start with the famous "Koch" fractal:
- Start with a straight stick.
- Replace the middle third of the stick with a little "tent" -- two third-length sticks.
- Do it again and again!
This fractal becomes infinitely long, but in a confined space! Very strange.... but this type of strange behavior is typical of fractals.
- Here
are some more lovely examples (including the Koch
fractal, but check out the beautiful bushes!).
- The Sierpinski triangle is an area fractal.
- First deterministically:
- After we're finished removing all that stuff, how much area is left in the triangle?
- First deterministically:
- We'll start with the famous "Koch" fractal:
- Now let's introduce a little bit of randomness: Sierpinski's
triangle again
The Chaos game - generating fractals using random movement!
- 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....
- 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!
- 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!
