- Homework for 4.7 is still being graded... sorry!
- New assignment:
Wed 3/3 Fractals and Dynamics - Read 6.3
- No new problem assignment.
- Dynamics.... What do we mean by dynamic models?
- The Game of Life:
- Is generally played on an infinite checkerboard, or on a
world that wraps back around on itself.
- The rules are:
- A living square will remain alive in the next generation
if exactly two or three of the adjoining eight squares
are alive in this generation; otherwise, it will die.
- A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.
- A living square will remain alive in the next generation
if exactly two or three of the adjoining eight squares
are alive in this generation; otherwise, it will die.
- Create your own creatures!
- Can you think of one that will lead to extinction?
- Can you think of one that will lead to a stable pattern?
- What's the smallest "animal" that can survive?
- Why are simple rules that result in complex phenomena important?
- Some people doubt the "theory of evolution", because they don't believe that simple rules can result in the complexity we see around us.
- If scientists can demonstrate that complex phenomena can arise from simple phenomena, then that argument holds less water.
- The upshot from the Game of Life:
"It is possible to construct logic gates such as AND, OR and NOT using gliders. It is possible to build a pattern that .... is theoretically as powerful as any computer with unlimited memory and no time constraints.... For example, the entire complexity of Microsoft Word could be programmed into the Game of Life with one starting state, and some inputs and outputs that would represent information like a document - all of the information would be there." (from this Wikipedia article).
All from those silly little rules on squares on a grid!
- Is generally played on an infinite checkerboard, or on a
world that wraps back around on itself.
- Other Dynamic models (e.g. population models): how many people do you suppose will be on the earth a hundred years from now?
- Start Somewhere (collect data!)
- Apply a process and get a result
- Apply the same process to that result to get a new result
- Apply the same process to the new result to get a newnew result
- Repeat patiently and persistently, forever.
- The Game of Life:
- Fractals
- 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!
- Building deterministic fractals with sticks
- Let's see the process, but a little slower (check out the beautiful bushes...).
- Now let's introduce a little bit of randomness:
- Sierpinski triangle (p. 436):
- 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....
- Sierpinski triangle (p. 436):
- Mindscape #14, p. 452.
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 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:
