Day 37, MAT115

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Announcements:
- Reminder: our next exam is Monday, 11/21. It will cover from Platonic solids through Mobius bands.
- Your Mobius bands assignment is due today! Any questions, as you prepare for your exam?
- Remember to try my funhouse mirror generator: make your own funhouse mirror image, using my web interface and your own image, and I'll post them on our "website gallery".
Today:
- Question of the day: What is the collage fractal process?
- Fractals
  - Our intuitive definition of a Fractal:
    A world within a world!
    
    This key notion is "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)."
  - So what is a fractal? A couple of things we'll generally see:
    1. It's a process or graph that possesses "self-similarity".
    2. 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!
    1. Mandelbrot Set
    2. Newton's Fractal
  - Building deterministic fractals with sticks
    - Last time we looked at a simple example of a stick fractal.
    - Let's see the process, but with the aid of the computer:
      - The Koch Curve
      - The Koch Snowflake
      - The Sierpinski Triangle
      - Check out the beautiful bushes!
      - Heed the warning: "be careful about taking them too far." You can drive a computer to its knees....
  - Now let's introduce a little bit of randomness (nature must use randomness -- it can't get everything perfect!):
    - Sierpinski triangle:
      - 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
        
        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!
        An on-line version of the game
        Some sample output:
        
        with 10 points
        with 100 points
        with 500 points
        with 1000 points
        with an infinite number of points!
        
        Here's an on-line version that makes a better version of those plots, and that has nice general options.
        From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose....
  - The Sierpinski Triangle is found in the strangest places (e.g. Pascal's Triangle! -- see pa. 54 of your book)
  - The collage making process for generating fractals
    - The fractal images of our gallary could called "collage fractals": they're made of separate images, composed in some nice way. How does it work?
      - Start with an original image.
      - Embed the original image inside the original image itself (call that the new image).
      - Embed the new image inside the original image (call that the new image).
      - Repeat forever, and ever, hallelujah, hallelujah, forever, and ever, ....
    - The stick fractals can really be thought of as collages, too, as is the Sierpinski triangle.
    - Let's make another collage, using a grid.
    - Let's make another collage, using a knot (the trefoil).

Links:

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