Day 37, MAT115

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)."

1. It's a process or graph that possesses "self-similarity".
2. It often possesses a simple and recursive (or iterative) definition.

• a simple rule
• repeated over and over and over.... but
• simple rules, intense content!

1. Mandelbrot Set
2. Newton's Fractal

• 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....

• 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 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).