Day 39, 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).

Two sets have the same cardinality if there is a one-to-one correspondence between them.

And if the sets are finite, the proper subset is always smaller....
but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!

• Welcome to the Hotel Cardinality
• Hotel Cardinality and the Infinite Life Insurance Co.
• Hotel Cardinality and the Limitless Bus Barn Co.