Day 23, MAT115

For some of you, it's time to get real with yourself -- can you overcome a low grade in the early going? Can you start actually handing in homework? That sort of thing....

1. The golden rectangle (and ratio) is a fractal -- amazing what you can do so simply!
2. A4 paper (and A0, A1, etc.) is defined as a fractal:
   
   In both cases there's "the world within the world". Both are fractals.
   1. Fractals
      • 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 another like Bessie the fractal cow:
          

        • World's Most Famous Fractal (Benoit Mandelbrot, Novel Mathematician, Dies at 85: 10/14/2010 -- a good image of him)

      • 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 -- 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
        • Consider the famous "Koch" fractal:
          1. Start with a straight stick.
          2. Replace the middle third of the stick with a little "tent" -- two third-length sticks.
          3. 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.

          Let's make a variation of that. Use your graph paper, and a stick along the edge that's 27 units long. Best to use a pencil with a good eraser!

          Intstead of triangular tents, we'll make square tents.

        • 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:
            • fractals by addition
            • fractals by subtraction
            • fractals by subtraction (a local copy)

          • After we're finished removing all that stuff, how much area is left in the triangle?

      • 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!
          • with 10 points
          • with 100 points
          • with 500 points
          • with 1000 points
          • with an infinite number of points!

          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....

      • 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!