Day 26, MAT115

• You have a new assignment on Fractals (due next Thursday).
• Your first homework on knots is not yet graded. Here's the solution, however.
• Your second homework on knots is due.
• Don't forget that your logo day is coming!
• I've assigned one more reading from your text.

We looked at two fractals in particular:

• Koch fractal
• We ended last time on "Emma's fractal", which can be characterized using a stick initiator, and her generator:

  I've found a Java program (from here) that we can use to visualize fractals using what is known as the "L-System" (Lindenmayer Substitution Fractals).

  So Emma's fractal is f=f++f---fff+++f--f (sort of).

• 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 21st 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:

  See how these arise out of the idea that we carry out a process, then "do it again, do it again"....

  • Our Fibonacci (and golden) spirals
  • Dual Platonics: e.g. a tetrahedron within a tetrahedron within a tetrahedron.... Maybe cooler are the other pairs, where we alternate solids (cube then octa), then repeat the process.
    

  • The tree diagram of the rabbit problem that gave rise to the Fibonacci numbers.

  • Additional examples you might have seen before:
    • Here's a suitable mascot for this unit: Bessie the fractal cow:
      

    • Russian Dolls
    • World's Most Famous Fractal (Benoit Mandelbrot, Novel Mathematician, Dies at 85: 10/14/2010):
    • The A0 Paper Fractal

      "A paper" is something you've no doubt encountered before: it's the long sheets we occasionally use (usually "A4" paper), which is far more common in Europe.

      "A paper" is constructed so that, if folded in half (do we say the long way or the short way?), you get a sheet which has exactly the same shape -- that is, the ratio of its side lengths is the same as the original sheet.

      

      Let's see what the dimensions of A paper must be so that this requirement is fulfilled....

      Call the long dimension of the A1 paper above , and the short dimension 1.

      • Then what are the dimensions of the A2 paper, according to the design of A paper?
      • What is the value of ?

      Why not an A0 paper spiral? Let's try that.... This is the spiral, created using non-squares each time:

      

      In both cases there's "the world within the world". Both are fractals.

• 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 (the "initiator").
    2. Replace the middle third of the stick with a little "tent" -- two third-length sticks (the "generator").
    3. Do it again and again!
    

    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!

    Triangular paper (you will want to try making one of those).

    What happens to the length of the Koch fractal? Let's compute it for the first few cases.

    How big of an area contains the Koch fractal? A finite area. And how long is it? Infinitely long! This is kind of puzzling.... But, in a way, it helps to explain things like DNA: if you twist something up enough, you can get a very long thing in a tight space!

    Alternatively: instead of triangular tents, we could make square tents. What happens if we do that? Again, use a power of 3 to make your fractal (because we're cutting things into threes).

  • Here are some more lovely examples (including the Koch fractal, but check out the beautiful bushes!).

• Building deterministic fractals with areas:

  The golden spiral is a fractal generated with areas rather than sticks (as are the Fibonacci spiral fractal and the A-paper spiral).

  The Sierpinski triangle is a more famous area fractal.

  Again, there is an initiator and a generator: the object, and then what one does to it to create new initiators.

  • Triangular paper is good for Sierpinski....

  • 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 (this means that the fractal is not formed deterministically anymore): Sierpinski's triangle again:

    The Chaos game - generating Sierpinski's triangle using random movement!

    Start with an equilateral triangle. Pick a point, anywhere. Roll a die. If you roll

    • 1 or 2, move half way to vertex A, and make a mark;
    • 3 or 4, move half way to vertex B, and make a mark;
    • 5 or 6, move half way to vertex C, and make a mark;
    do it again!
    • 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!

• Let's have a look at a recent (January, 2015) addition to the fractal family: the Harriss fractal). It starts with the golden rectangle, and then makes a slight change.

• I've found a Java program (from here) that we can use to visualize fractals using what is known as the "L-System" (Lindenmayer Substitution Fractals).
  • a simple rule
  • repeated over and over and over.... but
  • simple rules, intense content! Fantastic creations are a snap.

  You might try creating some of your own.

  The Koch fractal is this:

  1. Axiom: f
  2. Rules: f=f+f--f+f
  3. Start Angle: 90
  4. Turn Angle: 60

• Now back to Pascal's triangle. What does it have to do with Sierpinski?

  Let me re-introduce you to Vi Hart, who will talk us through the relationship between Pascal and Sierpinski. She talks really fast, however!

  Then we'll try her method to see how, within Pascal, there's Sierpinski....

  • even plus even is even
  • odd plus odd is even
  • even plus odd is odd
  So Vi remarks that we don't need to actually calculate the numbers in Pascal's triangle to see whether the next number will be even or odd. Let's do an E/O Pascal's....

• L-System Based Fractals (and a local copy).