Day 24: Heart of Mathematics

• Biological populations. Example: How many people do you suppose will be on the earth in 2109 (in a hundred years)?
  • Start Somewhere (collect data!)

    World Population clock

  • Apply a process and get a result.
  • Apply the same process to that result to get a new result.
  • Apply the same process to the new result to get a newer result.
  • Repeat patiently and persistently, forever.

• The Game of Life:
  1. Is a poster child to prove something important: simple rules, repeated over and over, can give very complicated dynamics.
  2. Is generally played on an infinite checkerboard, or on a world that wraps back around on itself.
  3. The rules are:
     • A living square will remain alive in the next generation if exactly two or three of the adjoining eight squares are alive in this generation; otherwise, it will die.
     • A dead square will come to life if exactly three of its adjoining eight squares are alive; otherwise, it will remain dead.
  4. the website's version
  5. Today let's check I.15, p. 423.
  6. Then let's have a look at the Wikipedia site for Life to learn about gliders, breeders, and some oscillators.
  7. Create your own creatures!
     • Can you think of one that will lead to extinction?
     • Can you think of a new one that will lead to a stable pattern?
     • For the final, make sure that you know how to go from generation to generation!

Here's our mascot for this unit: Bessie the fractal cow:
	We might have easily taken the Japanese artist Hokusai.

So what is a fractal? A couple of things we'll generally see?

1. It's a graph that possesses "self-similarity".
2. It often possesses a simple and recursive (or iterative) definition.
• Let's start by having a glimpse at the beautiful gallery of section 6.1, for these are the subjects of our unit.

• 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!
  • Can you explain how these fractal images illustrate chaos?
  • The pictures remind me of the "acres of dominoes" from Robert Graves's Warning to Children.... Here's the poem in manyeyes.

• Building deterministic fractals with sticks
  • Fractal ferns (p. 443), and even names!
  • Let's see the process, but a little slower.

• Now let's introduce a little bit of randomness:
  • Fractal Islands, and coastlines.
  • Sierpinski triangle (p. 436):
    • 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! (p. 447)

      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....

• The collage making process -- see page 437.

• Mindscape #14, p. 452.

Links:

• www.bitstorm.org/gameoflife/
• www.math.com/students/wonders/life/life.html
• www.radicaleye.com/lifepage/
• grid paper