Day 20 in Mat115

Today:

• Reminders:
  • we've got a test coming up next Wednesday, 11/9 (through chapter 4).
  • You've got a preliminary version of your project paper is due the following Monday (11/14).

• Collect problems 4.5.

• Questions/Comments on old stuff?
  • #18, section 4.5
  • The game of life
  • The game of life is deterministic: that is, if you play the game again, starting from the same initial pattern, you'll get precisely the same outcome.
  • Even so, some marvellous patterns can arise.
  • How could we make the game of life more realistic?
  • There a lots of actual problems in life for which we want this property: for example, if you put $1000 in the bank at 5% interest compounded annually, you want that to give the same amount after 10 years, no matter when you start.

    If you put a buck in a machine and hit the "Rootbeer" button, you want a Rootbeer. You don't want a Dr. Pepper (because it's "close")!

    We like deterministic situations.

• Section 6.5:
  • Chaos is roughly defined as "sensitive dependence on initial conditions", but what this means is that if you start at two different points in a system (for example, in weather you might be thinking of a state of three variables, temperature, barometric pressure, and wind speed), very close to each other but distinct, then eventually the system will produce wildly different behaviors (even those the starting conditions were extremely close).

  • Some links:
    • The Lorenz attractor in 3-D (here's a local copy)

    • Make your own butterfly! (here's a local copy)

  • Cobwebbing - an example (#6)

Links:

• grid paper