Day 34, MAT115

• This session will be captured on Zoom, if I remember to turn it on, and record it.

  Our page of zooms and the play-by-plays.

• Just keep your eyes open and you'll find cultural references to material in this course: I was looking at my issue of Science last night, and here was a reference to a Mayan calendar from 300-200 B.C.E.

• I'm still putting together your new IMath problems. To come soon.... Just to check you on symmetry, and Platonics, and fractals.

• You have an assignment: a reading for next time, and making your own set of Platonic solids. We'll start in on Mobius bands next week, but I'll give you a sneak preview.

• We talked about the definition of fractals, looked at examples we'd seen in the course, and some that nature provides (e.g. Romanesco broccoli).

  As I looked for a link to "Romanesco broccoli", I grabbed an image from a seed catalog, and checked it for spirals. Looking for my friend Fibonacci....:)

We'll start with another stick fractal, or two, and then we'll attack a fractal based on areas (and you're not surprised to find it in Pascal's triangle, because everything is found in that!).

• The big fractal picture is this:
  1. You start with an object (the "initiator")
  2. You do something to it (the "generator"), which results in more initiators.
  3. Do it again.
  Nature uses this idea all the time, in broccoli and elsewhere:
  • a simple rule
  • repeated over and over and over.... but
  • simple rules, intense content!

  On that note, let's take a look at the Fractal Microscope (click on the image!)

• I'll let Vi Hart give us an entertaining intro to making fractals -- she's quite fun, but she talks too fast!

• Last time we made (or tried to make!) the Koch snowflake:
  • Today we'll make an alternate version of that, but with squares. I've got a handout that you can use to do this. Start with the side with the squares on it.

    Here's the generator (a stick 32 units long):

    1. Cut half the line from the middle, and rotate it 90 degrees.
    2. Then add four new quarter lengths to connect it all back together, so that it looks like you've added two square bumps to the original.
    3. One important point is that there are then six smaller sticks, and one longer stick. So they get treated different at every step.

  • And then we'll discover Sierpinski's triangle inside of Pascal's triangle. A fractal who's initiator is a triangle!
    • The initiator's an equilateral triangle.
    • The generator is to take an equilateral triangle from the initiator, the largest you can that's pointing in the opposite direction.
    • You're left with three equilateral triangles.
    • Do it again....
    • What do you notice about the numbers inside the triangles?