Day 18 in Mat115

Today:

• Announcements:
  • Remember, we're going to be going to the polls next week: you have a reading about our electoral system (and some failings) for next time: please read!

  • Homework set(s) returned:
    1. Platonic Euler
    2. Can't find a soccer ball?
    3. Easier edges.
    4. Harder edges!
    5. Applications of Platonics
  • Graph homework, too:
    • All simple graphs with five vertices (there are 34!) This image is poorly drawn, in that we can't see the duality very well. The graphs from six edges on up should have been drawn with arcs exactly where there aren't arcs in the preceding graphs....
    • For the drawing of the knots:
      If we can't see the over/under pattern, then we can't identify the knot!
      Didn't I ask for a nice drawing, on a separate page? It's for your own good, for your own study as you prepare for an upcoming exam some day....
    • The tree: it's just vertices and arcs. And there are no cycles!
      • Famous country song: "I'm my own grandpa"
    • Different five knots.... (see The Rolfsen Knot Table)
      • One knot has radial symmetry (rotational symmetry); the other doesn't
      • One knot can be turned into the unknot with a single change in cross-over; the other can't
      • Label the crossings. Different patterns can only achieved with different knots
• Review of Fractals:
  • Thanks for the 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.

  • 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!

  • Building deterministic fractals with sticks and areas:
    • A fractal worksheet

  • Now let's introduce a little bit of randomness:
    • 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....

• Voting methods and sampling.

  We'll start with an example: the "Sixth grade vice-presidential vote"

  	6 boys prefer	4 boys prefer	8 girls prefer	4 girls prefer
  First Choice	Chad	Chad	Gwyn	Courtney
  Second Choice	Courtney	Gwyn	Courtney	Gwyn
  Third Choice	Gwyn	Courtney	Chad	Chad

  Three voting schemes, three different winners.

  1. Plurality voting -- Chad wins
  2. Vote for Two -- Courtney wins
  3. Borda Count Method -- Gwyn wins

• My favorite system is instant run-off, until we have a majority. Who wins?

  This one was proposed for situations like the "Nader voting" that helped defeat Al Gore and so elect George Bush in 2000: if Nader voters could have used instant run-off, they could have voted for their favorite (Nader), but then fallen back to Gore when Nader didn't take the vote.

• Voting Scheme worksheet
• Borda Count wins