Day 20 in Mat115

Today:

• Announcements:
  • Your Platonic homework is returned.
    • Some of you gave very compelling arguments about why hexagons and octagons can't be used to make Platonic solids.
    • Planar graphs are graphs that can be drawn such that no two edges intersect. Every Platonic graph can be drawn this way, so they are planar graphs.

    • The Mercedes logo is like a tetrahedron:

    • Euler's formula works for every planar, connected simple graph: F-E+V=2. This includes the soccer ball.

  • We have a test next time. It will cover the following topics:
    • Fibonaccis and the golden mean
    • Fibonacci spiral
    • Golden rectangle
    • More on Pascal's triangle (and extensions like "Leibniz's hexacomb")
    • Egyptian multiplication
    • Egyptian division
    • Symmetry
    • Platonic solids

  • For the test you'll need a calculator, and you may use the Platonic solids (created from this sheet).
  • Remember that there are also some readings that you should have done. That was a killer on the last exam. Do your readings!

• So we'll start with review, featuring any of those topics. As long as you have questions, we'll review. Then we'll move on....

• Today's Question of the day: What's the difference between a donut and a coffee cup?
  • Today's new topic is the Mobius band: Mobius bands are a part of geometry, but relate to a special area called "topology" -- the study of form and shape.

    The Topologist's favorite riddle: What's the difference between a donut and a coffee cup?

    Answer: There isn't any! (to a topologist....)

  • What is a Mobius band?
    • Let's make some! A Mobius band is made from a regular band using only a single twist.
      • There are two different Mobius bands? They're mirror images of each other! (Symmetry, again....)
        • Check yours against someone else's, and convince yourself that they're really different.
        • Make one of each! Use your clean paper for this.
        • Now deface one side of the band, or write a message along just one side of the band. What do you notice?

      • Properties of a Mobius band:
        • A Mobius band has only one side.
        • A Mobius band has only one edge.
        • A joke: "Why did the chicken cross the Mobius band?"

      • Applications of Mobius bands:
        • Conveyor belts
        • Kazakhstan's National Library in Astana
        • Recycling Topology
          • The official recycling symbol is Mobius, but over time an alternative has crept in:

            Now fold either of your two symmetric, mirror image bands flat, to make the recycling symbol.

            

          • Mobius, or only Seemingly Mobius?
          • My dad loved Mobius bands so much....
          • a history of the recycling symbol (and an AARP article on the history of the creation of the recycling symbol).

    • Cutting a band:
      • What happens when you cut a Mobius band "in half"? Check out a bug on a band....
      • What if you cut it into "thirds"? (Premonitions of fractals....) Geez, we could do this forever!

    • For the next three weeks we will be studying links and knots. How does the Mobius band relate to these?
      • Draw the edge of a Mobius band. (Maybe we can get a good artist to do it!) When you cut it "in half", it remains whole, and results in another band.
      • Draw the edge of a twice-twisted Mobius band. Now cut a twice-twisted band down the middle. What do you get?
      • Draw the edge of a thrice-twisted Mobius band. Now cut a thrice-twisted band down the middle. What do you get?
      This class of objects will lead us to the notions of links and knots....

  Links:
  • The Mobius handout
  • Konigsberg bridges page
  • There are also "star Platonic solids", as we see in this lovely stellated dodecahedron.
  • A lovely treatise on symmetry
  • The "Spirograph" is a wonderful tool for making symmetric objects:
    • http://cgibin.erols.com/ziring/cgi-bin/spiro.pl/spiro.html
    • http://wordsmith.org/~anu/java/spirograph.html