Day 19, MAT115

• Fun activities
• Free lunch! (Yes, there is!)

This complicates the calculation of changes in number of initiators, and in lengths/areas.

But I hope that you realized that this is a creative process, and that, once the rules are established (e.g. the Kellie fractal, or the Rachel fractall), it is easy to create something interesting and complex.

• Key
• Median: 2.5/5
• I find this website particularly good for help in distinguishing the wallpapers.

• Gasket on-line
• Furthermore, you can do it all in three dimensions! (Or four, or five, or....)

• 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.
  • By the way, did you know that there are two different Mobius bands? They're mirror images of each other! (Symmetry, again....)

    Make one of each! When you connect it, think of screwdrivers: righty tighty and lefty loosy:

• Drawing on a band: draw a "center line" down the middle of your band. What happens? Remember to just stay on one side!

• Properties of a mobius band:
  • A mobius band has only one side.
  • A mobius band has only one edge.

• Cutting a band:
  • What happens when you cut a mobius band "in half"? Check out a bug on a band....
  • Cut that "half" in half again: what happens?

  • What if you cut a mobuis into "thirds"?

  • Other bands:
    • What if you cut a twice-twisted band?

      You get a link!

    • What if you cut a thrice-twisted band?

      You get a knot!

    • If you keep twisting (four times), you'll create something that the ancient Romans certainly knew about, and found lovely:
      Solomon's "knot" (we'll see why this is poorly named, and why I use quotes!).

    • And the most we'll ever twist together is five times, resulting in the third of our five knots.

• 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:
      

      One of these has rotational symmetry: which one?

    • 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).
  • A joke: "Why did the chicken cross the Mobius band?"

  • Vi Hart's Mobius Music Box: for playing Mobius music.

• Our next topic is 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!)
  • Draw the edge of a twice-twisted Mobius band.
  • Draw the edge of a thrice-twisted Mobius band.
  This class of objects will lead us to the notions of links and knots....

1. Snowflake on-line
2. Gasket on-line
3. The Fractal Microscope (click on the image)

• Vi Hart draws fractals