Day 35, 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.

Nature is definitely weird and cool!

1. Stick Fractal 1
2. Stick Fractal 2, and Sierpinski's fractal (found within Pascal's triangle)

• the length of a fractal might grow to infinity by doubling each time, as in one of our stick fractals (Stick Fractal 2). If the initial length is $L_0=1$, then \[ L_n=2^n \] whereas the number of sticks increases by a somewhat surprising formula (because I counted the middle stick as a longer stick -- twice the length of the other 6 sticks). The generator breaks a stick into 7 sticks: \[ S_n = 7^n \]
• the area of the Sierpinski triangle was heading to 0, by the formula \[ A_n=\left(\frac{3}{4}\right)^n \] where the initial area is $A_0=1$. Meanwhile, the number of triangles is headed for infinity, by the formula \[ T_n=3^n \]

  In each case, we see the power of exponential growth (or decay, in the case of the area of Sierpinski's triangle still in play -- it keeps getting chewed up!).

• We also noticed how all the evens ended up getting thrown out in this process. If we expand Pascal's triangle, that will continue to happen! Pascal's triangle is the coolest math I know....
• Fractals in Pascal's Triangle

Gasket on-line

• 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 happens is you start to create something that the Romans knew about evidently: Solomon's "knot" (we'll see why this is poorly named, and why I use quotes!).

  • What if you cut it into "thirds"?

• 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:
      

    • 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)