Day 20, MAT115

• Your quiz today is over Fractals.
• We have an exam next week, Thursday. It will be over fractals, wallpapers, Platonic solids, and symmetry.

• Reminder: You're invited! 2024 Math/Art Pop-Up Conference We'd love to have you:
  • Fun activities
  • Free lunch! (Yes, there is!)
  • Here's the link to sign up!
  This could be an opportunity to create something that could serve as the focal point of your project.
• Let's take a vote on our favorites in the Fractal Art Gallery

  Vote for three. Put your name on your ballot (there will be no ballot stuffing!:).

• We talked about projects, and visited Mathemalchemy to see where we're headed next: the Knotical Bay, and then to Infinity!
• The Mobius Band was then the focal point, "the band that never ends".

  We learned that

  • A Mobius band is made by giving a strip of paper a half twist.
  • It only has one side, and one edge. Today we'll draw some (hopefully some of you are better draw-ers than I!).
  • The Mobius band has left- and right-handed versions, which cannot be transformed into each other.
  • Cut one down the middle, and you get a longer band (twice-twisted); but cut that down the middle, and you get two bands, linked together (in a Hopf Link).

  That's because one had an odd number of twists and the other had an even number of twists.

• We cut a thrice-twisted band, and got our first interesting knot: the trefoil knot.
• We cut a frice-twisted band (that's four twists!), and got our first interesting knot: the trefoil knot.
• Then we squashed Mobius bands, to make the official recycling symbol and the other (fake!) version that has rotational symmetry of order three:
  

  Which reminds me of a logo (remember your logos, which will be presented in just a few weeks?)

  

Today is just a "gentle introduction to links and knots". Next time we'll start in on how to distinguish them, which involves a little more mathematics.

• Time to create some knots and links in a methodical fashion.

  Let me begin by mentioning that a belt is a great tool for creating mobius bands, and trefoil knots, and the like....

  How does the Mobius band relate to links and knots? Let's try to draw some edges:

  • Draw what I call "the wedding band" -- an untwisted band.
  • Draw the edge of a Mobius band (once-twisted).
  • Draw the edge of a twice-twisted Mobius band.
  • Draw the edge of a thrice-twisted Mobius band.
  • Draw the edge of a fourice-twisted Mobius band.
  • Draw the edge of a fiveice-twisted Mobius band.
  This class of objects will lead us to the notions of links and knots....

• Links:
  • Links are like the links you find in chains: constructed by "linking" loops (or rings) together.

  • Perhaps the most famous links in the world are the Olympic rings:

    Wikipedia	Here are the originals

    Are they the same?

  • Borromean Rings: "Is it possible to create a link of three loops (or rings) in such a manner that they are indeed linked, yet if we remove any one of the rings the other two become unlinked?"

  • Yes, indeed! Let's try it with rubber bands first.

    Standard diagram of the Borromean rings

    A realization of the Borromean rings as ellipses

    Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration

    
    (Thanks Wikipedia!)

  • More on the Borromean rings:
    • I always think of them in the context of this quote:

      "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.

    • Led Zeppelin, in the album sleeve of their album Led Zeppelin IV used it:

      "Drummer John Bonham's symbol, the three interlocking rings, was picked by the drummer from [Rudolf Koch's Book of Signs]. It represents the triad of mother, father and child, but also happens to be the logo for Ballantine beer." (from the Wikipedia article on Led Zeppelin IV).

      

      (thanks Matt!).

      Ironically I just discovered John Paul Jones's symbol in another brewing company (while enjoying one of their products); Arcadia Brewing company of Kalamazoo, Michigan, has this as their logo:

      

      Maybe every Led Zepellin symbol is on a beer somewhere?

    • If you want to draw the Borromean rings, you'd draw three circles, as in the first figure above:	but you'd want to indicate, somehow, that one ring is below another ring (aka the "Irish Trinity"):

      One more:

      

      Scene from Stora Hammar stone

      with a closeup of the rings:

  • Now it's back to knots. Let's see how to create our last two knots, which are called "twist knots".
    • Knot Theory
      • We begin with the "unknot", of course: a circle. This is the simplest knot, but not very interesting.

      • Trefoil Knot -- the simplest interesting knot. It has three crossings, and there's no way to make it any simpler! We will also call it the 3-knot:
        

      • Can we make a simpler knot -- one with either 1 or 2 crossings? Why not?

      • There are two different ways to think about making the trefoil knot: as a
        1. twist knot, and as a
        2. torus knot.
        Let's draw it in these two different ways.

      • The figure-eight knot:
        

    • Let's continue by talking about the video from Vi Hart, about snakes and graphs.
      • This was assigned because it relates to graphs; but also because it relates to knots and links. Vi is doing double duty for us here! (I hope that you also watched her video about the Mobius Music Box).
      • Notice her graphs in the first part of the video: those are the two non-planar graphs, and the graph of an octahedron!
      • By making a closed-curve squiggle graph, you're creating a knot. (A knot is a single piece of string, in which the two cut ends are ultimately tied back together to create a single continuous piece of string -- which may be "knotted"!)
      • "Borromean snakes": "No two snakes are actually linked to each other" (1:22).
      • Just before the Borromean snakes there is a trefoil snake (1:14).
      • "What kind of knots are you drawing and can you classify them?"

    • Now let's migrate our discussion of knots over to "Knot Mosaics":
      

      (an example of Solomon's Knot -- which you might notice is actually a link!)

      • Celtic Knot Mosaic Examples: (not quite the same idea, but pretty! And again, you'll see that there are links contained therein -- including Borromean rings)
        • Triangular paper (in case you want to try making some of those).

      • Knot Mosaics
        • help us to draw Borromean Rings, for example: use your Mosaic Graph Paper to create your own version of the Borromean Rings.
        • Rectangular Mosaic Tiles and symmetry
        • We can use them to make any knots: try your hand at
          • the trefoil knot:
            

          • The figure-eight knot:
            

    • Something fun to do with a few friends: can we make a human knot in the shape of a trefoil knot? If so, then we won't be able to undo it -- you and your friends will be knotted together forever! I hope that's a good thing....

• Triangle and rectangle paper
• Shodor:
  1. Snowflake on-line
  2. Gasket on-line
  3. The Fractal Microscope (click on the image)

• L-Systems in PostScript, from which I generated this image:
  

• Vi Hart
  • Vi Hart draws fractals
• L-System Fractals
• Nice website with examples of Cantor, Koch, and other fractals
• Here are some more lovely examples
• l-system generator: world's simplest math tool
• I first encountered the L-Systems fractals at britton.disted.camosun.bc.ca/fractals_arcytech/lsystems.html, but that's a dead link. However, I copied the files, and have a local copy here. However, web browsers these days don't seem to like the applet tag in javascript, so this doesn't work anymore in a standard web browser...:(