Day 19, MAT115

• Your graph homework is due tonight (Monday, 3/22): I've fielded a few questions about it, and am happy to hear others.

  I'll get it graded ASAP, so that you'll have feedback before this week's exam.

• Your newer assignment, on squiggles, etc. is due on Thursday. I'll get it turned right around, too, so that you'll have feedback for the exam.

  Note that, although you are to draw some links and knots for this assignment, these will not be on the exam.

• This week (on the weekend of the 26th) we'll have our second exam. It will be like the last one -- involving an IMath part, and a Canvas part.

  It will feature everything from Fibonacci and golden spirals on. In particular we learned a little of the Egyptian's mathematics (multiplication and fractions), then about symmetry, the platonic solids, and more on graphs. Ending with planar graphs.

  Make sure that you've done all the associated readings, since you'll see some questions about those as well.

  If you have any specific questions that you'd like covered in a review, please let me know. I'll begin next time by reviewing -- and maybe that's all we'll do. But it will depend on your input.

• Last time we started something new, and highly tactile and visual: -- the amazing mobius band. That will not be covered on this upcoming exam, however. In the last few weeks we'll be looking into links and knots (today and next time), then fractals, on to infinity, and more! Perhaps miracles or angels, I'm not sure which. You probably didn't know that both miracles and angels can be given a mathematical treatment.

• Materials for today:
  1. Here is our video:
     1. Part I (links).
     2. Part II (knots).
     3. The "Knot work" (by hand)

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.

• Let me begin by mentioning that a belt is a great tool for creating mobius bands.

• How does the Mobius band relate to links and knots? First we'll create these, and then we'll try to draw some edges:
  • Draw the edge of a Mobius band.
  • 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....

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

  • 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 on to knots. Today we'll just look at a couple of the simplest 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....