Day 37, 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.

I added that as an explicit assignment on Canvas. Those are due Monday, 5/2 -- and some of you will be presenting that day.

The rest of the folks will present on Wednesday, 5/4.

These are cousins of the Platonic solids, with the wonderful name of Archimedes (my favorite mathematician) tied to them (my succulent came in A13).

It turns out that only two of these solids (Platonic and Archimedean) can tile the whole of space:

1. the cube (of course!); and
2. the truncated octahedron:
   

   

• If you cut a thrice-twisted band down the middle, you'll get a knot -- our first interesting knot, called a "trefoil" (three leaves).
• We learned how to draw twisted bands (well, one way: another way today!).

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

We want to get to know the cast of characters. We already know the following:

• Links:
  1. The unlink (two circles lying on top of each other!)
  2. The Hopf link
  3. Solomon's Knot (link)

• Knots:
  1. The unknot (a circle)
  2. The trefoil knot (three crossings)

• 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:

  • Knots:

    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.

      • There are two more "toroidal" objects we should draw:
        1. Solomon's knot (actually a link)
        2. the Cinquefoil: a 5-crossing toroidal knot
        Let's draw those.

        Then there are two more twist knots:

      • The figure-eight knot (the only four-crossing 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....