Day 23, MAT115

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!)

• Use symmetry!
• How far off golden are we?
• Do you see the icosahedron?

• 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!).

• 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:

• 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.
• "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?"

(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

• 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.

• 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....

• 5-Knot
  • Actually, there are two kinds of 5-knots
  • To help you, we have the Rolfsen Knot Table
  • The cinquefoil knot
  • The 3-twist knot
  • How can we tell them apart? Are they really different? For that we need to study knot equivalence, and our primary tool, the Reidemeister moves.