Day 31, MAT115

• Complete graph with 8 vertices
• All simple graphs with five vertices
• All simple graphs with four vertices
• The house layouts are just Konigsberg problems: this page and relate it to Euler's Konigsberg problem. Try some of the puzzles yourself.

So it will start with Fibonacci (Nim, and the rest), and continue with the golden rectangle, Pascal's triangle and Fibonacci, Fractals, and the beginning portions of graphs.

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!).

  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:

• 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.