Day 22, MAT115

• Announcements:
  • Homework is returned.
    • Why oh why oh why do you use words when pictures would be so much better? (#2)

  • We have an exam next Tuesday: the topics will go from Fibonaccis to Graphs, and include golden rectangles, Platonic solids, Fibonacci Nim. Everything Fibonacci to Euler!

  • You have a new reading assignment for next time: visit this website and read about "human knots": have you ever played this before?

• Question of the Day:
  How are knot mosaics and links related?

• 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!)

  • icosahedra, golden rectangles, and interlocking rings. It turns out that there are golden rectangles in the heart of an icosahedron
    Let's make one using 4x6 index cards.
    • Use symmetry!
    • How far off golden are we?
    • Do you see the icosahedron?

  • 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!).

    • 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 (e.g. the "Irish Trinity", as Cody called it):

      One more:

      

      Scene from Stora Hammar stone

      with a closeup of the rings:

• 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

  • 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

• Knot Theory
  • Trefoil Knot -- the simplest interesting knot.

    From the "Why Knot Art?" gallery of Cliff Long:

    • Spanning Surface of a Trefoil Knot (wood)
    • Same thing, in Marble
    • Some twists on a Trefoil Knot....

  • Figure-Eight Knot (aka the "four-knot")
    • Here's the figure-eight knot, in more detail.

  • Now: let's make some human knots! The trefoil and figure-eight knot.

  • 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.

Links:

• Make a mosaic, using triangular tiles:

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