Day 14 in Mat115

Today:

• Announcements:
  • An old homework returned....
    • A beautifully done Fibonacci Spiral....
  • You have an assignment due next time, as well as a reading assignment. Please check the website!

• Platonic solids: a final take
  • How do we depict the Platonic solids? It's hard to draw three-dimensional objects, but it turns out that it's easy to draw them (well, some of them!) projected into two-dimensions -- as graphs.
    • Easy: tetra and hexa
    • Harder: octa
    • Pretty hard: icosa and dodeca
    (Here they are -- from source) These representations are examples of graphs, which is a section of our text that I've asked you to read for your homework (p. 116).

  • Last time we linked the Platonic solids back to the golden rectangle (and hence Fibonacci numbers!) by showing that there are golden rectangles in the heart of an icosahedron.

    Now: as I said then, there's a third concept contained in the same interlocking golden rectangles: that of the Borromean Rings

    • This topic reminds me of a famous quote from Benjamin Franklin, made in the Continental Congress just before the "rebels" signed the Declaration of Independence, 1776: "We must, indeed, all hang together, or most assuredly we shall all hang separately."

    • Here is the same idea mathematically: "Is it possible to link three rings together in such a manner that they are indeed linked, yet if we remove any one of the rings the other two become unlinked?"
    • Answer: Yes!

      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

    • We can make a set of Borromean rings with rubber bands (although we'll have to cut one, and re-tie it). Let's try that....

• Knot Mosaics: (an example of Solomon's Knot)
  • Examples: (not quite the same idea, but pretty!) Celtic Knot Mosaics
  • 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

• The Rolfsen Knot Table
• Making an icosahedron with golden rectangles (well, 3x5 cards....)