Day 16 in Mat115

Today:

• Announcements:
  • Sorry: I haven't graded the homework set yet.
  • Reminders:
    • There's no class next Tuesday. Fall Break!
    • There's a homework due next Thursday.

• Quick review of graphs:
  • The Bridges of Konigsberg
    • Solved by Euler!
      • The degree of a vertex is the number of edges coming into it.
      • To be traversable, there must be at most two vertices of odd degree.
    • An alternative solution -- thanks to spiked math
    • Which of the Platonic solid graphs can be traversed without backtracking? (here they are)

  • You've probably been making graphs for a long time....

  • Definition of a graph: a graph is
    • a collection of vertices,
    • a collection of edges, and
    • a set of rules, one for each edge, telling how each edges connected to its pair of vertices.

  • Graphs don't change by bending edges, but breaking them or detaching edges from their vertices (and hence creating new vertices) gives new graphs.

  • A graph is simple if it doesn't have any loops (edges connected from a vertix to itself), or multiple edges with the same two vertices.

  • Exercise: draw all the simple graphs with three vertices.
    • A cycle is a route from a node back to itself that doesn't retrace steps.
    • One of these graphs is a tree: a graph that has no cycles.
    • Where have we seen trees used before?

  • Exercise: you draw all the simple graphs with four vertices.

  • NOTE: we can use duality to reduce the work!

  • Planar graphs are graphs that can be drawn such that no two edges intersect. Every Platonic graph can be drawn this way, so they are planar graphs.

  • Complete graphs are simply graphs with connections between every pair of vertices.

  • Exercise: show that
    • The complete graph with five vertices is not planar. What happens?
    • The three utilities and three houses graph is not planar.
    • Any non-planar graph has a copy of one of these two graphs in it somewhere, as a subgraph.

• Glance back at knots:
  • Let's freehand a couple of knots first:
    • Trefoil
    • A five-knot that "falls" out of the trefoil
  • Then we'll use the basic five mosaic tiles to make a couple more knots:
    • Here's some good graph paper for this exercise.
    • the trefoil knot (try the Heart Knot version)
    • Figure eight (and other versions of the the figure-eight knot)
    • Cinquefoil Knot (perhaps this one is easier to see how to do with mosaic tiles)

• Move on to Moebius bands: the Band that keeps on playing.
  • Moebius bands are a part of geometry, obviously, but relate to a special area called "topology".

    Riddle: What's the difference between a donut and a coffee cup?

    Answer: There isn't any! (to a topologist....)

  • What is a Moebius band?
    • Let's make some!
    • How do we know that the moebius band is not "topologically equivalent" to a regular band?

  • What happens when you cut a moebius band "in half"? Let's check out a bug on a band....

  • What if you cut it into "thirds"?

  • By the way, did you know that there are two different Moebius bands? They're mirror images of each other! (Symmetry, again....)
    • Check yours against someone else's, and convince yourself that they're really different.
    • Make one of each!

  • My dad loved moebius bands so much....
  • Applications of Moebius bands:
    • Conveyor belts
    • Kazakhstan's National Library in Astana
    • Recycling Topology
      • a history of the recycling symbol
      • Recycling -- Moebius, or only Seemingly Moebius?
    • A joke: "Why did the chicken cross the Mobius band?"

• The Rolfsen Knot Table