Day 21, MAT115

• Your exams are returned.
  • The exam
  • The curve: add 7 to your raw score, then divide by .7 to get your percentage.
  • The key
  • comments...

• Your Platonic solid homework is due, but I'll collect it next time. We're not in a hurry, and I want to say a little more about Platonics today.

• Here they are, in all their glory:

  The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia!
  Tetrahedron	Hexahedron or Cube	Octahedron	Dodecahedron	Icosahedron
  (Animation)	(Animation)	(Animation)	(Animation)	(Animation)

  A few more "applications" (or examples) of Platonic solids in our world:

  • Icosahedron
    • amoeboid protozoa (Circogonia icosahedra)
      

    • Viruses:
      • The Cliff Notes Version...
        

        "An array of viruses. (a) The helical virus of rabies. (b) The segmented helical virus of influenza. (c) A bacteriophage with an icosahedral head and helical tail. (d) An enveloped icosahedral herpes simplex virus. (e) The unenveloped polio virus. (f) The icosahedral human immunodeficiency virus with spikes on its envelope."

      • Bacteriophages and Virions

    • Dodecahedral
      • Viruses:
        "The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA"

  • More critters:
    • Canine Parvovirus
    • Enterobacteria Phage

• Now let's fill in the following table about the Platonic solids:

  	# of Vertices	Edges	Faces	faces at each vertex	sides at each face
  Tetrahedron
  Cube
  Octahedron
  Dodecahedron
  Icosahedron

  What conclusions can we draw from this data? Is there a pattern? (Of course there is!:) The table possesses a certain symmetry. The pattern leads to the concept of "Duality":

  • Each Platonic solid has a "twin" -- called its dual -- which we can discover from the table.

The Topologist's favorite riddle: What's the difference between a donut and a coffee cup?

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

• Let's make some! A Mobius band is made from a regular band using only a single twist.
  • There are two different Mobius 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! Use your clean paper for this.
    • Now deface one side of the band, or write a message along just one side of the band. What do you notice?

  • Properties of a Mobius band:
    • A Mobius band has only one side.
    • A Mobius band has only one edge.
    • A joke: "Why did the chicken cross the Mobius band?"

  • Applications of Mobius bands:
    • Conveyor belts
    • Kazakhstan's National Library in Astana
    • Recycling Topology
      • The official recycling symbol is Mobius, but over time an alternative has crept in:

        Now fold either of your two symmetric, mirror image bands flat, to make the recycling symbol.

        

      • Mobius, or only Seemingly Mobius?
      • My dad loved Mobius bands so much....
      • a history of the recycling symbol (and an AARP article on the history of the creation of the recycling symbol).

• Cutting a band:
  • What happens when you cut a Mobius band "in half"? Check out a bug on a band....
  • What if you cut it into "thirds"? (Premonitions of fractals....) Geez, we could do this forever!

• For the next two weeks we will be studying links and knots. How does the Mobius band relate to these?
  • Draw the edge of a Mobius band. (Maybe we can get a good artist to do it!) When you cut it "in half", it remains whole, and results in another band.
  • Draw the edge of a twice-twisted Mobius band. Now cut a twice-twisted band down the middle. What do you get?
  • Draw the edge of a thrice-twisted Mobius band. Now cut a thrice-twisted band down the middle. What do you get?
  This class of objects will lead us to the notions of links and knots....

• The Mobius handout

• There are also "star Platonic solids", as we see in this lovely stellated dodecahedron.
• The "Spirograph" is a wonderful tool for making symmetric objects:
  • http://cgibin.erols.com/ziring/cgi-bin/spiro.pl/spiro.html
  • http://wordsmith.org/~anu/java/spirograph.html