Exam 2 Study Guide

Here are a few comments about the exam:

• Your exam will cover the following material:
  • Platonic Solids
  • Rings and Knots
  • Mobius bands
  • Graphs
  • Fractals
  • Voting
  • Chaos

• You should have done all the assigned reading: you will be responsible for the content, even if we have not explicitly studied it in class. I will put a few modest questions on the exam, to check to see if you did your reading.

• Platonic Solids
  • We started with regular polygons, and defined Platonic solids from them.
  • You should know all the solids' names.
  • Know the properties of each solid (e.g. number of edges, vertices, faces)
  • Understand duality
  • Know some applications of each.
  • You may have your paper collection of Platonic solids with you for the exam (but you may not share -- BYOPS!).

• Rings and Knots
  • Borromean rings
  • Borromean rings, icosohedra, and the golden rectangle
  • There are five knots you should know:
    1. The unknot
    2. The trefoil knot
    3. The four knot
    4. The five knot 5_1, the "Cinquefoil Knot"
    5. The five knot 5_2, the 3-twist knot
  • Knot Mosaics, and using them to represent knots

• Mobius bands
  • topological equivalence (a donut is the same as a coffee cup!)
  • How are they defined?
  • What are their amazing properties?
  • What happens when you cut one down the middle?
  • What happens when you cut one into "thirds", by cutting down the middle but starting only a third of the way in?

• Graphs
  • The Bridges of Konigsberg
  • Euler Paths
  • How do the Platonic solids look projected as graphs?
  • Simple graphs
  • Planar graphs
  • Complete Graphs
  • Duality in graphs

• Fractals
  • "Worlds within worlds"
  • Making fractals with sticks (simple rule, do it again, do it again, do it again....)
  • Making fractals by subtraction or addition of areas
  • Know some examples:
    • Koch snowflake
    • Sierpinski triangle
  • Infinite length and areas of fractals

• Voting
  • Know various voting schemes and how to use them.
  • Arrow's impossibility theorem
  • Know that the result of an election is often determined by the voting scheme used.
  • You should be able to describe our strategy in approaching voters in the election. What were our objectives, and how did we attempt to achieve them?

• Chaos:
  • History: Lorenz and his weather experiment
  • Other examples:
    • Hexstat probability generator
    • Double pendulum
    • The "swinger"
  • Informal definition: sensitive dependence on initial conditions
  • Iterative maps, such as the logistic map
  • Cobwebbing
  • The game of life (know the rules, various interesting patterns -- leading to extinction, persistance, periodicity, growth, movement)
  • The relationship between fractals and chaos