Exam 2 Review

Readings/Viewings:

"Working Your Quads" (p. 67 of our text)
Some readings (found here -- view each image separately for easier reading):
- Background History of "Pascal"'s triangle (and Arabic Mathematics)
- Properties of "Pascal"'s triangle
This beautiful summary of number systems, from The History of Mathematics, By Anne Rooney.
A symmetry handout: featuring definitions of rotational and reflectional symmetries, and examples.
The Enemy of My Enemy (complete graphs, labelled graphs, weighted graphs)
Group Think (complete, directed graphs)

Topics/Concepts:

Fibonacci numbers and spirals
- Know your Fibonacci numbers: 1,1,2,3,5,8,13,21,34,55,89,....
- Fibonaccis within Pascal's triangle
- attaching squares to squares to make a spiral
The golden ratio and rectangle
- ratios of Fibonaccis lead to gold: 3/2, 5/3, 8/5, 13/8, ...., $\phi=\frac{1+\sqrt{5}}{2}\approx{1.618}$
- Removing squares is how we think about deriving the value of $\phi$. But you can also spiral
- Quadratic formula -> phi
Egyptian Multiplication
- Powers of 2 -- doubling
- Binary decomposition of numbers
Egyptian Division
- Powers of 2 -- doubling/halving
- multiplication table backwards
- unit fraction table
- Why did they do it this way? Fair division?
Symmetry
- rotational -- order 1 if no rotational symmetry
- reflection: it must be exactly the same on the two sides, like when you hold an object up to a mirror.
- Symmetry is frequently associated with beauty.
The Platonic Solids
- Starting with regular polygons (as faces; very symmetric polygons)
- Know their names: Tetrahedron, Hexahedron (or Cube), Octahedron, Dodecahedron, Icosahedron
- Know their characteristics
- Know their duals
- Know the Platonic solid graphs -- we project them onto the page.
Graphs (and trees are graphs)
- Definitions:
  - What is a graph?
  - Vertex/Node
  - Edge/Arc
  - degree of a vertex
  - Directed graph
  - Simple graphs (no loops or parallel edges)
  - Complete graphs (simple, every pair of nodes connected)
  - Planar graphs (no false intersections)
  - $K_5$, $K_{3,3}$, and Kuratowski's theorem
- Examples:
  - Facebook graphs (where nodes are labelled)
  - Directed graphs (e.g. mattresses)
  - The enemy of my enemy is my friend
  - Pascal's triangle applications
- Rules:
  - Euler paths -- can you trace the graph without picking up your pencil, and without going over any edge twice?
  - Euler's formula for planar graphs (ran-to: R-A+N=2)
  - Pascal's triangle applications for counting edges in the complete graph of $n$ vertices; or types of each type of simple Facebook graph -- those with each different possible number of edges.

Website maintained by Andy Long. Comments appreciated.