Day 19, MAT115

• You have a homework due next time
• Our next test is Tuesday, 11/4
• Return of homework:
  1. Chinese Bamboo Counting Rods
  2. The ratios of the rectangles' longer to shorter sides should be compared with
  3. Use Pascal: add up the elements of the row "1 7 21 35 35 21 7 1" to get
  4. Use Pascal: How many ways to put 4 in one van, and 3 in the other? Pick the entry 35 from the row "1 7 21 35 35 21 7 1"

A few more answers:

Fig. 4.6 The platonic solids have been found living in the sea. The tetrahedron, somewhat rounded as if from internal pressure, is embodied in a protozoan called Callimitra agnesae, the cube is Lithocubus geometricus, the octahedron Circoporus octahedrus, the dodecahedron Circorrhegma dodecahedrus, and the icosahedron Circognia icosahedrus.

Duality:

"In the Hindu tradition, the icosahedron represents purusha, the male, spiritual principle, and generates the dodecahedron, representing prakriti, the female, material principle."

Here is an interesting note from The Parsimonious Universe

• Today we begin talking about graphs.

• Graphs:
  • The Bridges of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one?"
    
    • Solved by Euler (1735), the beginning of graph theory.
      • Graphs are made up of vertices (points) and edges that connect the vertices. More formally, a graph is
        1. A collection of points, called vertices.
        2. A collection of edges, each of which connects two vertices.

      • The degree of a vertex is the number of edges coming into it.

    • Euler's solution:
      • Konigsberg: To be traversable, there must be at most two vertices of odd degree.
      • Euler noticed the hand-shaking theorem: in any graph, the number of vertices with odd degree must be even.

    • 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.... (the pentagram)

  • 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 edge is 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?
    • Duality -- again!

  • Exercise: you draw all the simple graphs with four vertices.
    • Have you ever seen any of these before? Could we give any of them names?

  • 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 (but no loops).

  • Exercise: Let's see if we can show that the complete graph with five vertices is not planar:
    • We can show that the complete graph with four vertices is planar.
    • Why not? 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: