- A homework set is due...
- I hope that you managed to read your section on graph, p. 116. We'll be talking about that subject a lot today.
- I'd like to shake up our schedule some:
I'd forgotten about the midterm election: we should do a unit on voting and surveying.
We'll follow that up with a day of "mathematics action" -- or "mathemaction" -- on election day. We will not have class on election day, Tuesday, November 2, but rather you will have an "outside" assignment for that day. More about that coming soon....
- Freehand (all of these have some nice symmetry)
- Using triangular graph paper (and this model, found at this Celtic Knot Mosaics site)
- tetrahedron and the Mercedes sign
- 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)
- Solved by Euler!
- 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.
- 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.
- Let's freehand a couple first:
- Trefoil
- A five-knot that "falls" out of the trefoil
- Let's use the basic five mosaic tiles to make a couple more knots:
- 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)
