- There is an assignment (due this Friday, 11/4)
- Fibonacci Spiral Images:
- The winners!
- Winners get a free extra get-out-of-homework-free card.
- Jessie donated a bit of artwork to the cause: what kind of rings are these?
- You have a new reading assignment for next time: visit
this website and read about "human knots": have you ever played this before?
- Question of the day: How do we know when we can't avoid false
intersections (where edges cross over each other with no vertex
there)?
- Graphs (final thoughts), and then Knots:
- The Bridges
of Konigsberg problem
- Solved
by Euler (1735), the beginning of graph theory.
- Graphs are made up of vertices (points) and edges that connect the vertices.
- The degree of a vertex is the number of edges coming into it.
- Hand-shaking theorem: in any graph, the number of vertices with odd degree must be even.
- Konigsberg: To be traversable, there must be at most two vertices of odd degree (hence either two or zero odd degree nodes)
- An alternative solution -- thanks to spiked math
- Which of the Platonic solid graphs can be traversed without backtracking? (here they are)
- Solved
by Euler (1735), the beginning of graph theory.
- 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?
- We can show that the complete graph with four vertices is
planar.
- 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:
- The Bridges
of Konigsberg problem
(an example of Solomon's Knot -- which you might notice is actually a link!)
- Celtic Knot Mosaic Examples: (not quite the same idea, but pretty! And again, you'll see that there are links contained therein -- including Borromean rings)
- Knot Mosaics
- help us to draw Borromean Rings, for example: use your Mosaic Graph Paper to create your own version of the Borromean Rings.
- Rectangular Mosaic Tiles and symmetry
- We can use them to make any knots: try your hand at
