- The key for the exam is on-line.
- Curve: add 6 to your score for your curved grade on exam 2.
- You had a reading assignment: Graphs, Idea 29, p. 116 --
have you done it?
- There is no assignment due Friday -- take a break! (But do your reading!).
- Links:
- Links are like chains: constructed by "linking" loops (or
rings) together, into a necklace holder, say.
- Borromean
Rings: "Is it possible to create a link of three loops (or
rings) in such a manner that
they are indeed linked, yet if we
remove any one of the rings the other
two become unlinked?"
- Yes, indeed! Led Zeppelin, in the album sleeve of
their album Led
Zeppelin IV used it:
"Drummer John Bonham's symbol, the three interlocking rings, was picked by the drummer from the same book. It represents the triad of mother, father and child, but also happens to be the logo for Ballantine beer." (from the Wikipedia article on Led Zeppelin IV).
Standard diagram of the Borromean rings
A realization of the Borromean rings as ellipses
Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration
(Thanks Wikipedia!) - icosahedra, golden rectangles, and
interlocking rings. It turns out that there
are golden rectangles in the heart of an
icosahedron
- Use symmetry!
- How far off golden are we?
- Do you see the icosahedron?
- I want you to think about drawing Borromean rings: you'd draw three circles, as in the first figure above:
You'd want to indicate, somehow, that one ring is below another ring (e.g.
The Borromean rings as a symbol of the Christian Trinity, from a 13th-century manuscript. - I'm not sure that Led Zeppelin did, which means that the symbol is really
the Borromean rings -- just looks like them:
- Links are like chains: constructed by "linking" loops (or
rings) together, into a necklace holder, say.
- Graphs:
- 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.
- 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?
- 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
