- There is a new homework assignment (due next Friday, 11/4)
- Links: review of Borromean rings
- Borromean rings: "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.
- 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 [Rudolf Koch's Book of Signs]. 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).
- The RCO (Really Cool Object): icosahedra, golden rectangles, and
Borromean rings
-
If you want to draw the Borromean rings, you'd draw three circles, as in the first figure above: but you'd want to indicate, somehow, that one ring is below another ring (e.g. the "Irish Trinity", as Cody called it): One more:
Scene from Stora Hammar stone
- 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 (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?
- 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
- Borromean rings: "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.
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