SSL Notes for meeting #10 (10/9)
Today's note-taker: Sam (Stephen next time)
Today's snack-provider: none (Hal next time)
NOTE: Tuesday meetings will be switched (until further notice) to 2:30-4:30 PM
to accomodate Melania.
Jim will update the posted minutes page with the minutes from the past three
meetings or so.
On Friday at 4:00 PM there is a talk on the Ninth Floor of Van Vleck being
given by Kirillov on the general topic of noncommutative bases and fractals,
and also how these tie in with Farey sequences.
Jim: A Farey sequence is . . .
(proceeds to draw Brocot tree)
0/1 1/0
0/1 1/1 1/0
0/1 1/2 1/1 2/1 1/0
0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0
(etc.)
Proceeding in this manner gives all the positive rationals.
We can define the "mediant" of fractions a/b and c/d (expressed in lowest
terms) as the fraction (a+b)/(c+d). This gives a "grown-up" name to the
sort of childish construction of the tree.
A Farey sequence is an ordered sequence of rationals where every rational
has denominator less than or equal to some given number, say k.
For example, letting k=4 we obtain the Farey sequence
0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1
A Farey sequence divides the interval into pieces. (Why do the fractional
sizes of these pieces have numerator 1?)
Jim draws the "Apollonian gasket," a fractal consisting of three tangent
circles, and infinitely many other circles tangent to three original circles.
Remarks that this ties into Markoff numbers, and that he can tell that this
should have some combinatorial structure; that is, the reciproal radii of
the circles in the gasket should be counting *something*).
Paul: Are we supposed to understand the combinatorics behind this?
Jim: No! Remember this periodic transformation (x,y) --> (y, (y+1)/x). This
is a fact about triangulations of polygons and has some combinatorial
significance for snake graphs as well. We will begin to look at these
kinds of things soon.
Hal: What is combinatorial about this? (Referring to picture of
triangulations of the pentagon).
Jim: It's about snakes! Replace triangles with "Y"s.
[Jim will tie this picture together in the next few weeks]
Jim will e-mail SSL the link to a new article by Itsara, et al., about
triangulations and Markoff numbers. Also, Jim will link to the Maple
worksheets from Tuesdays's meeting (10/7) on the main SSL page (or perhaps
the private page).
Carl: "Y"-graphs? What do we know about them?
Jim: I don't know. Relatively new ideas. General picture: Frieze patterns
article by Coexeter and Conway, but not a combinatorial interpretation.
On second thought, there isn't such a real connection here. We will say more
about Y-graphs and triangulations, Catalan numbers, etc.
Vote for collaborative, "free-for-all" work:
We will spend five minutes on . . .
Abby: Snakes of cubes and matchings
Martin: Working towards research problems discussed on Tuesday
Jim: Think also about knowledge transfer
Paul: Non-bipartite graphs?
Jim: Maybe next semester. Not much work done on perfect matchings.
Carl: A and B matrices, were we done? I have more work on this if people are
interested.
Jim: Let's reconvene and summarize our work at 5:20.
[Collaborative work on chalkboard]
Abby: Conjectures that the sequence consisting of perfect matchings of n cubes
satisfies the recurrence M_n = 2M_(n-1) + 3M_(n-2). Initial conditions?
Carl: We showed that the arithmetic and Fibonacci rules for counting perfect
matchings of (square) snakes using A and B matrices are valid.
Sam: Would like to know more about the significance of the *entries* of the
A,B matrices with matchings of snake graphs.
Carl: I've noticed something from playing around with these matrices, but I
can't quite put it into words yet.
Jim: Hint--use weights.
Hal: Jim and I will be sure to talk about non-SSL topics outside of SSL.