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.