Meeting Thursday Oct. 30, 2003
Note-taker: Sam
Snacks: Paul
Hal will take notes next time.
Emilie will bring snacks next time.
Hal: Stephen, could you explain your suggestion about simplifying the calculations I
was doing in Maple regarding the Soddy equation and the Apollonian gasket?
Stephen: Sure. What you want to do is to use four variables instead of three. [Stephen
draws the tangent circles and writes Soddy's equation 2(a^2+b^2+c^2+d^2) = (a+b+c+d)^2]
[Hal will work on this during the meeting and report back]
Stephen: One we thing we still need to resolve is the assertion in the Itsara article
which Sam brought up on Tuesday that an arbitrary snake can be transformed into a
2-by-k straight snake for some k, with only single and double edges.
Martin: Do we believe this to be true?
Paul: We do not currently have a way of doing it.
[Hal goes up to the board. He has an idea of extending snakes instead of collapsing
them. It amounts to having straight snakes with double, single, or *zero* edges.]
Hal: Clearly doesn't match up with claim, but might be useful somehow.
Martin: I'd also like to work on the alternating pattern in the matchings of the
2-by-2-by-n snakes.
Emilie: I have been working on Jim's idea, where we split a triangle into six and have
some way of ascribing rational values to the sides. I have a system of nine (non-linear)
equations that I am going to try to solve in Maple.
Stephen: We'll work in groups or individually and then reconvene at ten till to
summarize for the minutes.
[At 5:20 . . .]
Stephen: I'll start. I mostly talked with Sam about the Itsara assertion, and I read
some of Cluster Algebras II.
Abby: Talked with Carl about what he has been working on - matchings of faces of cubes.
Also talked with Paul about some possible bijections (or even well-defined maps) of
2-by-2-by-n snake graph matchings and pairs of tilings of the 3-by-2n grid.
Downloaded graph.tcl and got it to work sufficiently well.
Martin: Worked with Hal and his Apollonian gasket cluster algebra equations in Maple.
Worked with Paul a little also on the above. Perhaps my previous bijection between
partial sums of the one sequence (1,3,11,41,153,...) and the other sequence
(1,4,15,56,209,...) could be of some help.
Hal: Worked on the Apollonian gasket and programming in Maple, I still ended up with
lots of nested radicals.
[Stephen interjects and draws Soddy's equation on the board]
Stephen: It's like the Markoff recurrence where we have been substituting.
Hal: Oh, I see it. I'll have something for next time for sure.
Sam: Worked with Stephen. Realized a few things. First, the assertion is not true for
arbitrary snakes, in fact, it is not true for "staircase snakes" at all (that is,
snakes with n boxes and n+1 matchings). Martin was able to show exhaustively that
there is no such configuartion of a straight rectangle with double and single edges
having six perfect matchings.
Second, Stephen and I sketched a proof of why this identifying vertices around a
corner maneuver gives the same number of matchings as the original graph. I can
write this up. What's left is to understand why the graphs in question in the Itsara
article are never staircases (or never even have staircase pieces).
Emilie: Worked on my system of equations.
Carl: Conversed with everyone and played around a bit with matchings of faces of cubes.