SSL Notes for meeting #11 (10/14)
Today's note-taker: Stephen
Next Time: Hal
Today's snack-provider: Hal
Next Time: Abby
Everybody's favorite constant:
Jim: pi^2/6
Sam: i
Stephen: pi^(1/2)
Melania: i
Martin: 1
Carl: empty matrix
Hal: 6
Emily: e
Paul: 0
Abby: 24
Hal describes why straight line cube snakes are planar: Imagine drawing each
cube as a pair of concentric squares with appropriate vertices attached. The
straight cube snake with n boxes becomes 2n concentric squares. He also gives
an example that suggests that arbitrary cube snakes are planar.
Jim: Goal: Have some collaboration happen for Melania to watch
A list is drawn up of possible topics for discussion and further investigation:
1) Graph condensation (Kuo's article)
2) Itsara (i.e., article by Carroll, Itsara, Le, and Propp)
3) 3D (cube) snakes and their matchings
4) Interpretation of the entries of the "snake matrices"
5) Coeff's of Fibonacci polynomials
6) Computer algorithms to count matchings
7) The four tangent circles problem
8) Somos sequence
Hal offers to put an X-server on the lab computers.
Emily offers a conjecture on the interpretation of the #'s in the snake
matrices, and solicits advice on proving it. The conjecture:
if C= (a b) is the matrix corresponding to a snake, then a is the number of
(c d)
matchings "beginning and ending with a y", b is the number of
matchings "beginning with a y and ending with an x", c is the number of
matchings "beginning with an x and ending with a y" and d is the number of
matchings "beginning and ending with an x".
She also suggests that it would be interesting to figure out how to factor a
snake matrix into a product of A's and B's (A=(1 1), B=(0 1) ).
(1 0) (1 1)
Paul offers a recursive formula for the number of matchings of a straight cube
snake with n cubes. If A_n is this number, then
A_{n+3}=3A_{n+2}+3A_{n+1}-A_n, with A_0=2 and A_1=9.
Then everyone breaks into small groups to talk about whatever they're
interested in.
Thursday: Group discussion, maybe some presentation
(e.g., discuss Abby's problem about 3D snakes;
discuss condensation article, Markoff number article)
Suggestions for Thursday/ What people did Tuesday:
Sam: Suggests looking at Kuo's graph. condensation proof. He worked with
Emily and Martin on the factoring problem for snake matrices mentioned above.
Emily: see Sam
Martin: see Sam
Carl: Worked with Hal on the four tangent circles problem from Kirillov's talk.
Hal: see Carl. Also suggests that the Doit website electronic shelf has
instructions for installing an X-server.
Jim: says Yvonne will send him an email re. computer lab availability, and
that he will put a link in the minutes for the graph.tcl program. (There
are two versions: Matt Blum's http://jamespropp.org/graph.tcl and
Billy Hillegass' variant http://jamespropp.org/alt/graph.tcl .
The latter is better for Windows systems. For other software by Jim and
his students, see http://jamespropp.org/software.html .)
John and Paul: worked on twisty cube snakes
Abby: Worked on transformatrices
Both Paul and Sam suggest that establishing common base of knowledge for the
group would be a good idea for next time (Itsara/Somos sequences?)
Jim would like to see a thorough written analysis of straight cube snakes that
includes the kind of helpful asides that a textbook might leave out.