SSL Notes for meeting #8 (10/2)
Today's note-taker: Emilie and Sam (on deck: Paul)
time: Emilie and Sam on Thursday, Paul on Tuesday
Today's snack-provider: Stephen (on deck: Carl)
We found a proper weighting of a chain of hexagons which "wiggles" that
corresponds to a chain of straight square snakes. A matching on the
squares corresponds to an analagous matching on the hexagons with the
same weight. (see picture 2003-10-02-003.jpg on
http://www.math.wisc.edu/~jvano/ssl/picts/ for the weighting scheme).
In that picture the colored edges correspond to eachother. Meaning that
the pink edges in the hexagons correspond to the pink edges in the
squares, the blue group of three edges in the hexagons corresponds to
the single blue edge in the squares, and the green edges in the
hexagons corresponds to the green edges in the squares.
Next we found a weighting of hexagon chains that go straight that
corresponds to a chain of squares like in a staircase. (see picture
2003-10-02-002.jpg on http://www.math.wisc.edu/~jvano/ssl/picts/ it's
the picture on the left..the pink one). A matching of a straight
hexagonal snake can be easily turned into an analagous matching of a
staircase of squares and both matchings will have the same weight.
After this we decided to put the two ideas together and find some
specific weight of a hexagon snake which bent and went straight and
corresponded to a square snake which went straight and bent. With a
specific example we found a weighting (see picture 2003-10-02-002.jpg
on http://www.math.wisc.edu/~jvano/ssl/picts/ and it's the green one on
the right). A matching on the hexagons corresponds to a matching on the
squares. We did a specific example and you can see that the weights are
the same on both graphs.
After finding this we decided it was time to look at "the blob" (the
thing we used to prove the matchings on a square snake followed the
fibonacci/arithmetic progression rules). We looked at it and decided
that it would not work because we needed to know more information about
the blob that what we had.
Right about then Jim came in and gave us some information that we
didn't know what to do with. He wrote this on the board:
ABAAB
A=[1 1]
[1 0]
B=[0 1]
[1 1]
He said that that had something to do with the matchings on a snake of
either hexagonal or square shapes. Some people decided to start
multipilying this out, however I did not quite get what they were doing
and thus did not write it down. However here is what ABAAB equals:
[3 7]
[1 2] I'm not sure what it means, but that's the matrix that ABAAB
equals.
Now it was time to leave and we decided that we should get together
outside of the rergular Tuesday and Thursday meetings. We thought that
Monday at 3:30 would be a good time for an optional meeting. We'd do
similar things to what we did this Thursday (brainstorm together). This
monday we're meeting in the Physics club room in the Physics building.
I can't remember the room number, email hal for more information. After
we go there we'll probably go find an empty classroom in Van Vleck. If
you have to come late go to the physics club room and look on the
board. Hal will write where we all are if we aren't in that room.
The End.