# SSL Minutes for Tuesday 12-2-03
#
# Today's notetaker: Carl
# Today's snackbringer: Hal
#
# Notetaker for Thursday: Paul
# Snackbringer for Thursday: Steven
#
# NOTE: Jim is taking us to DINNER next tues.
# the bringing of snacks for that day is not yet decided...
SSL STARTS and Jim doesn't immediately say anything. SSL begins bustling,
then someone notices the time (SSL: hey--it's past 2:30)
Jim: actually I was just going to see where this energy goes
Melania: Please lend me your signatures; I'm running for the school board
SSL: are you in favor of math?
Melania: Yes!
SSL: the new math? with set theory?
Melania: yes! here are my views on how math should be taught: ...
[ Let's get started ]
Jim: I'm curious what you've been doing; I'll start the ball rolling. I
sent the email on David Speyer's stuff -- it leads into what we'll be
doing in the spring (elaborates) ... that's what I've been doing, and
writing emails...
Talks about "Frieze patterns." (we'll be getting into in spring)
And I'm going to try to come up with some concrete things for you to
chew on over winter break. .. And hopefully you'll find / prove some
stuff with Newton's method...
Hal: I'd like help with Maple, having trouble with some stuff with large
fields (Jim: what size?) like 2^5 (Jim: alright, that's not that big..)
I can try rewriting my program in C ... I'm not actually that
experienced with Maple...
Jim: it's probably a good idea to try to rewrite a program in another
language, to make sure it's doing the right thing, etc. check to make
sure they get the same answers.
Martin: I've already started that (with his program?)
Jim: the other thing [for us to code] is the automorphism group.
We can use an adjacency matrix (explains) ... that style algorithm
should consider less things than brute force... The other thing that
would automate this is GAP; there's a link in my email about that
program on 12-1 ("the Markoff equation over finite fields"). (Jim
emails a URL)
Carl: Looked a bit more at last triangulation with degree 4-6-12 -- still
nothing really interesting. Made copies of Math192 lectures on CDs.
(distributes them.)
Sam: Newton method looks interesting, people wanna work on it?
plus the 4-6-12 triangulation from before -- I'll post stuff about it's
progress on my website..
Jim: I thought we finished encouraged before when we found the 1-7-11
triple, but it seemed to stop after that -- did we find anything else?
perhaps we should look more at Neil Harriot's example, since it's
simpler..
Martin: I have conjectures for the number of Markoff brother triples on
the field mod p. Need help with number theory stuff for proving this.
Was going to try to program girth, had trouble implementing efficiently.
The girth of k-regular graphs has been looked at, especially 3-regular
(cubic) graphs...
Jim: I'm surprised there aren't canned algorithms for calculating girth..
...eventually for bigger examples we're going to need to write stuff in
C or something, to make it more efficient... if you have stuff, put up
a link on your page, if it's not stable, mark it as such...
The hope is getting at Expanders.
Paul: looked more at triangle graphs... interested in seeing more on the
1,1,2,4 markoff brother. (Jim sends email correcting broken link)
Emilie: in 4-6-12 triangulation, found some triangles with all 3 sides in
tree, but not triples. (hrmm...) will post it.
Jim: Skewing of plane that preserves lattice:
. . p . . . . . p .
. o q . --> . o q .
. . . . . . . . . .
where opq is our triangle, and the transformation shifts rows of points
this doesn't work for this triangulation though:
+-*-+-*-+
|/|\|/|\|
*-+-*-+-*
|\|/|\|/|
+-*-+-*-+
since it doesn't preserve labeling (of degrees)
ie.,
8 4 8 4 8 4 8 4 8 4
4 8 4 8 4 --> 4 8 4 8 4
8 4 8 4 8 4 8 4 8 4
is not what we want.
Maybe move 2 to the right..?
+-*-p-*-+ +-*-+-*-p
|/|\|/|\| |/|\|/|\|
*-o-q-+-* --> *-o-q-+-*
|\|/|\|/| |\|/|\|/|
+-*-+-*-+ +-*-+-*-+
We'll call the weights for the sides of the triangle (opq) "price"
(after Gregory Price). Remember that the scheme for price here is to
(1) take all the triangles the line between two points passes through,
(2) make 'Y's in all of those triangles, (3) remove the edges of the
original triangles, (4) erase the 2 tail segments from the Y graph,
and (5) count the # of perfect matchings on the remaining graph.
By convention, the price of two adjacent vertices is 1.
In the above transformation (shifting rows by 2), the prices of the
triangles are 1-1-2 and 1-3-5.
We want to be able to apply some transformation, and have original
formula (ax^2 + by^2 + cz^2 = dxyz) still hold...
Jim: Emily, is the 1-3-7 triangle in same orbit as the 1-7-11 triangle?
If so (i.e., if some transformation can connect those two), then perhaps
we can get infinitely many. That would be very encouraging...
(but, are the labels (4,6,12) of equal "density" in the grid? if not,
I'm not sure if we can have a transformations...)
[ Let's eat ]
Hal provides us with two varieties of delicious cookies, and an assortment
of hot tea!
Meanwhile SSL begins to wonder and ask about "the Project".
(SSL: can we do something along the lines of our web-page?
Jim: Yeah, that's OK. A nice 'package' wrapping up what's been done
this semester...)
[ Munching can only keep SSL from math for so long... ]
Jim: (read article about stuff like apollonian gasket)
Shows us a variation of the gasket: this time, instead of packing just
one circle in the middle of three, packing three inside of three.
Starting with circles A,B,C, all tangent, pack inside circles A',B',C'
where A tangent to B' & C', B to A' & C', and C to A' & B'.
-- A',B',C' seem to be unique given A,B,C
This somehow seems similar to the Descartes / Soddy equations... Would
they be satisfied in this? Or would there at least be some link?
Look for Geometer's Sketchpad -- at least to empirically investigate...
Also, centers of circles in the apollonian gasket satisfy a complex
equation that is similar to the descartes eq...
One thing to consider: Inversion. Take a circle with radius R tangent
to some circle A, and draw its inverse inside A with radius 1/R.
There is some more discussion about encoding circle packings as
centerpoints and radii -- and how not all the radii are necessary
information...
Emilie: Found some more stuff that worked with 4-6-12 triangulation.
People break up into groups -- Sam, Carl & Paul investigate Newton's
method stuff. Emilie continues work with what she just found.
Together, the body of renegade mathletes discovers more exciting
MATH STUFF in the ongoing saga of SSL ...
TO BE CONTINUED...