SSL Minutes
2.17.04
Notes this time: Sam
Notes next time: Martin
Snacks this time: Martin
Snacks next time: Carl
Reminder that on Thursday Chris Henley (Cornell) will be visiting SSL to get an
impression of how an undergraduate research lab such as ours operates. Chris is
giving talks in both the Physics and Math Departments. You can catch his talk on
"Open Problems in Random Tiling Quasicrystals" in the Probability Seminar on the
9th Floor lounge in Van Vleck from 2:25-3:25, right before SSL. Interestingly,
this work is somewhat related to discussions we had today (Tuesday) on aperiodic
tilings such as the Penrose tiling. The Penrose tiling has "zero entropy" that
is, it is highly ordered. One important open question is "Can a quasicrystal
tiling have nonzero entropy?"
Paul: Can a one-dimensional recurrence which gives rise to an unfaithful (i.e.,
coefficients are not restricted to +-1) Laurent polynomial always be lifted to
a higher-dimensional recurrence, giving rise to a faithful Laurent polynomial?
Jim: Unknown! I would recommend using growth as an (albeit somewhat weak)
heuristic.
Sam: Might we consider other tilings of the plane (besides what we have
referred to as 'Arpo, Chico, and Groucho because of the standard A2, C2, G2)
in investigating the Markoff brother equations? Such as one with triangles
whose *sides* correspond in ratio to the coefficients?
Jim: Well, we have only considered reflection tilings. There are other ways
to tile a plane, such as non-reflection tilings.
[Proceeds to draw what turns out to be a reflection tiling - with 30-30-120
angles]
Jim: Note this can be viewed as either a splitting of A2 or a merging of G2.
[Barycentric subdivisions? I lost this thread while taking notes . . .]
[[The G2 picture can be obtained as a barycentric subdivision of the A2
picture, says Jim, but that's not really important for our purposes.]]
Reference - Grunbaum and Shepherd, Tilings and Patterns
Jim: We should review the work of Neil Herriot on Chico and use it as hints for
Groucho. Special triangles result in tilings arising from a single triangle
and reflecting.
[Realizes we can construct an exhaustive list using the following properties]
We can determine all such triangles, since they must satisfy two properties:
Let A,B,C be the angles of the triangle in question. Then
(i) Each of A, B, and C divide 2*pi
(ii) A+B+C = pi
>From which we derive the relation
1/a + 1/b + 1/c = 1/2 (where A = 2 pi / a, B = 2 pi / b, C = 2 pi / c)
And we find the triples (a,b,c):
(6,6,6)
(5,5,10)
(4,8,8)
(4,5,20)
(4,6,12)
(3,12,12)
(3,9,18)
(and maybe a few others? Jim wasn't sure)
Not all of these will actually lead to reflection tilings. For instance, the
case (5,5,10) fails.
Tilings that are aperiodic also exist, such as the Penrose tiling. We can
exhibit an example of a recurrence which gives to rise to an aperiodic
combinatorial model.
Namely, the recurrence a(n)*a(n-3) = [a(n-1)]^2 + [a(n-2)]^2 for n>2,
a(n)=1 for n<3. (Attributed to Dana Scott)
This sequence corresponds (bijectively) to a the perfect matchings of a
sequence of snakes. However, these snakes are aperiodic in their growth.
[This concludes this discussion]
Brendan: I have derived some interesting facts/identites regarding Newton's
Method. Firstly, I have cleaned up the formulas a little which should help
in proving it. Some interesting series expansions arise from setting a=b=c=1
(recall we are applying Newton's Method to the generalized quadratic
ax^2+bx+c). For instance, the double sum taken over t and k
binomial(2^n-k-t-1,2^n-k-2t)*x^(2^n-k) = 1/(1-x-x^2).
Jim: I have spent time looking at this new 3D recurrence which I have been
e-mailing about.
Martin: I am looking forward to obtaining MAGMA and to start learning it.
Brendan goes back to the board to elaborate upon something. The sequences he
has been seeing with Newton's method are tied to the generating function
1/(1-x)^n, which counts the number of combinations of the n-set with
repetition allowed.
Martin and Jim and others discuss aspects of Martin's investigations into
computing the complexity of the Fibonacci numbers. Martin will also address
Jim's response to his paper.
[We adjourn to the Physics building]
[Discurison: Fermi-Condensates? What about the exclusion principle for
fermions?]
[[Surprise guest appearance by former SSL-member Mark Chapman, says Jim!]]
Paul has tried some more mixed (linear and quadratic) recurrences, none of
which have turned out to exhibit Laurentness; all of these were "palindromic."
Might we try looking additionally at non-palindromic recurrences? There might
not be any known examples of such a recurrence.
Jim proposed that the more quadratic terms you try to put into the recurrence
(as opposed to linear terms or constant terms), the less chance you have of
having the recurrence satisfy the Laurent property. Cf. Somos sequences:
Once you get out to Somos-8, you have 'too many' quadratic terms and the
Laurent property fails.
There should actually be a slight modification of this, as there do exist
recurrences such as A(n)*A(n-10)=[A(n-1)]^2 + [A(n-2)]^2 + . . . + [A(n-9)]^2
which are Laurent.