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Do you know this number?
2.685452001065306445309714835481795693820382293994462953051152....
They say that mathematics is not really about numbers, and they're right. But sometimes it's fun to play around with the darn things!
Given any positive number you can work out its continued fraction expansion, like this:
sqrt(2) = 1 + 1
---------
2 + 1
--------
2 + 1
-------
2 + 1
------
2 + 1
----
2 + 1
----
.
.
.
But normally it won't look so pretty! A number is rational if
and only if the continued fraction stops after finitely many steps.
If its continued fraction expansion eventually repeats, like this:
sqrt(3) = 1 + 1
---------
1 + 1
--------
2 + 1
-------
1 + 1
------
2 + 1
----
1 + 1
----
.
.
.
then it satisfies a quadratic equation with integer coefficients.
So the continued fraction expansion of e can't ever repeat... but
it's cute nonetheless:
e = 2 + 1
---------
1 + 1
--------
2 + 1
-------
1 + 1
------
1 + 1
------
4 + 1
------
1 + 1
------
1 + 1
----
6 + 1
----
.
.
.
It continues on predictably after that initial hiccup. The number π,
on the other hand, gives a random-looking mess. This is a hint that
π is number-theoretically more complicated than e, which is also
apparent when you compare the proofs that e and π are transcendental
- the proof for e is much easier.
Pondering all this, it's natural to ask about the "average" behavior of the continued fraction expansion of a number. What's the average behavior of the series a1, a2, a3, .... that we get this way:
x = a1 + 1
---------
a2 + 1
--------
a3 + 1
-------
a4 + 1
----
.
.
.
?
It turns out that if we take the geometric mean of the first n terms
and then let n approach ∞, the mean almost always converges to
"Khinchin's constant" - the number at the beginning of this article!
Here by "almost always" I mean that the set of exceptions has measure
zero. One can prove this using some ideas from ergodic theory.
Now, there is much more to say about continued fraction expansions, but my real goal is simply to point out that there are lots of interesting constants in mathematics besides π, e, the golden ratio, and Euler's number. Where can you read about them? Here:
1) Steven Finch, MathSoft Constants, http://pauillac.inria.fr/algo/bsolve/constant/constant.html
This is a great place to learn about Khinchin's constant, Feigenbaum's number, Madelung's constant, Artin's constant, Grothendieck's constant, and many other fun numbers!
Speaking of fun websites, here's another:
2) The Mathematics Genealogy Project, http://hcoonce.math.mankato.msus.edu/
My advisor's advisor's advisor's advisor's advisor's advisor's advisor's advisor was Gauss. If you think I'm showing off, you're right! But I couldn't have done it without this website, and if you're a mathematician, there's a good chance you use it to track down your academic lineage. And if you can't, you can at least add your information to the database.
Before Demian Cho showed me this site, I'd gotten stuck 3 generations back in my attempts to discover my academic ancestors. Now I can go back 11 generations. I know it's annoying, but I'm gonna tell you the whole story:
My advisor was Irving Segal, the guy who helped prove the Gelfand-Naimark-Segal theorem. This is a basic result about C*-algebras, a kind of gadget he invented to formalize the notion of an "algebra of observables" in quantum theory. The GNS theorem implies that every C*-algebra sits inside the algebra of all bounded operators on some Hilbert space, so it's a kind of justification for using Hilbert spaces in quantum physics. But even better, it gives a procedure for representing a C*-algebra as operators on a Hilbert space starting from a "state" on the C*-algebra. The upshot is that while Hilbert spaces are important, the right Hilbert space to use can depend on the state of the system you're studying. At first people thought Segal was nuts for saying this, but by now it's well-accepted.
Segal also did work on quantum field theory, nonlinear partial differential equations, and other topics at the borderline between physics and functional analysis. His students include Isadore Singer and Bertram Kostant, whose work on geometric quantization generalized Segal's ideas on the "Bargmann-Segal representation". I worked with Segal because I liked analysis and wanted to understand quantum field theory in a rigorous way.
Segal's advisor was Einar Hille, the guy who helped prove the Hille-Yosida theorem. Hille did a lot of work on integral and differential equations, but later he became interested in functional analysis: the study of infinite-dimensional vector spaces equipped with nice topologies, such as Hilbert spaces, Banach spaces and the like. At the time, he was rather special in his emphasis on applying these abstract ideas to concrete problems. In his book "Methods in Classical and Functional Analysis," he wrote:
If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist. His problems come from analysis and his results should throw light on analysis....The Hille-Yosida theorem shows how to write a large class of one-parameter semigroups of linear operators on Banach spaces in the form exp(-tH). These so-called "contraction semigroups" naturally come from the heat equation and its relatives. Segal was fond of this idea, and he generalized it to semigroups of nonlinear operators, which arise naturally from nonlinear partial differential equations. He used this idea to prove global existence of solutions for various nonlinear classical field theories.
Hille's advisor was Marcel Riesz, the guy who didn't prove the Riesz representation theorem. Marcel's brother Frigyes was the guy who did that. Marcel worked on functional analysis, partial differential equations, and mathematical physics - even Clifford algebras and spinors!
The advisor of Marcel Riesz was Lipot Fejer, the guy who discovered the Fejer kernel. This shows up when you sum Fourier series. If you just naively sum the Fourier series of a continuous function on the circle, it may not converge uniformly. However, if you use a trick called Cesaro summation, which amounts to averaging the partial sums, you get uniform convergence. The average of the first n partial sums of the Fourier series of your function is equal to its convolution with the Fejer kernel. Fejer also worked on conformal mappings. His students included Paul Erdos and Gabor Szego.
Fejer's advisor was Karl Herman Amandus Schwarz, the guy who helped prove the Cauchy-Schwarz inequality. That's a wonderful inequality which everyone should know! But Schwarz also worked on minimal surfaces and complex analysis: for example, conformal mappings from polyhedra into the sphere, and also the Dirichlet problem. Don't mix him up with Laurent Schwartz, the guy who invented distributions.
(Actually, Lipot Fejer's name was originally Leopold Weiss. He changed it to seem more Hungarian. This was a common practice at the time in Hungary, but when he did it, his advisor Schwarz stopped speaking to him!)
Schwarz's advisor was Karl Weierstrass, the guy who proved the Weierstrass theorem. This theorem says that every continuous real-valued function on the unit interval is a uniform limit of polynomials. Weierstrass also has a function named after him: the Weierstrass elliptic function, which I explained in "week13". But his real claim to fame is how he made analysis more rigorous! For example, he discovered the importance of uniform convergence, and found a continuous function with no derivative at any point. Besides Schwarz, his students include Frobenius, Killing, and Kowalevsky.
Now, Weierstrass doesn't have an advisor listed in the mathematics genealogy. However, by using this website full of mathematician's biographies, I can go back further:
3) John J. O'Connor and Edmund F. Robertson, The MacTutor History of Mathematics Archive, http://www-groups.dcs.st-andrews.ac.uk/~history/index.html
According to this, Weierstrass had an erratic career as a student: his father tried to make him study finance instead of math, so he spent his undergraduate years fencing and drinking. He learned a lot of math on his own, and got really interested in elliptic functions from the work of Abel and Jacobi. I can't tell if he ever had an official dissertation advisor. However, in 1839 he went to the Academy at Muenster to study under Christoph Gudermann, who worked on elliptic functions and spherical geometry. Gudermann strongly encouraged Weierstrass in his mathematical studies. Weierstrass asked for a question on elliptic functions, and wound up writing a paper which Gudermann assessed "... of equal rank with the discoverers who were crowned with glory." (When Weierstrass heard this, he commended Gudermann's generosity, since he had strongly criticized Gudermann's methods.)
Given all this, and the fact that Weierstrass seems to have had no other mentor, I'll declare Gudermann to be his advisor, de facto even if not officially.
But who was Gudermann? He's the guy they named the "gudermannian" after! That's this function:
gd(u) = 2 arctan(exp(u)) - π/2.
Now, if you're wondering why such a silly function deserves a name, you should work out its inverse function:
gd-1(x) = ln(sec(x) + tan(x)).
And if you don't recognize this, you probably haven't taught freshman calculus lately! It's the integral of sec(x), which is one of the hardest of the basic integrals you teach in that kind of course. But it's not just hard, it's historically important: a point at latitude gd(u) has distance u from the equator in a Mercator projection map. If you think about it a while, this is precisely what's needed to make the projection be a conformal transformation - that is, angle-preserving. And that's just what you want if you're sailing a ship in a constant direction according to a compass and you want to know where you'll wind up.
If you don't see how this works, try:
4) Wikipedia, http://en.wikipedia.org/wiki/Mercator_projection
Gudermann's advisor was Carl Friedrich Gauss, the guy they named practically everything after! Poor Gudermann, who was content to mess around with special functions and spherical geometry, seems to have been one of Gauss' worst students. But that's not so bad, since three of the other four were Bessel, Dedekind and Riemann.
Gauss' advisor was Johann Pfaff, the guy they named the "Pfaffian" after. If the matrix A is skew-symmetric, we can write
det(A) = Pf(A)2
where Pf(A) is also a polynomial in the entries of A. Pfaffians now show up in the study of fermionic wavefunctions. Pfaff worked on various things, including the integrability of partial differential equations, where the concept of a "Pfaffian system" is important. Unfortunately I've never gotten around to understanding these.
Pfaff's advisor was Abraham Kaestner. I'd never heard of him before now. He wrote a 4-volume history of mathematics, but his most important work was on axiomatic geometry. His interest in the parallel postulate indirectly got Gauss, Bolyai and Lobachevsky interested in that topic: we've already seen that he taught Gauss' advisor, but he also taught Bolyai's father, as well as Lobachevsky's teacher, one J. M. C. Bartels. In fact, Kaestner was still teaching when Gauss went to school, but Gauss didn't go to Kaestner's courses, because he found them too elementary. Gauss said of him, "He is the best poet among mathematicians and the best mathematician among poets". Perhaps this faint praise refers to Kaestner's knack for aphorisms.
At this point I got stuck until my student Miguel Carrion-Alvarez helped out. It appears that Kaestner's advisor was one Christian A. Hausen. He's the guy they named the Hausen crater after - a lunar crater located at 65.5 S, 88.4 W. He did his thesis on theology in 1713, but became a professor of mathematics in Leipzig. He worked on electrostatics, but made no memorable discoveries.
At this point the trail disappears into mist. For some conjectures, see this page:
5) Anthony M. Jacobi, Academic Family Tree, http://www.staff.uiuc.edu/%7Ea-jacobi/tree.html
It's interesting how the same themes keep popping up in this genealogy. For example, Weierstrass invented uniform convergence and proved that the limit of a uniformly convergent series of continuous functions is continuous. The Fejer kernel shows up when you're trying to write functions on the circle as a uniformly convergent sum of complex exponentials. Segal's C*-algebras generalize the notion of uniform convergence to operator algebras. I guess these things just go from generation to generation....
A little while ago John McKay visited me and told me about all sorts of wonderful things: relations between subgroups of the Monster group, exceptional Lie groups, and modular forms... a presentation of the Monster group with 2 generators, a way to build the Leech lattice from 3 copies of the E8 lattice... a way to get ahold of the Monster group starting with a diagram with 26 nodes....
Unfortunately, I'm having trouble finding references for some of these things! It's possible that the last two items are really these:
6) Robert L. Griess, Pieces of eight: semiselfdual lattices and a new foundation for the theory of Conway and Mathieu groups. Adv. Math. 148 (1999), 75-104.
7) John H. Conway, Christopher S. Simons, 26 implies the Bimonster, Jour. Algebra 235 (2001), 805-814.
Anyway, I need to read about this stuff.
Speaking of exceptionology: in "week163" I explained how Spin(9) sits inside the Lie group F4, thanks to the fact that Spin(9) is the automorphism group of Jordan algebra of 2x2 hermitian octonionic matrices, and F4 is the automorphism group of the Jordan algebra of 3x3 hermitian matrices. But in fact, since there are different ways to think of 2x2 matrices as special 3x3 matrices, there are actually 3 equally good ways to stuff Spin(9) in F4. Since I'd been hoping this might be important in particle physics, it was nice to discover that Pierre Ramond, a real expert on this stuff, has had similar thoughts. In fact he's written two papers on this! Let me just quote the abstracts:
8) Pierre Ramond, Boson-fermion confusion: the string path to supersymmetry, available at hep-th/0102012.
Reminiscences on the string origins of supersymmetry are followed by a discussion of the importance of confusing bosons with fermions in building superstring theories in 9+1 dimensions. In eleven dimensions, the kinship between bosons and fermions is more subtle, and may involve the exceptional group F4.
9) T. Pengpan and Pierre Ramond, M(ysterious) patterns in SO(9), Phys. Rep. 315 (1999) 137-152, also available as hep-th/9808190.
The light-cone little group, SO(9), classifies the massless degrees of freedom of eleven-dimensional supergravity, with a triplet of representations. We observe that this triplet generalizes to four-fold infinite families with the quantum numbers of massless higher spin states. Their mathematical structure stems from the three equivalent ways of embedding SO(9) into the exceptional group F4.
Pippi stood on her head on the horse's back and waved her legs in the air. "Heigh-ho," said she, "then why are the other people here?"
Astrid Lingren, Pippi Goes on Board
© 2001 John Baez
baez@math.removethis.ucr.andthis.edu
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