#### by Ethan D. Bolker

#### 50__\( + \)__ years…

I first met Andy in 1956, when he taught sophomore abstract algebra at nine in the morning (even on fall football Saturdays). He agreed to let me audit his course and submit homework papers.

It took me several years and two graduate courses to realize how deceptive a lecturer he was. The proofs scrolled by. You could read his writing. He literally dotted his i’s and crossed his t’s. I know; I recently found the purple dittoed handwritten linear algebra notes he wrote for us in the spring of 1957. Strangely, those notes were sometimes subtly hard to study from. Now I know why. He took such care preparing and searched so hard for economy and elegance that the rough places were made plain. The hard parts didn’t seem so in the seamless flow, so it could be hard to find the crux of a proof. George Mackey once told me it was good that one of his teachers (I choose to forget who) was disorganized because it forced him (George) to master the material for himself. What Andy’s style proves is that disorganization may be sufficient but is not necessary.

Andy was the reader for my undergraduate thesis on multiplicity theory
for eigenvalues of bounded self-adjoint operators on Hilbert space. In
those days (perhaps still) each senior was set a special exam on the
thesis topic. One question on mine asked me to apply my theorems to
the multiplication operator __\( g \mapsto fg \)__ for __\( g \)__ in a Hilbert
space __\( L^2 (\mu ) \)__.
The function __\( f \)__ was a cubic polynomial, and, __\( \mu \)__ was Lebesgue measure on
__\( [0, 2] \)__ with an extra atom of weight 1 at 1. Fortunately, I’d thought
of putting an example like that in the thesis, so I knew how to do the
problem. What mattered was where the cubic was __\( 1 : 1 \)__, __\( 2 : 1 \)__, or __\( 3 : 1 \)__.
But my answer seemed not to need the atom at 1. When I asked Andy
later about that, he gently pointed out how he’d carefully constructed
his cubic with a local maximum at 1, so there was a set of positive
measure on which the cubic was __\( 2 : 1 \)__. I missed that, because when
finding the critical points I calculated __\( 2 \times 3 = 12 \)__. He graciously
said only that I’d spoiled a good problem.

When I chose Andy as a doctoral thesis advisor I had neither a topic
nor a direction. I thought I was an analyst and thought he was one and
knew him, so I opted to try to work with him. I struggled with __\( p \)__-adic
groups for a year, going nowhere. But I did have one idea about a way
I might prove the Radon–Nikodym theorem for measures on lattices like
those that come up in quantum mechanics. The idea didn’t work, but I
did manage to say some new things about measures on Boolean algebras
even while the generalizations to lattices eluded me. Andy encouraged
that play and said after a while that what I was working on was in
fact my thesis. He told me he liked it better when his students
found topics than when he had to suggest one.

In the spring of 1964 I thought my thesis was done. I found the central theorem in February, wrote it up, and sent it off to Andy. When I telephoned to ask what he thought of it, he asked if I needed my degree in hand to accept my new job at Bryn Mawr. When I said “no” he said, “Work on it another year.” I know that if I’d said “yes” he’d have accepted what I’d written. But then I’d have had a thesis with just a theorem. The central mechanism for producing examples and counterexamples showing the theorem was sharp came later that spring. Moreover, I think the idea was his, although I didn’t give him due credit then. So Andy was right to care about the quality of the work and to ask for the extra year. The thesis was better and better written and ready for publication soon after the degree was awarded — and my year-old daughter got to go to my commencement. (He and Jean sent her a Raggedy Andy when she was born.)

Eighteen years later Andy employed her as a painter. That’s how I learned how he applied logic outside mathematics. She saw him eating breakfast hurriedly one day — peanut butter spread on bread right out of the freezer. He said the nutritional value was the same.

I was telling Andy once about a bijection I’d found for counting
permutations with particular cycle structures. He was interested and
had some further ideas and references. When he suggested a joint paper
[1]
I jumped at the chance to earn a Gleason number of 1. When I
wanted to say something numerical about the asymptotics which called
for __\( \Gamma (1/3) \)__, I looked up nearby values in a table and interpolated. In
response to a draft I sent Andy he wrote back:

There is one not terribly important thing where I can’t check you. You obtain\[ \frac{3^{1/6} e^{\pi \sqrt{3}/18}}{\Gamma (1/3)} \approx 0.6057624. \]With my hand calculator I found\( \Gamma (1/3) \approx 2.678938543 \)(of which at least 8 figures ought to be right)and hence the above number comes out 0.6065193. Hand calculators make substantial errors in exponentials, so I really don’t know which is right.

Andy’s “With my hand calculator I found…” is a little
disingenuous. There’s no __\( \Gamma \)__ key on the calculator — he
programmed the computation. Today Mathematica quickly finds __\( \Gamma
(1/3) \approx 2.67893853471 \)__ with twelve significant figures, so
Andy’s intuition about eight was right.

Over the years I had lunch with Andy often, sampling Chinese, Vietnamese, and Indian food in Cambridge and nearby towns. Over lunch once, thinking about geometry, he told me he’d give a lot for “one good look at the fourth dimension.” Any mathematical topic, at any level of sophistication, was fair game. I’d tell him why I thought the convention for writing fractions was upside down; he’d tell me he was thinking about the foundations of geometry or the Riemann hypothesis. Often in the past year I’ve wanted to ask him about something that came up in my teaching or while editing these essays and was stunned anew by the realization that I couldn’t ever do that again.

#### Solving cubics by trisecting angles

Andy was a problem solver more than a theory builder. He liked hard problems, like Hilbert’s Fifth, about which you can read more below. Others less deep interested him no less. I think he even enjoyed the problems in spherical trigonometry and navigation on the exams he took to maintain his naval commission while in the reserves.

Once he set out to discover which regular polygons you could
construct if you add the ability to trisect angles to the tasks
available with Euclidean straightedge and compass. His answer, in
“Angle trisection, the heptagon, and the triskaidecagon”
[2]:
just
the __\( n \)__-gons for which the prime factorization of __\( n \)__ is of the form __\( 2^r
3^s p_1 p_2 \cdots p_k \)__, where the __\( p_i \)__ are distinct primes greater than 3,
each of the form __\( 2^t 3^u + 1 \)__. His proof depends on the observation
that these are precisely the primes for which the cyclotomic field has
degree __\( 2^t 3^u \)__ and so can be constructed by a sequence of adjunctions of
roots of quadratics and of cubics, all of whose roots are real.

You solve such a cubic by trisecting an angle, because when the cubic
has three real roots (the *casus irreducibilis*), finding them with
Cardano’s formula requires extracting the cube root of a complex
number. To do that you trisect its polar angle and find the cube root
of the modulus. For the particular cubics that come up in the
construction of these regular polygons, the modulus is the __\( 3/2 \)__ power
of a known quantity, so a square root computes the cube root.

Andy’s solution to that problem allowed him to indulge several of his
passions. The paper is full of historical references, including the
corollary that the ability to trisect angles doesn’t help you
duplicate the cube. That requires solving the other kind of cubic. He
cites (among others)
Plemelj,
Fermat,
Euler,
and
Tropfke
and concludes
with a quote from
Gauss’s
*Disquisitiones Arithmeticae*.1
The “triskaidecagon” in the title, where most of us would be
satisfied with “13-gon”, exemplifies Andy’s love of language. He had
lots of ideas he never got around to publishing. I wonder if he wrote
this paper in part just so he could use that word.

Andy loved to compute too. About his construction of the triskaidecagon he writes:

After considerable computation we obtain\[ 12 \cos \frac{2\pi}{13}= \sqrt{13}-1 + \sqrt{104-8\sqrt{13}} \cos \frac{1}{3}\arctan \frac{\sqrt{3}(\sqrt{13} +1)}{7-\sqrt{13}}. \]

Mathematica confirms this numerically to one hundred decimal places. I don’t think there’s software yet that would find the result in this form.

I first explicitly encountered Andy’s passion for precision of expression when in graduate school he told me that the proper way to read “101” aloud is “one hundred one” without the “and”. That passion stayed with him to the end: when he was admitted to the hospital and asked to rate his pain on a scale of 1 to 10, he’s reputed to have said first, “That’s a terrible scale to use….”

Andy told me once that he knew he wanted to be a mathematician just as soon as he outgrew wanting to be a fireman.2 He succeeded.

#### Stories

In the essays that follow you’ll find more about Andy’s mathematics and more stories. I’ll close here by quoting some that aren’t included there.

Persi Diaconis writes about Andy’s legendary speed:

Andy was an(unofficial)thesis advisor. This was illuminating and depressing. My thesis was in analytic number theory, and I would meet with Andy once a week. A lot of the work was technical, improving a power of a logarithm. I remember several times coming in with my current best estimates after weeks of work. Andy glanced at these and said,“I see how you got this, but the right answer is probably…”I was shocked, and it turns out he was right.

Jill Mesirov describes a similar experience:

I remember quite clearly the first time that I met Andy Gleason. I was working at IDA in Princeton at the time, and Andy was a member of the Focus Advisory Committee. The committee met twice a year to review the work being done, and I had been asked to give a presentation of some work on speech I had done jointly withMelvin Sweet.I worked hard on the presentation, and designed it to give some idea of how we were led step by step to the answer. The groundwork was laid for revealing each insight we had gained, but in such a way that it should come as a surprise to the audience and thus make them appreciate the sense of discovery we had enjoyed as we did the research and solved the puzzle ourselves. Needless to say, I hadn’t counted on Andy’s“infamous”speed!

Twice I carefully led the audience through some twisted trail to end with the question, “So, what do you think we tried next?”Twice, before the words had begun to leave my mouth, Andy was saying,“Oh, I see, then you want to do this, this, and this, after which you’ll observe that…”While I appreciated his quick grasp of the issues, I was beginning to see my carefully laid plans falling by the wayside. Therefore, as I was reaching the next crescendo, and I saw Andy leaning forward in his seat, I turned around, pointed my finger at him and shouted,“You, be quiet!”He smiled, and left me to lead the rest of the crowd through the revelations.

Finally, Victor Manjarrez, a graduate school contemporary of mine, offers this summary:

In the late fifties and early sixties I took graduate algebra and a reading course from Andrew Gleason. Whenever we spoke at meetings in later years I was struck by how unfailingly polite he always was. The English word“polite” (marked by consideration, tact, or courtesy)evokes the French“politesse” (good breeding, civility),and the Greek“polites” (citizen — of the mathematics community and the world),all of which Andrew Gleason exemplified to the fullest. This, of course, in addition to his amazing erudition.