# John F. Hughes - Mathematics/Computer Science Autobiography

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## Beginnings

I was born in 1955. By the time I was 4, my father would keep me amused by giving me pairs of 3-digit numbers to add in my head (although he was kind, and never gave me ones where I had to "carry"). My report card for kindergarten says "...is doing very well in arithmetic. ... want to know when we'll begin 'minus numbers'," which was my father's term for "negative numbers," something he'd already taught me about at home. One of the earliest scientific questions I can recall asking him was whether "hot is hotter or cold is colder." His answer, which was to compare the extremes of each, pointing out that there was a "coldest possible," but not a "hottest possible," now seems to me like a pretty decent way to go, although I might be inclined, now, to say that temperature should be defined logarithmically with "0" set at "absolute zero," so that his answer would amount to begging the question. Anyhow, I recall finding it an unsatisfactory answer at the time, and I can still bring to mind, the way one can sometimes recall a smell or taste, the particular sensation of dissatisfaction associated with that answer.

## Later

I went to a remarkable elementary school -- Shady Hill School, in Cambridge, MA -- where my interest in mathematics and computers was encourage, and where the "new math" was in full swing -- we learned about sets, about counting in different bases, about all sort of stuff. My father worked at various computer-related companies over the years. When I was about 7, I played "space wars" on a computer at Bolt, Beranek, and Newman, where he was then working. It had, as I recall, a drum memory (spinning incredibly fast), and space-wars was read into the machine through a blindingly fast paper-tape reader. One's ship was controlled by four toggle switches: one to turn left, one to turn right, one to thrust, and one to shoot bullets. There was a fifth switch, I believe, that let you "warp" away from the center of the screen, where there was a "death star" with high gravity that kept trying to suck you into it. Space-wars had a nifty notion of "hyperspace," which meant that when you went off the right-hand side of the screen, you came back on the left, and similarly for the top and bottom. It wasn't until years later that I realized this meant I was playing on a torus. Its strikes me as odd now that I recall this so vividly, since I did it only once, at age 7, for an our or so.

At school I loved science and math. At home, I played with electronics -- fixing broken TVs, getting interested in Ham Radio (I never learned Morse code well enough to get a license, alas) -- and mechanical things, and mathematics. I read "math puzzle books" voraciously, and did all those "Chinese wooden puzzles" where a bunch of bits of wood fit together to make a mouse or a block or a ball.

When I was 12, I came upon an interesting math problem: the 15 balls on a pool table fit into a triangle, but if you include the cue-ball, you have 16 balls that fit into a square. I asked "what other triangular numbers, when incremented once, give squares?" Clearly a triangle of 3 balls, when one is added, make a 2 x 2 square. Were there other examples? That summer, I wrote my first computer program, in FORTRAN. It was not very clever: for each integer n, it computed n-triangular (i.e., 1 + 2 + ... + n = n(n+1)/2), added one, and called the result U. Then, for each integer k between 1 and n it computed k-squared to see if it was U; if so, it wrote out n, k, and k-squared. I got about 10 (n, k) pairs before exceeding the MAXINT on the IBM 1130. By the way, the program was written on punch-cards

I wondered about the pattern of the numbers, and for several years the discovery of the pattern attempting to prove that my discovery was correct occupied a good deal of my time.

In 10th-grade, I got a remarkable opportunity: my math teacher was a PhD student at MIT who taught at my school for a couple of hours each morning before going in to work. He taught us calculus from Mike Spivak's remarkable calculus book ("Calculus," Michael Spivak, 3rd Edition, Publish or Perish Publication, ISBN: 0914098896) -- a book I still refer to on a weekly basis! -- and then, over the next two years, taught me a huge amount of mathematics. By the time I reached Princeton, I was able to skip the first two years of the "honors" math sequence. This teacher and I are still close friends, 25 years later.

## Later Still

I went to college, got interested in topology, continued programming the way I had in high-school (I wrote a program, in SNOBOL, that took a fortran assignment statement and produced a FORTRAN subroutine that computed the derivative of the target with respect to any variable), and eventually wrote an undergrad thesis on "Transversally intersecting submanifolds and knotted intersections."

I went to grad school at Berkeley, and studied geometric topology under Rob Kirby, a wonderful and indulgent advisor. I wrote my dissertation in troff on a PDP-11 computer; this was radical at the time -- everyone else paid typists to type up the dissertations. I also used symbolic mathematics programs (precursors of Maple and Mathematica) to do some computations -- another radical idea.

Then I taught at Bryn Mawr for two years -- a wonderful and radicalizing experience -- and finally, in 1984, came to Brown as Tamarkin Assistant Professor of Mathematics, primarily because of Tom Banchoff's influence: his work on visualization of surfaces and solids in higher dimensions was exactly the sort of thing that interested me. I started using the equipment in the graphics lab to make pictures of the mathematical objects that interested me, and soon found I was answering more questions than I was asking. In late 1985, I wrote a paper with two Masters students on constructive solid geometry, and I was on my way to becoming a graphics person rather than a math person. I gradually drifted "across the street" to the CS department, becoming a full-time tenure-trackCS faculty member 3 years ago.I continue, however, to have a strong interest in mathematics, and still am working on a couple of math problems that intrigue me, when I get a few free minutes.