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Finding a solution to an open problem about projective planes


Getting my PhD and the problem of symmetric block designs with lambda equals two
Donald Knuth Scientist
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Here I am then, earning money as a computer person, and… and enjoying… and then studying mathematics so that I can be a math teacher as a… as a career. And the idea that there could be any connection with this, or that there could be… ever be an academic field of computer science or something, never entered my mind whatsoever as a possibility. Very few people in the world had thought of it at that time either.
Now, at Caltech, I… okay, somebody had told me early on in life that you go to grade school for 8 years, and you go to high school for 4 years, then you go to college for 4 years, and then you might want to go to graduate school for another 3 years, and get your… and get a PhD. I don't know who had told me that early on, but somebody had said that it takes 3 years to get a PhD. And I had believed that and so when I got to Caltech in 1960, I'm thinking, oh, okay, in 1963 I'll graduate. And I've just sort of set my… my sights that way. Now if somebody had told me it was going to take 5 years, I'm sure I would have graduated in 1965, but… but I… I had always been thinking I would graduate in ‘63, so I… so I just sort of planned ahead. It was… it didn't occur to me until the day of graduation, that none of my other colleagues were there. You know, Al Hales was there, but just two of our 12 had done it in 3 years, and… and I think it was partly that… just that we had… we had set ourselves up that way, to do it.

My thesis… I was working with Marshall Hall, who I hadn't met before coming to Caltech, but I got to enjoy. His lectures were very disorganized, but he really knew the subject, so he would be teaching us about permutations, what actually I found was that I would take notes of what he said during class, and the problem was to figure out a permutation of what he had said that made sense. What… how to re-order, you know, I mean every… every important idea had been presented some time during that hour, but… but they didn't come to him in… in the right order.  Just like now, I'm not thinking of everything in the right order, well he, you know… so his classes were like that, but I learned a heck of a lot this way, by… by trying to unscramble the notes of what he said in class.  And so I learned about the things that he was good at, and… and I decided that you know, one of the unsolved problems that he posed to us, technical term is symmetric block designs with lambda equals two. It's a… a technical term that… that really means – it's something like geometry, except instead of having two… instead of having one line through every two points, you have two lines through every two points. And, you know, instead of saying that every… every two lines intersect in one point, every two lines intersect in two points. So… so this is a system called the symmetric block design with lambda equals two, and at the time I started working on this unsolved problem I was going to try to find infinitely many such designs, such symmetric designs, and… and six were known altogether. And so I thought, okay, but I had a new way to approach the subject, maybe I'd be able to find infinitely many. Well, it… it's a good thing that I didn't take that as my final project though, because I believe up to this day only… only one more has ever been found, so… so nobody knows that… that there is only finitely many, but… but it… it seems to be extremely hard to construct these designs.

Born in 1938, American computing pioneer Donald Knuth is known for his greatly influential multi-volume work, 'The Art of Computer Programming', his novel 'Surreal Numbers', his invention of TeX and METAFONT electronic publishing tools and his quirky sense of humor.

Listeners: Dikran Karagueuzian

Trained as a journalist, Dikran Karagueuzian is the director of CSLI Publications, publisher of seven books by Donald Knuth. He has known Knuth since the late seventies when Knuth was developing TeX and Metafont, the typesetting and type designing computer programs, respectively.

Tags: Caltech, Alfred Hales, Marshall Hall

Duration: 4 minutes, 14 seconds

Date story recorded: April 2006

Date story went live: 24 January 2008