The Universe of Discourse

Wed, 23 Apr 2008

Recounting the rationals
I just read a really excellent math paper, Recounting the rationals, by Calkin and Wilf.

Let b(n) be the number of ways of adding up powers of 2 to get n, with each power of 2 used no more than twice. So, for example, b(5) = 2, because there are 2 ways to get 5:

5 = 4 + 1
= 2 + 2 + 1

And b(10) = 5, because there are 5 ways to get 10:

10 = 8 + 2
= 8 + 1 + 1
= 4 + 4 + 2
= 4 + 4 + 1 + 1
= 4 + 2 + 2 + 1 + 1

The sequence of values of b(n) begins as follows:

1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 ...

Now consider the sequence b(n) / b(n+1). This is just what you get if you take two copies of the b(n) sequence and place one over the other, with the bottom one shifted left one place, like this:

    1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 ...
    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
    1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 ...

Reading each pair as a rational number, we get the sequence b(n) / b(n+1), which is 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, ... .

Here is the punchline: This sequence contains each positive rational number exactly once.

If you are just learning to read math papers, or you think you might like to learn to read them, the paper in which this is proved would be a good place to start. It is serious research mathematics, but elementary. It is very short. The result is very elegant. The proofs are straightforward. The techniques used are typical and widely applicable; there is no weird ad-hockery. The discussion in the paper is sure to inspire you to tinker around with it more on your own. All sorts of nice things turn up. The b(n) sequence satisfies a simple recurrence, the fractions organize themselves neatly into a tree structure, and everything is related to everything else. Check it out.

Thanks to Brent Yorgey for bringing this to my attention. I saw it in this old blog article, but then discovered he had written a six-part series about it. I also discovered that M. Yorgey independently came to the same conclusion that I did about the paper: it would be a good first paper to read.

[ Addendum 20080505: Brad Clow agrees that it was a good place to start. ]

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Fri, 18 Apr 2008

A few notes on "The Manticore"

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This past week I reread Robertson Davies' 1972 novel The Manticore, which is a sequel to his much more famous novel Fifth Business (1970). I've read Fifth Business and its other sequel, World of Wonders (1975), several times each, but I found The Manticore much less compelling, and this is only the second time I have read it.

Here are a few miscellaneous notes about The Manticore.

Early memories

Here is David Staunton's earliest memory, from chapter 2, section 1. (Page 87 in my Penguin paperback edition.)

Dr. Von Haller: What is the earliest recollection you can honestly vouch for?

Myself: Oh, that's easy. I was standing in my grandmother's garden, in warm sunlight, looking into a deep red peony. As I recall it, I wasn't much taller than the peony. It was a moment of very great—perhaps I shouldn't say happiness, because it was really an intense absorption. The whole world, the whole of life, and I myself, became a warm, rich, peony-red.

Here is the earliest memory of Francis Cornish, the protagonist of Davies' novel What's Bred in the Bone (1985):

It was in a garden that Francis Cornish first became truly aware of himself as a creature observing a world apart from himself. He was almost three years old, and he was looking deep into a splendid red peony.

That is the opening sentence of part two, page 63 in my Penguin Books copy.

The sideboard

This is from chapter 3 of The Manticore, David's diary entry of Dec. 20:

Inside, it is filled with ... gigantic pieces of furniture on which every surface has been carved within an inch of its life with fruits, flowers, birds, hares, and even, on one thing which seems to be an altar to greed but is more probably a sideboard, full-sized hounds; six of them with real bronze chains on their collars.

The following quotation is from Davies' 1984 New York Times article "In a Welsh Border House, the Legacy of the Victorians", a reminiscence of the house his father lived in after his retirement in 1950:

Until my father had it dismantled and removed to a stable, the Great Hall was dominated by what I can only call an altar to gluttony against the south wall. It was a German sideboard of monumental proportions that the Naylors had acquired at the Great Exhibition of 1851. Every fruit, flower, meat, game, and edible was carved on it in life size, including four large hounds, chained to the understructure with wooden chains, so cunningly wrought that they could be moved, like real chains.

This is reprinted in The Enthusiasms of Robertson Davies, Judith Skelton Grant, ed.

What do Canadians think of Saints?

Davies has said on a number of occasions that in Fifth Business he wanted to write about the nature of sainthood, and in particular how Canadians would respond if they found that they had a true saint among them. For example, in his talk "What May Canada Expect from Her Writers?" (reprinted in One Half of Robertson Davies, pp. 139–140) he says:

For many years the question occurred to me at intervals: What would Canada do with a saint, if such a strange creature were to appear within our borders? I thought Canada would reject the saint because Canada has no use for saints, because saints hold unusual opinions, and worst of all, saints do not pay. So in 1970 I wrote a book, called Fifth Business, in which that theme played a part.

Fifth Business does indeed treat this theme extensively and subtly. In The Manticore he is somewhat less subtle. A perpetual criticism I have of Davies is that he is never content to trust the reader to understand him. He always gets worried later that the reader is not clever enough, and he always comes back to hammer in his point a little more obviously.

For example, Fifth Business ends with the question "Who killed Boy Staunton?" and a cryptic, oracular answer. But Davies was unable to resist the temptation to explain his answer for the benefit of people unable or unwilling to puzzle out their own answers, and the end of The Manticore includes a detailed explanation. I think there might be an even plainer explanation in World of Wonders, but I forget. I have a partly-finished essay in progress discussing this tendency in Davies' writing, but I don't know when it will be done; perhaps never.

What would Canada think of a saint? Fifth Business is one answer, a deep and brilliant one. But Davies was not content to leave it there. He put a very plain answer into The Manticore. This is again from David's diary entry of Dec. 20 (p. 280):

Eisengrim's mother had been a dominant figure in his own life. He spoke of her as "saintly," which puzzles me. Wouldn't Netty have mentioned someone like that?

David's old nurse Netty did indeed mention Eisengrim's mother, although David didn't know that that was who was being mentioned. The mention appears in chapter 2, section 6, p. 160:

She had some awful piece of lore from Deptford to bring out. It seems there had been some woman there when she was a little girl who had always been "at it" and had eventually been discovered in a gravel pit, "at it" with a tramp; of course this woman had gone stark, staring mad and had to be kept in her house, tied up.

If you want to know what Robertson Davies thinks that Canada would make of a saint, but you don't want to read and ponder Fifth Business to find out, there you have it in one sentence.

[ Addendum: The New York Times review of The Manticore is interesting for several reasons. The title is misspelled in the headline: "The Manitcore". The review was written by a then-unknown William Kennedy, who later became the author of Ironweed (which won the Pulitzer Prize) and other novels. Check it out. ]

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Thu, 17 Apr 2008

Is blood a transitive relation?
When you're first teaching high school students about the idea of a relation, you give examples of the important properties of relations. Relations can be some, none, or all of reflexive, symmetric, antisymmetric, or transitive. You start with examples that everyone is already familiar with, such as the identity relation, which is reflexive, symmetric, and transitive, and the ≤ relation, which is antisymmetric and transitive. Other good examples include familial relations: "sister-in-law of" is symmetric on the set of women, but not on the larger set of people; "ancestor of" is transitive but not symmetric.

It might seem at first glance that "is related to" is transitive, but, at least under conventional definitions, it isn't, because my wife is not related to my cousins.

(I was once invited to speak at Haverford College, and, since I have no obvious qualifications in the topic on which I was speaking, I was asked how I had come to be there. I explained that it was because my wife's mother's younger brother's daughter's husband's older brother's wife was the chair of the mathematics department. Remember, it's not what you know, it's who you know.)

I think I had sometimes tried to turn "related to" into a transitive relation by restricting it to "is related to by blood". This rules out the example above of my wife being unrelated to my cousins, because my relationship with my wife is not one of blood. I don't quite remember using "related by blood" as an example of a transitive relation, but I think I might have, because I was quite surprised when I realized that it didn't work. I spent a lot of time that morning going over my counterexample in detail, writing it up in my head, as it were. I was waiting around in Trevose for the mechanic to finish examining my car, and had nothing better to do that morning. If I had had a blog then, I would probably have posted it. But it is a good thing that I didn't, because in spite of all my thought about it, I missed something important.

The example is as follows. A and B have a child, X. (You may suppose that they marry beforehand, and divorce afterward, if your morality requires it.) Similarly, C and D have a child, Z. Then B and C have a child Y. Y is now the half-sibling of both X and Z, and so is unquestionably a blood relative of both, but X and Z are entirely unrelated. They are not even step-siblings.

Well, this is all very well, but the reason I have filed it under oops/, and the reason it's a good thing I didn't post it on my (then nonexistent) blog is that this elaborate counterexample contains a much simpler one: X is the child and hence the blood relative of both A and B, who are not in general related to each other. C, D, Y, and Z are wholly unnecessary.

I wish I had some nice conclusion to draw here, but if there's something I could learn from it I can't think would it might be.

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