Friday, November 27, 2009

mathematical lapses

Every man has somewhere in the back of his head the wreck of a thing which he calls his education.
- Stephen Leacock, A Manual of Education

I recently stumbled upon some of Stephen Leacock's writings on wikisource (also available at Project Gutenburg). Stephen Leacock is well known (in Canada at least) as a humourist who wrote in the early part of the 20th century. It is less often noted that a few of his pieces were inspired by (or made use of) school mathematics. Now about a century since they were first published, some of these don't stand up so well, but overall they retain flashes of humor and insight that make them well-worth reading.

The best known of Leacock's school-math inspired writings is A, B, C: The Human Element in Mathematics, republished recently in Mathematics Teacher. Others include Boarding House Geometry, and Aristocratic Education.

What I like the most about Stephen Leacock's math-related humour is that it represents the sort of writing that restores mathematics to its rightful place within the humanities. In the Leacock universe, mathematics is something that every educated person knows about (at least up to a point). It is, or should be, a subject that people have enough facility with that they can find humor in it. Leacock makes it seem completely normal that you would simile knowingly at a mathematical reference just as you might a literary one, and that Euclid would be as familiar a name as Shakespeare.

Thursday, November 19, 2009

interesting and uninteresting numbers

In a paper published this week on arXiv, Dann van Berkel mentions the famous proof (or joke, depending on how you view things) that all natural numbers are interesting. Essentially, if there are uninteresting natural numbers, consider the smallest such uninteresting number (which is guaranteed to exist, since the naturals are well-ordered). This number, because it is the smallest uninteresting number, is interesting, providing us with a contradiction.

As Wikipedia points out, Nathaniel Johnson has given a slightly different formulation of the question, that seems more objective, and yet also more paradoxical. And currently, this formulation actually  produces 'uninteresting' numbers. The formulation goes something like this:

A number is interesting if and only if it belongs to an interesting sequence.
The Online Encyclopedia of Integer Sequences (OEIS) contains all interesting sequences.
Consequently, the numbers that do not appear in OEIS are uninteresting.

This gives us (as of last week, according to Johnson) that the number 12407 is currently the smallest uninteresting natural number.

Unfortunately, the sequence of uninteresting numbers seems interesting, and should be considered for inclusion in OEIS... but this would contradict its definition. So far, the sequence has not been included, so it still exists.

Does this formulation give a proof that OEIS cannot contain all interesting sequences? I think there are probably a bunch of statements we can generate with this sort of "interesting" logic, such as:

If an encyclopedia contains all interesting sequences, then every natural number must occur at least once in the encyclopedia.

But then again, you could argue that any number that occurs only once is interesting, and that the sequence of these once-occurring numbers is interesting also, causing them to be included again... In any case, this way of thinking about "interesting numbers" shows a lot of affinity with the Barber paradox, Russell's paradox, and other classics.

But van Berkel was not discussing uninteresting numbers in his paper, but rather a particularly interesting number, 3435.

If you take a natural number and calculate the sum of its digits raised to themselves, you get another natural number. Can this operation ever give you the original number back? Yes. Consider the number 1, which has a single digit, 1, when raised to itself gives 1^1=1. Here are some other calculations:

n ... sum of digits raised to themselves
1 ... 1^1 = 1
2 ... 2^2 = 4
3 ... 3^3 = 27
4 ... 4^4 = 256
5 ... 5^5 = 3125
6 ... 6^6 = 46656
7 ... 7^7 = 823543
8 ... 8^8 = 16777216
9 ... 9^9 = 387420489
10 ... 1^1 + 0^1 = 2
11 ... 1^1 + 1^1 = 2
12 ... 1^1 + 2^2 = 5
13 ... 1^1 + 3^3 = 28
14 ... 1^1 + 4^4 = 257
15 ... 1^1 + 5^5 = 3126
16 ... 1^1 + 6^6 = 46657
17 ... 1^1 + 7^7 = 823544
18 ... 1^1 + 8^8 = 16777217
19 ... 1^1 + 9^9 = 387420490
20 ... 2^1 + 0^0 = 5
21 ... 2^1 + 1^1 = 5

If we call this function $\theta(n)$, do we ever get another number than 1 that is a fixed point for $ \theta(n)$? Well, yes, and 3435 provides us with the next example:

\[3435 = 3^3 + 4^4 + 3^3 + 5^5\]
The graph below shows a plot of theta for n up to 5000 - the two fixed points that are found in this range, 1 and 3435, are shown as blue squares (data file here). The huge scale on the y-axis makes the graph look more constant in places than it really is.

Numbers for which $n = \theta(n)$ for a given base $b$ are called Munchausen numbers, and are interesting, according both to common conceptions of what 'interesting' means, and according to Johnson's OEIS-based definition (OEIS entry: A166623, also submitted by van Berkel). Making 3435 even more interesting is the fact that, according to van Berkel's paper, 1 and 3435 are the only Munchausen numbers, base 10.

Munchausen numbers are (presumably) named after the semi-fictional Baron Munchausen, who interestingly enough may not have actually been interesting, but was intent on convincing others that he was. You may know Munchausen from the Terry Gillam movie, and he has also lent his name to the curious Munchausen syndrome, and to the disturbing Munchausen-syndrome-by-proxy. The image at the top of the post is from a 1902 edition of the book, The Adventures of Baron Munchausen.

Monday, November 9, 2009

symmetry on TED

To see any pure math lecture on TED is exciting. Why are these things so rare? A recent talk by Marcus du Sautoy, Symmetry, reality's riddle, is now availalbe here. Marcus du Sautoy is a prolific mathematics expositor, and was recently appointed as Oxford's Simonyi professor of the public understanding of science (a post he takes over from Richard Dawkins). I enjoyed his Story of Maths on BBC, and also the one book of his that I've read, The Music of the Primes. His talk touches on topics that he, reportedly, explores more fully in his new book, Finding Moonshine.

For many, the story of Galois that du Sautoy used to launch his talk will be new, but to me it seems to be a tale that mathematics popularizers have stretched a little thin. The story of Galois's duel must rank along side the story of Gauss's defiance of his school master as the top two most popular mathematical biographical capsules of all time. It would be great if someone would go through the popular re-tellings of the Galois story the way that Brian Hayes has for the Gauss story, to see how different authors have embellished and expanded it (perhaps someone has?).

One thing that I found striking and appropriate about du Sautoy's talk is that although it is about group theory it never mentions the phrase "group theory," instead it always stays focused on the motivating notion of symmetry. The mathiest parts of the talk were the examples drawn from the wallpaper groups of Spain's Alhambra fortress (which, along with Galois's duel, are also frequently mentioned in popular literature on symmetry).

Watching this TED lecture made me feel a little depressed that I haven't yet taken the time to really start working through the beautiful and quirky book The Symmetries of Things, by  John Conway, Heidi Burgiel, and Chaim Goodman-Strauss - definitely the next place to go if you are inspired by the lecture.

Tuesday, November 3, 2009

The Calculus of Friendship - a review

In The Calculus of Friendship, Steven Strogatz, a Professor of Mathematics at Cornell, gives us a memoir of the friendship he has maintained with his former high school teacher, Don Joffray, as revealed through fragments of their 30 year correspondence. Their friendship was sparked and sustained through their shared enthusiasm for mathematics, and their long correspondence was, almost exclusively, framed by math problems.

The story of the correspondence between these two men is at once charming and subtly powerful.  Strogatz writes directly and honestly, telling the story of a slow-growing friendship that was at once somewhat stilted and yet deep and sustaining. The immediacy and intimacy of Strogatz's writing transform the pleasures and tragedies of normal life into the elements of a compelling narrative, and because the book works so well on this human level, it also very effective in presenting some important lessons about education and about mathematics.

Joffray is presented as the kind of teacher who taught through encouragement and obvious enthusiasm. In many respects, Strogatz's book teaches in a similar manner. Never does Strogatz tell us unequivocally how mathematics should be taught, how we should think about mathematics, or what mathematics is all about. And yet, Strogatz provides us with some distinct impressions on all these questions, so that like the students in Joffray's classes, you leave having learned something important without realizing that you were being taught to.

What does the book tell us about mathematics education? Although Joffray and Strogatz taught and learned in a very privileged environment, the essential lessons of Joffray's teaching seem universal. A good teacher, we learn through Joffray's example, is humble, enthusiastic, respectful, and encouraging. Consider how these qualities reveal themselves in a problem-oriented mathematics curriculum: problems are approached by both teachers and students as co-investigators, the successes of students are celebrated, rich problems are revisited again over time, and the teacher is rarely simply presenting results to passive students. Perhaps the most important quality that Joffray seems to embody is authenticity, and this, more than anything else, seems to have sustained him through his long career and won the admiration of his students. Here again, a subtle lesson - you have to be true to yourself, your instincts, and to what brings you enjoyment if you are going to succeed in education (or in life generally).

What does the book tell us about mathematics? Through the letters of Strogatz and Joffray, mathematics is revealed as a social activity, a language to express excitement and big ideas, and a subject whose growth is fueled by creative (and perhaps somewhat eccentric) enthusiasts who are compelled to do mathematics because of the pleasure it brings them. Strogatz seems always willing to return to the source of his original excitement about mathematics - solving interesting problems and sharing the excitement of discovery, while Joffray never gives up his own explorations and problem solving.

In their letters, Strogatz and Joffray touch on many interesting mathematical topics. Perhaps because he was writing to an enthusiast who was not engaged in "serious" research, much of what Strogatz wrote about in his letters are classics of recreational mathematics. The letters, and the additional references provided at the end of the book, provide many starting points for learning about these topics. Among other things, Stogatz and Joffray discussed pursuit problems (see the MathWorld entry here for an overview and some nice animations) the Monty Hall problem (see this post), and a geometric proof of the irrationality of the square root of two (see "proof 7" at Cut-the-Knot's list of 19 proofs).

One problem that was new to me was the "monk and the mountain" problem, which is related to a whole family of intermediate value and fixed point theorems. Charles Wells has written several interesting posts on the monk and the mountain problem (a recent one here, and another here), explaining how it exemplifies a class of problems that have "naive proofs." The proof that Wells presents is identical to the one that Strogatz describes, and at first it seems too lacking in rigor to be considered truly mathematical. Wells maintains, and I think that Strogatz would agree, that the proof, despite its naiveté, is still a solid mathematical proof. Naive proofs, like this one, have a special place in mathematics education - they can help broaden a learner's perception of what mathematics is and how it can be applied, show that good proofs cut through the unnecessary in order to get to the heart of why something is true, and remind us that mathematics often relies on metaphor in order to communicate its essential ideas.

The Calculus of Friendship is an accessible, engaging, and useful book. Strogatz has taken an engaging personal story and through it has taught some important lessons about friendship, teaching, and mathematics.

The images in this post are from a GSP iteration (file here) that I was reminded of while reading Strogatz's description of a pursuit problem involving 4 dogs chasing each other from the corners of a square.