Friday, January 29, 2010

Appolonian Gaskets and Ford Circles

At first I didn't want to read the latest issue of American Scientist - Brian Hayes, the regular author of the  "Computing Science" column is currently on sabbatical, and his articles are my usual motivation for reading AS. It turns out that Dana Mackenzie has provided a really nice article in Brian's absence. Writing about Apollonian Gaskets, as usual for this column (which is about computing, but not necessarily computers), Dana Mackenzie's article combines elements of recreational mathematics, physics, and, of course, computing science.

One of the particular Apolonian-ish gaskets that the article mentioned (one involving a circle with zero curvature - a straight line) is even more fundamental to Number Theory than the article indicated - the circles that are tangent to the line in the diagram below are the sequence of Ford Circles that can be thought of as representations of the Farey Sequence.

I haven't yet had the chance to really find out, but I am thinking that some of the the general Appolonian circle formulas that Mackenzie discusses in his article can be shown to be consequences of the mediant formula in this "Ford Circle" case.

I tried to recreate the Ford Circle component of the nice diagram above in Processing, but my limited skills in this area lead to the numbers (that represent the curvature of the circles) being somewhat off center and not so well-sized.

I prefer the version below that does not display the numbers - if the applet loads properly, clicking on the applet-area will cause a new generation of the Farey sequence / Ford circles to be drawn (plus and minus keys give a primitive zoom).

This browser does not have a Java Plug-in. Get the latest Java Plug-in here.

As Mackenzie mentions, "the whole gasket is like a kaleidoscopic image of the first four circles, reflected again and again through an infinite collection of curved mirrors." Which might convince you to do some experiments with kaleidoscopes - if so, you might want to check out this recent post at Maxwell's Demon.

Some earlier blog posts on Farey sequences and Ford circles are here and here.

Update: Please check out the great post The Circles of Descartes on the Wolfram blog.

Thursday, January 28, 2010

off-topic: progressive education in Ottawa

If you know people in Ottawa who care about public education, please let them know about an event that is coming up in support of Churchill Alternative School, and in support of alternative public education generally.

From the promotional literature about this public talk by Alfie Kohn:

"Our knowledge of how children learn – and how schools can help -- has come a long way in the last few decades. Unfortunately, most schools have not: They’re still more about memorizing facts and practicing isolated skills than understanding ideas from the inside out; they still exclude students from any meaningful decision-making role; and they still rely on grades, tests, homework, lectures, worksheets, competition, punishments, and rewards. Alfie Kohn explores the alternatives to each of these conventional practices, explaining why progressive education isn’t just a realistic alternative but one that’s far more likely to help kids become critical thinkers and lifelong learners."

Alfie Kohn is the author of eleven books, including Punished by Rewards, The Schools our Children Deserve, Unconditional Parenting, and, most recently, The Homework Myth. He has been described by Time magazine as America's “most outspoken critic of education’s fixation on grades [and] test scores.”

The lecture will be on February 11th, 2010 from 7:00 to 9:30 pm at the auditorium of Woodroffe High School (2410 Georgina Dr., Ottawa).

Tickets are available online through Dovercourt at a cost of $20 CDN each.

If you are going to be in Ottawa on Feb 11 - please come out to hear Alfie's talk.

Thursday, January 21, 2010

a most loved and hated theorem

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the Infinte.
- Jorge Luis Borges, Avatars of the Tortoise

Young man, in mathematics you don't understand things. You just get used to them.
- John von Neumann

There have been a lot of nice posts recently about favorite theorems (at f(t), SumIdiot, and Research in Practice, for example).

I found it interesting that one of them (Kate Nowak of f(t)'s favorite), the uncountability of the Real Numbers via Cantor's Diagonalization Argument, also seems to be one of some people's most hated theorems.

Witness the raging debate that ensued just a short while ago when Mark CC of Good Math, Bad Math did the public service of explaining why a recent Cantor-denier crank was mistaken. More recently, over at P=NP there is an excellent post about Cantor's diagonal method that is also generating plenty of comments.

Cranks aside, a good number of smart people have been troubled by Cantor's argument, and how it is used. Ludwig Wittgenstein was not a Cantor denier, but my reading of his Remarks on the Foundations of Mathematics is that he felt that the way that we talk about the results of the argument causes us to skirt close to mathematical and linguistic nonsense. Wittgenstein refers to the diagonalization method as a "puffed up proof" because people claim that it shows more than it really does - to him, it shows us that "the concept real number has much less analogy with the concept natural number than we, being mislead by certain analogies, are inclined to believe" and that we are further mislead to believe that it reveals a property of the set of real numbers, rather than a limitation of the concept of "set." Wittgenstein put much greater limitations on mathematical discourse (and on sets) than most practicing mathematicians would, but his discomfort at the idea of 'larger' infinities should be noted, and we should realize that others who share this discomfort are in good company.

Cantor's diagonal argument is certainly one of my favorites, and over time I've come to appreciate it more as I have slowly understood (or gotten used to) its centrality to a surprising ecosystem of theorems and proofs that include the undecidability of the Halting Problem (which I think gets a vote for one of my  favorite theorems, whichever proof you use), fixed point theorems, Godel's incompleteness theorem, and Russell's paradox. (There is an interesting and accessible development of Cantor's diagonal argument in the context of fixed point theorems via category theory in the book Conceptual Mathematics by Lawvere and Schanuel; the less accessible version of the same is here.)

We can gain a lot of perspective on the Cantor-haters by reading an interesting paper by Wilfrid Hodges (mentioned by Richard Lipton of the P=NP blog). Hodges recounts his experience fielding dozens of papers attacking Cantor's diagonal argument while he was a journal editor. (Hodges paper is in postscript here, and here is a pdf.)

An observation by Mark CC  in his post, mentioned above, is also noted by Hodges: "many of our authors failed to realize that to attack an argument, you must find something wrong in it. Several authors believed that you can avoid a proof simply by doing something else." Or as Mark puts it: "You can't refute Cantor's proof using an enumeration without addressing Cantor's proof. This is just yet another stupid attempt to refute Cantor without bothering to actually understand it."

But why do some people react so badly to the non-countability of the Reals via Cantor's diagonalization method? Perhaps it is the same reason that some react to it so favorably. Hodges suggests:

It's nothing more than a guess, but I do guess that the problem with Cantor's argument is as follows. This argument is often the first mathematical argument that people meet in which the conclusion bears no relation to anything in their practical experience or their visual imagination... and even now we accept it because it is proved, not for any other reason.

It seems that we are often limited in our appreciation of what mathematics can prove, how proof works, and what we should expect to completely comprehend. Cantor's argument tests the limits of all of these, and sometimes we don't pass the test.

Tuesday, January 19, 2010

check out math teachers at play

You really should visit Math Hombre and check out the 22nd edition of Math Teachers at Play.

After doing that, please submit your posts for the 23rd edition, to appear next month right here at mathrecreation.

Friday, January 8, 2010

collecting old math texts

Perhaps the most deeply hidden motive of the person who collects can be described in this way: he takes up the struggle against dispersion.

There are now many resources available that allow us to look at primary sources in mathematics - the seminal documents that first introduced a new concept, or that changed the course of thinking and research. For example, the MAA has its mathematical treasures site, and the Royal Society has recently publicicized many of its treasures through its Trailblazing exhibit.

But what about old 'secondary' sources, like old textbooks? Primary sources are often interesting in how they stand out from their time - how they provoke and anticipate future directions. Secondary sources are often interesting because they do the exact opposite by encapsulating and reflecting the milieu in which they are produced.

School mathematics textbooks, individually and collectively, sum up many aspects of the times that produced them. They tell us about school mathematics, but also pedagogy, curriculum, who was expected to attend school, and the kind of people who taught at them. They tell us about typesetting, bookbinding, who authored (and authorized) texts, how they were distributed, and much more. For math teachers, it is interesting to see how perpetual the notion of reform is in mathematics education, and how current debates about the place of technology in mathematics education have previously played out in discussions about calculators, slide rules, and trigonometry tables.

In his notes on collectors in The Arcades Project, Walter Benjamin explains that "for the collector, the world is present, and indeed ordered in each of his objects... and for the true collector, every single thing in this system becomes an encyclopedia of all knowledge of the epoch, the landscape, the industry, and the owner from which it comes."

It would be interesting, for example, to chart how textbooks have tried to justify and encourage the study of mathematics over the years - the image below is an example taken from a text published in the 1930s; today what image would we use?

Pat Ballew (of Pat's Blog) has put together an interesting survey of the history of high school math texts that highlights the emergence of some key concepts that we currently take for granted. His overview shows how interesting and useful looking at old text books can be. For your own collecting, if you don't like the mustiness of old books, Google Books has made many very old math texts available.

Thursday, January 7, 2010

Turing 2012

Provided that we are still kicking around, those interested in mathematics and computer science will note 2012 as the Alan Turing centenary. The organizers of the Alan Turing Year celebrations have started planning, and their website already has many interesting resources for anyone who wants to learn more about Turing's life and work.

British PM Gordon Brown's apology to Turing from September 2009 provides a moving reminder of why Turing's life is viewed in a somber as well as a celebratory light.

Tuesday, January 5, 2010

odd Catalans

One thing that jumps out of the Catalan Number Triangle fractal is the infrequency and regular spacing of the odd Catalan numbers. The Catalan numbers show up in the right-most column of the Catalan Number Triangle, and the odd ones are the ones show up in black in the diagram above (those that are congruent to 1 mod 2 are displayed as black, and those congruent to 0 mod 2 are displayed as grey).

Mathworld tells us that odd Catalan numbers are those of the form $C_{2^k-1}$, in other words, the indicies of the odd Catalans are Mersenne numbers. The first bunch of odd-Catalans are 1, 5, 429, 9694845, ... (A038003 in OEIS).

Odd Catalans grow pretty rapidly. Coincidentally, there is an even more rapidly growing sequence known as the Catalan-Mersenne sequence (A007013 in OEIS). The Catalan-Mersenne sequence is not mathematically connected to the Catalan numbers - it is a subsequence of the Mersenne numbers which Catalan discovered, and is defined by this recursion:

The values of $c_n$ grow so quickly that OEIS only displays the first five terms.