Division by Zero

Posted by: Dave Richeson | July 3, 2009

Quantifier soup

Anyone who has tried to teach Calculus I students the $\varepsilon-\delta$ definition of the limit knows that students have a difficult time working out definitions with multiple quantifiers. In fact it is something that we must come back to again and again with our mathematics majors.

While doing a little research this week I came across the definition of something called the specification property for a discrete dynamical system—it was introduced in the early 1970’s by Rufus Bowen. The logical complexity of the definition made me laugh out loud the first time I read it. I had to reread it about 5 times to figure out what it was saying.

The dynamical system $f:X\to X$ is said to satisfy the specification property if for any $\varepsilon>0$ there exists an integer $M>0$ such that for any $k\ge 2$ , any $k$ points $x_1,\ldots,x_k\in X$ , any integers $a_1\le b_1\le a_2\le b_2\le\cdots\le a_k\le b_k$ with $a_i-b_{i-1}\le M$ for $2\le i\le k$ and for any integer $p$ with $p\ge M+b_k-a_1$ , there exists a point $x\in X$ with $f^p(x)=x$ such that $d(f^n(x),f^n(x_i))\le \varepsilon$ for $a_i\le n\le b_i$ , $1\le i\le k$ .

(In case you’re wondering, roughly speaking it says that for a dynamical system with this property, given a finite number of finite orbit segments, there is a periodic orbit that follows one orbit segment very closely, then a little while later follows the next one, then a little while later follows the next one, etc.)

Leave a Comment

Posted in Math | Tags: quantifiers, Rufus Bowen, specification property

Posted by: Dave Richeson | July 1, 2009

Ambigram: pie≠3.14

I thought the first comment on this article was funny. It says that pie and 3.14 are mirror images, if written in a certain way. Regular readers may recall that I had fun creating ambigrams a few months ago. This blog comment inspired me to whip up a quick ambigram exhibiting this symmetry.

pie

I like this ambigram because it has two meanings. On the one hand $\pi\ne\text{pie}$ and also $\pi\ne 3.14$ .

4 Comments

Posted in Humor, Math | Tags: Ambigram, pi, pie

Posted by: Dave Richeson | July 1, 2009

Last Sunday was a perfect day

Most geeky math types (like me) already know about pi day (March 14… 3/14, get it?).

Writing in The Times Online, Marcus du Sautoy suggests a new math holiday: June 28. He suggests calling this day the World Math Day (actually, he suggests World Maths Day).

Why? What is so mathematical about June 28?

June 28 can be expressed as 6/28 and 6 and 28 are the first two perfect numbers. A perfect number is an integer that is the sum of its factors (not including the number itself). For example, $6=1\cdot 2\cdot 3=1+2+3$ and $28=1\cdot 2\cdot 4\cdot 7 \cdot 14=1+2+4+7+14$ . The next known perfect number is 496.

The timing for this pronouncement is especially fitting. A new perfect number was just discovered on April 12, 2009. Actually, what was found was the 47th Mersenne prime (as part of the Great Internet Prime Search). But I’m getting ahead of myself. Let me explain.

There is an intimate connection between perfect numbers and a special class of primes called the Mersenne primes. The ancient Greeks noticed that the first four perfect numbers (6, 28, 496, 8128) fit a pattern—they all had the form $2^{n-1}(2^n-1)$ for some $n$ . Moreover, in each of these cases $2^n-1$ was prime (3, 7, 31, 127, respectively). In his Elements, Euclid proved that if $2^n-1$ is prime, then $2^{n-1}(2^n-1)$ is a perfect number. Then, two thousand years later Euler proved that every even perfect number must have this form. Primes of the form $2^n-1$ are called Mersenne primes.

Thus, each new Mersenne prime corresponds to a new perfect number.

By the way, we know of no odd perfect numbers, but they may exist. Thus there are exactly 47 known perfect numbers. It is not known whether there are finitely or infinitely many Mersenne primes.

Leave a Comment

Posted in Math | Tags: Euclid, Euler, Great Internet Prime Search, Mersenne, Mersenne primes, perfect numbers, World Math Day

Posted by: Dave Richeson | June 30, 2009

Putting CVs online

I’ve had my CV online for quite some time. For a while I’d type it up, export it as a pdf, then upload it to my website. The problem was that I would only upload it once every six months or once a year—it was perpetually out of date.

Then I made the switch to Google Docs—I uploaded my CV there, then I had Google publish it as a web page. Now when I add items to the CV, Google Docs automatically republishes the CV with the changes. (I decided not to link to my CV from here, but there is a link to in on my personal web page if you want to see it.)

Here’s another fun thing I did with some of the information on my CV—I put it on a Google map. I have markers for my school and work history as well all the places I’ve given talks. Just click on the markers for more information.

View Larger Map

There are a lot of interesting place that I’d like to go. I just need an invitation to speak there…

2 Comments

Posted in Academic Technology | Tags: curriculum vitae, Google Docs, Google Maps

Posted by: Dave Richeson | June 29, 2009

Indeterminate form in The New Yorker

In Calculus II we teach our students about a variety of indeterminate forms: $\displaystyle\frac{0}{0}$ , $\displaystyle\frac{\infty}{\infty}$ , $\infty-\infty$ , etc. I was reminded of another indeterminate form when reading Malcolm Gladwell’s thought-provoking (negative) review of the book Free: The Future of a Radical Price, by Chris Anderson (editor of Wired). The review appears in The New Yorker (that you can read online for free…).

Gladwell describes premise of the book:

[Free] is essentially an extended elaboration of Stewart Brand’s famous declaration that “information wants to be free.” The digital age, Anderson argues, is exerting an inexorable downward pressure on the prices of all things “made of ideas.” Anderson does not consider this a passing trend. Rather, he seems to think of it as an iron law: “In the digital realm you can try to keep Free at bay with laws and locks, but eventually the force of economic gravity will win.”

The indeterminate form that came to my mind was $0\cdot\infty$ . If $\displaystyle\lim_{x\to a}f(x)=0$ and $\displaystyle\lim_{x\to a}g(x)=\infty$ , then $\displaystyle\lim_{x\to a}f(x)g(x)$ could be anything; $f(x)g(x)$ could be dominated by $f$ and be driven down to zero, or dominated by $g$ and blow up to infinity, perhaps they will will be evenly matched in their fight and $f(x)g(x)$ will tend to a finite nonzero number, or maybe they will fight eternally with no limit at all.

I was reminded of this indeterminate form when Gladwell was writing about the fact that Google has not made money from YouTube.

Why is that? Because of the very principles of Free that Anderson so energetically celebrates. When you let people upload and download as many videos as they want, lots of them will take you up on the offer. That’s the magic of Free psychology: an estimated seventy-five billion videos will be served up by YouTube this year. Although the magic of Free technology means that the cost of serving up each video is “close enough to free to round down,” “close enough to free” multiplied by seventy-five billion is still a very large number. [my emphasis].

I’m terribly over-simplifying the situation with this crude analysis, but here goes. Although it may be the case that $\displaystyle\lim_{x\to \infty} c(x)=0$ (where $c(x)$ is the cost to host each video if $x$ videos have been hosted), Gladwell takes issue with Anderson’s assertion that $\displaystyle\lim_{x\to \infty}(x\cdot c(x))\approx 0$ , pointing out that it may be the case that $\displaystyle\lim_{x\to \infty}(x\cdot c(x))=\infty$ .

1 Comment

Posted in Math | Tags: calculus, Chris Anderson, Free, indeterminate form, limits, Malcolm Gladwell, New Yorker, review

Posted by: Dave Richeson | June 25, 2009

The famous trick donkeys: a Sam Loyd puzzle

Several years ago a colleague and I made paper copies of this famous old puzzle to distribute to prospective mathematics students. It is a fantastic puzzle, so I thought I’d post it here again.

From what I can tell, the puzzle was invented in 1871 by Sam Loyd (1842–1911), probably history’s most famous “puzzler.” It was printed on a card which was used by P.T. Barnum to promote his circus. The puzzle was called “P. T. Barnum’s trick mules” or “The famous trick donkeys.” Apparently Barnum distributed millions of them, and Loyd made thousands of dollars in just a few weeks from this single puzzle.

The goal of the puzzle is to arrange the three cards so that each rider is in his/her correct riding position on top of a donkey. (Here’s the original color version of the card complete with the solution written in reverse—to be viewed in a mirror.)

Print out the image below, cut out the cards, and see if you can put the riders on the donkeys.

horses

Give up? Here the solution.

(Disclaimer: now that I look at these images again, I see that they aren’t Loyds’ original donkeys and riders. This may be a later version created by him, or a copy by someone else.)

If you like Sam Loyd puzzles, check out this fully digitized Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums.

2 Comments

Posted in Math, Puzzle | Tags: famous trick donkeys, P. T. Barnum's trick mules, Puzzle, Sam Loyd

Posted by: Dave Richeson | June 22, 2009

Carnot’s Theorem

Here’s a neat theorem from geometry.

Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c is negative if the segment joining the circumcenter to the side does not pass through the interior of the triangle (such as the value b shown below, represented by the teal segment), and it is positive otherwise.

Then we have the following elegant result:

Carnot’s theorem. a+b+c=R+r

Check out this GeoGebra applet that I created to see this theorem in action.

Recently I wrote about the Japanese Theorem. If you were unsuccessful in proving this beautiful theorem, try again using Carnot’s Theorem.

1 Comment

Posted in Math | Tags: applet, Carnot's theorem, circle, circumcenter, circumscribed, GeoGebra, geometry, incenter, inscribed, Japanese theorem, triangle

Posted by: Dave Richeson | June 19, 2009

Euler’s Gem is the #2 best-selling mathematics book for libraries

It was a nice surprise to read this blog post at the Princeton University Press blog. Apparently my book (Euler’s Gem) is currently the number 2 best-selling mathematics book for libraries (according to the Library Journal). Cool! It was second to The Princeton Companion to Mathematics, which was edited by Fields Medal winner Tim Gowers.

(By the way, anyone who is so-inclined can follow Euler’s Gem on Facebook.)

3 Comments

Posted in Math | Tags: best-selling books, Euler's Gem, Facebook, Library Journal

Posted by: Dave Richeson | June 18, 2009

Three cool facts about rotations of the circle

I was playing around with GeoGebra and made this applet about one of the simplest, but most intersting dynamical systems: the rigid rotation of a circle. Let me tell you a little about this fascinating subject.

Let $S^1$ denote a circle. For simplicity, let’s think of it as a circle with circumference 1. Let $a$ be any real number. Define a function $f_a:S^1\to S^1$ as follows: if $x$ is a point on $S^1$ , then $f_a(x)$ is obtained by rotating $x$ counterclockwise $a$ units.

There are a few other equivalent ways of thinking of this. You may want to think of $S^1$ as the interval $[0,1]$ with the points 0 and 1 glued together. Or you may want to think of it as $\mathbb{R}/\mathbb{Z}$ ; that is, take the real number line and wrap it up like a slinky so that all points that differ by an integer are glued together. In these cases we can write our function as $f_a(x)=x+a\text{ (mod }1)$ .

We may view this function as a dynamical system. Given a point $x$ , we obtain the orbit of $x$ by plugging it into $f_a$ repeatedly: $x, f_a(x), f_a(f_a(x)), f_a(f_a(f_a(x))),\ldots$ . As usual we will use the familiar notation that $n$ compositions of the function, $(f_a\circ\cdots\circ f_a)(x)$ , is denoted $f^n_a(x)$ . However, in this case $f^n_a(x)=f_{na}(x)$ . In other words, rotating $n$ times by $a$ is the same as rotating once by $na$ .

This is not a chaotic dynamical system—far from it. Points do not get mixed up at all, this is a rigid rotation of the circle.

Cool fact #1.

The behavior of the orbits differ depending on whether $a$ is or is not rational. If $a$ is rational, then every point is periodic. The period of each point is the same; it is the value of the denominator of $a$ (assuming $a$ is expressed as a reduced fraction). On the other hand, if $a$ is irrational, then not only are the orbits not periodic, they are dense in the circle. By this we mean that the orbit “fills up” the entire circle (or more precisely, given any point of the circle, there is a subsequence of the orbit that converges to this point).

Theorem. If $a$ is rational, then every point is periodic. If $a$ is irrational, then every point has a dense orbit.

Cool fact #2.

The previous theorem states that the orbit of any point under an irrational rotation is dense. However we can say more than this. It turns out that if we look at the limiting behavior of an orbit, we see that it fills up the circle uniformly (technically, we say that the function is uniquely ergodic).

Theorem. Let $a$ be irrational and $I$ be an arc in $S^1$ . Let $m(n)$ be the cardinality of $I\cap\{0,f_a(0),f_a^2(0),\ldots,f_a^{n-1}(0)\}$ . Then $\text{length}(I)= \displaystyle\lim_{n\to\infty}\frac{m(n)}{n}$ .

In other words, the fraction of the time that the orbit spends in $I$ is precisely the length of the interval $I$ .

Cool fact #3.

Consider a finite orbit segment: $\{0,f_a(0),f_a^2(0),\ldots,f_a^{n-1}(0)\}$ . These $n$ points divide the circle into arcs various lengths. It turns out that for a fixed $n$ , the lengths come in only one, two, or three sizes. Moreover, if there are three lengths, then one is the sum of the other two!

Three Gap Theorem (The Steinhaus Conjecture). The points $\{0,f_a(0),f_a^2(0),\ldots,f_a^{n-1}(0)\}$ divide $S^1$ into arcs of one, two, or three lengths. If there are three lengths, then one is the sum of the other two.

To see the three gap theorem in action, view this GeoGebra applet.

There is a lot more one can say about rigid rotations of the circle; they have interesting connections to dynamical systems, topology, the study of continued fractions, number theory, symbolic dynamics, etc. Go explore!

Leave a Comment

Posted in Math | Tags: circle, dynamical systems, irrational rotation, rational rotation, rotations, Steinhaus conjecture, three gap theorem, uniquely ergodic

Posted by: Dave Richeson | June 12, 2009

Keith Devlin totally looks like Graham Nash

I was watching (and loving) the PBS American Masters documentary on Neil Young last night. There were several interview snippets with Graham Nash. Every time he came on he reminded me of Keith Devlin (NPR’s “The Math Guy”). They look similar, are both British, are about the same age, and both want to teach our children (although I don’t care for Nash’s song and I’m not sure I agree with Devlin’s ideas for teaching multiplication).

So, again, in the spirit of the website Totally Looks Like: Keith Devlin totally looks like Graham Nash.

devlin nash