Posted by: Dave Richeson | September 9, 2013

## 2013: the year of pi

A couple days ago I saw this tweet.

Pretty cool! Let’s see why

$\arctan(2)+\arctan(0)+\arctan(1)+\arctan(3)=\pi.$

Two terms are easy to deal with:

$\arctan(0)=0$ and

$\arctan(1)=\pi/4.$

But why is $\arctan(2)+\arctan(3)=3\pi/4?$

One way to prove this is via the inverse trigonometric identity

$\displaystyle \arctan(a)+\arctan(b)=\arctan\left(\frac{a+b}{1-ab}\right)\, (\text{mod }\pi).$

In this case

$\arctan(2)+\arctan(3)=\arctan(-1)\, (\text{mod }\pi).$

Because $\arctan(2)$ and $\arctan(3)$ are between 0 and $\pi/2$,  $\arctan(a)+\arctan(b)=3\pi/4$.

I didn’t use this approach on my first attempt to solve the problem.  (To be honest, I didn’t know this identity existed before finding it online.) I used geometry and trigonometry. We are interested in the angle $\theta$ in the diagram below.

The law of cosines tells us that

$(2+3)^2=(\sqrt{5})^2+(\sqrt{10})^2-2\sqrt{5}\sqrt{10}\cos\theta.$

This implies that $\cos\theta=\frac{1}{\sqrt{2}},$ and hence $\theta=3\pi/4.$

(I wonder if there is a year of tau coming up sooner than the year 112233.)

Update: I just saw that Cut-the-Knot has a page devoted to this topic too.

Posted by: Dave Richeson | July 4, 2013

## Using a kayak to measure the perimeter of a lake

I’m on vacation this week on a lake in northern Michigan (hold up your right hand, palm toward you, point at the first knuckle of your middle finger—that’s where I am). Yesterday I paddled around the perimeter of the lake in a kayak. On a whim I brought my GPS-enabled phone. My route is shown below. Note that the cartographer didn’t do a perfect job—despite what you see, I can assure you that I paddled only on water.

When I got back, the GPS said that I had paddled for 2.74 miles (14470 feet). Let’s say that I stayed exactly 50 feet off the shore for the entire trip. What is the perimeter of the lake?

We should probably make some assumptions about the shoreline and about my path. For simplicity, let’s say that my path is differentiable, that it bounds a region $X$, and that the lake is the set ${N_{50}(X)}$, where $N_{r}(X)$
is the radius ${r}$ neighborhood of ${X}$. Intuitively, the boundary of $N_r(X)$ is always $r$ units off the port side of the boat. (See Ravi Vakil’s article “The Mathematics of Doodling” for more about this set.)

You may not think you have enough information to solve the problem, but you do! The clue is found in a well-known puzzler. Suppose we have an electrical wire running round the equator of the earth. We want to get it up off the ground and put it at the top of 50-foot electrical poles. How much longer must the wire be? [If you haven't seen this puzzler before, stop now and think about it.] If the radius of the Earth is ${R}$ feet, then the original wire must be ${2\pi R}$ feet long. The new wire is a circle with a radius 50 feet longer than before, so its length must be ${2\pi (R+50)=2\pi R+100\pi}$ feet. That is, it must be ${100\pi\approx 314}$ feet longer.

It turns out that this argument can be generalized in various ways (again, see Vakil’s article for more information). First of all, if ${X}$ is any convex set with a polygonal boundary, then, just like in the puzzle, the perimeter of ${N_{r}(X)}$ is ${2\pi r}$ units longer than the perimeter of ${X}$. Vakil illustrates this with the picture below.

It turns out that the exact same result holds if ${X}$ is nonconvex—such as the region enclosed by my kayak trip. As the kayak goes in coves and around peninsulas, the “extra” regions of convexity and nonconvexity cancel each other out. In the end it is the same as going around a circle one time. Thus the perimeter of the lake is ${2\pi\cdot 50\approx 314}$ feet longer than my kayak path: ${14470+314=14784}$ feet.

Posted by: Dave Richeson | April 16, 2013

## Countability of the rationals drawn using TikZ

I’m continuing my exploration of TikZ (here is my first post about TikZ). I will be showing my Discrete Math class how to “count” the positive rational numbers. (See this old blog post for more information about countable sets.) I used TikZ to create the picture below.

Here is the source code for this figure. If you click on the link you can get an editable copy of the document in WriteLatex

Posted by: Dave Richeson | April 12, 2013

## Greatest living mathematician and expositor?

On Twitter I posed the following question:

I got a great repsonse. Here is the complete—unedited—list of names (in alphabetical order).

Posted by: Dave Richeson | April 9, 2013

## Bubble diagrams for functions in LaTeX using TikZ

I am teaching Discrete Math this semester (our intro-to-proof course). One of the topics is functions. Not surprisingly my students and I have to draw “bubble diagrams” for functions between finite sets—and we have to include them in LaTeX documents. Rather than simply sketching them in Adobe Illustrator and importing them as graphics, I decided to try creating them in TikZ. After a lot of tinkering I came up with something pretty nice (see below). Moreover, I set it up so that all the complicated TikZ and LaTeX commands are in the header of the document. So my students and I can easily generate new bubble diagrams.

I’ve put a sample document on WriteLatex. If you click on this link you can get an editable copy of the document. You can edit it and it won’t change my copy—so go wild with it! (This cool feature of WriteLatex is described on their WriteLatex for Education page.)

Posted by: Dave Richeson | March 5, 2013

## Circular reasoning: who first proved that C/d is a constant?

I just uploaded an article “Circular reasoning: who first proved that $C/d$ is a constant?” to the arXiv. The abstract is below. It is on a topic that I’ve been thinking about and reading about off-and-on for the last year and a half. I’d be happy to hear people’s thoughts, reactions, and impressions.

Abstract. We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid’s Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes’s work coexisted with the 2000-year belief—championed by scholars from Aristotle to Descartes—that it is impossible to find the ratio of a curved line to a straight line.

Posted by: Dave Richeson | February 26, 2013

## Interview on Wild about Math!

I had a very nice conversation with Sol Lederman (of the Wild about Math! blog) on his “Inspired by Math” podcast series. Check it out and be sure to check out his other podcast episodes.

Posted by: Dave Richeson | February 10, 2013

## Editing mathematical writing

As I mentioned in a previous post, I’ve been assigning large-scale collaborative writing projects in my mathematics classes. I’ve had my topology students write a textbook for their class, and this semester I’ve been doing the same in my discrete mathematics class. As I mentioned in that post, the approach has been very successful, but one area that needs work is the editing/revising processes. Some  students are very reluctant to edit what has already been written. I think partly they are uneasy about editing the work of others, but partly they do not know how to revise/edit writing. So I’m trying a few new things this semester.

As I have in the past, I will give the students a document that I wrote called “The Nuts and Bolts of Writing Mathematics.” But now I’ve taken that document and turned it into a set of questions that the students should keep in mind while editing the document. (In fact, this checklist should work for any mathematical writing, not just collaborative writing.) You can download the pdf or view the HTML document below. (Note: If you have any suggestions for questions that I should add, please post them in the comments!)

I also plan to model good editing. I will give them a mathematical document (that I wrote) that needs a lot of revising, and we will spend some class time going through it, discussing how to revise it.

Finally, I’ve been reading this book on team writing. It discusses three ways to do team writing: face-to-face (you and I work together on the writing), divided (you do this, I do that), and layered (you look at it first, I look at it next). I think the layered approach will work best for my students. The book breaks up naturally into small parts—theorems and examples. For each problem I will practice the layered approach to writing and revising: Student 1 does the writing, student 2 does the editing, I give feedback, then student 3 does the final editing. (I may add one more layer of student editing before I see it.) I’ve created a Google Docs spreadsheet which gives, for each theorem/example, the names of the student writers and editors and the due dates for their work.

Editing mathematical writing: a checklist

The mathematical argument

• Is the mathematics correct?
• Is the argument easy to follow?
• Is there any extraneous information?
• Is the argument clear?
• Would it benefit from reordering the sentences?
• Is the level of detail appropriate for the audience?
• Are all proofs free of examples?

Mathematical writing

• Is the writing in first person plural? (Use “we,” not “I” or “one.”)
• Is the writing in the present tense? (Write “we find that,” not “we found that.”)
• Is the writing in the active voice? (Avoid “it was shown that.”)
• Are there symbols where there should be symbols and words where there should be words? (Use “x≤ 0,” not “x is ≤ zero” or “x is less than or equal to zero.”)
• Are the connecting words (thus, so, hence, therefore, moreover, furthermore, in addition, consequently, etc.) used appropriately and not repetitively?
• Is the writing free of flowery, imprecise, descriptive, and vague language?

LaTeX

• Are all mathematical expressions written in math mode? (We should see x, not x.)
• Is all text written in text mode? (We should see “such that,” not “suchthat.”)
• Are mathematical functions typeset correctly? (We should see sin(x), not sin(x).)

Figures

• Are the figures of high quality?
• Do they have captions?
• Does the text refer to them?

Mechanics

• Is the spelling, grammar, punctuation, capitalization, subject/verb agreement, sentence structure, etc. correct?

Style

• Does the layout reflect the style of the rest of the document?

Note: do not be afraid to scratch an entire proof and start over. Sometimes completely rewriting an argument is better than trying to fix a poorly written one.

Posted by: Dave Richeson | December 16, 2012

## How I teach topology: an inquiry-based learning approach

Recently I’ve had a number of people ask for more information about how I teach topology. I’ve taught it five times using a “modified Moore method” or “inquiry-based learning” approach. I’ve modified it each time, trying to work out the bugs. I think it is pretty successful now.

Context. At our college all math majors are required to take a 400-level class. Typically it is Complex Analysis or Topology (they are taught in alternating years). Thus, while topology is not a required class, many math majors have to take it in order to graduate. Some of them find it extremely challenging. On the other hand, I often have very talented students in the class. In fact, often the juniors (or even sophomores) who take the class are among the best students. Typically I have 8-15 students in the class (depending on whether we offer one or two sections). The other important piece of information is that the college wants the 400-level course to be writing intensive. Many departments have a “senior seminar” that fills this role. We don’t, so that’s an extra reason to teach topology in the way that I do.

Brief description. In this course the students do not have a textbook; in fact, they are forbidden from using outside sources at any time. Instead, they are given the skeleton of a textbook. It has definitions, statements of theorems, some explanatory text, and some problems. They must prove the theorems, solve the problems, and type their work into the empty textbook. By the end of the semester they have created their own textbook. I have it printed and bound at the college’s print center and they take it home with them. (I like to have 5-10 people contribute to each textbook, so if the class is on the large side I have them create two textbooks in parallel.)

Textbook. The textbook I use came from the topology course that I took as an undergraduate—from Dick Bedient at Hamilton College. I’ve modified the book quite a bit over the years, but I couldn’t have done this without the excellent book I began with. It began as a Word file, but I moved everything over to LaTeX years ago. I maintain two copies of the book—one has space for the students to write in (like a workbook) and the other has no spaces (this is the version that they edit). I have the college’s print center print the workbook one sided (in case they need to write on the backs of the pages), three-hole punch it, put it in a binder, and sell it in the bookstore (for the cost of printing only). Sample pages of both versions are shown below.

Homework. Each class period I tell the students what theorems/problems they should work on. It is a guessing game on my part—I have to estimate how far we’ll get in the next class period. They do the work in the workbook at home. They are free to work together, they are free to talk with me, but they are forbidden from looking at outside sources.

Presentations. When I first taught the class I had students volunteer to present the proofs/problems in class. The problem was that only the best students volunteered. So I moved to a system in which I assigned (randomly) presentations to the students. Now the students know ahead of time what “their problem” will be. If a student comes to class unprepared or presents an incorrect proof, then I take volunteers to complete the proof (or the student can stay at the board and repair the proof with the help of the rest of the class). This is also a time to talk about how the proof is written. They know that what goes on to the board will ultimately go into the textbook, so the class often gives wording, notation, or organization suggestions. (At the beginning of the semester I talk about how to give appropriate feedback. We don’t want hurt feelings.) Often the weaker students are nervous about presenting a solution that is wrong, so I tell them that they are free to come to my office in advance to “check out” their proof.

Daily sheets. The students are required to participate in class. Each day I bring the sheet below and keep track of who presents and who makes comments. I also leave myself notes, like “great proof!” or “volunteered to present the theorem when Billy was sick” or “came to class 15 minutes late.” (I number each column by the theorem/problem number.) I use these sheets to help assign a class participation grade at the end of the semester. (Excel file)

Green pen/homework. At the end of class, the students turn in the pages that we just discussed. I grade each problem on a simple V (very good work), G (good work), L (little work), N (no work) scale. Because we talked about each problem at length, I do not write any comments on their work. One great idea I had a few years ago was to require the students to bring a green pen to class. During class they can take as many notes and make as many corrections as they please, BUT they must use the green pen. This makes my grading especially easy. I can see exactly what they wrote at home and what they wrote in class.

Secretary. Each day one student is the “secretary”—this position rotates through the class. (If the class is broken into two groups, there will be two secretaries—one for each textbook.) The secretary is responsible for writing down everything clearly, taking note of all editorial suggestions. The secretary must type all of the new content into the textbook before the start of the next class.

Textbook. When I was a student the textbook was a MS Word file that lived on a 3.5″ floppy disk. The students would pass it around as needed (this was in 1992—before the web, and at a time when I had to go to the basement of the library to check my email). When I first taught the class the textbook was a Word file that students emailed back-and-forth. Later I turned it into LaTeX (which meant there was a whole directory of files). At first I had the secretaries edit the files on a dedicated computer in the math major’s research room. The last two times I taught the class I placed the files in a shared DropBox folder—the students could edit the book at home or on one of the school machines.  That worked very well, but not flawlessly—a few times we had problems with files being forked. But we’d come a long way. Next time I teach the class I plan to use one of the new online LaTeX websites. I think this will be fantastic—collaboration will be a breeze and the students won’t have to install LaTeX to do their work. (Note: I’ve had my students use LaTeX in many of my classes—by and large they seem to really like it. I made this LaTeX cheat sheet which helps flatten the learning curve a little.)

Editing. The textbook is broken down into chapters. At the start of the semester I create “editorial teams” for each chapter. After we complete a chapter (and the secretaries have typed in the proofs), the editorial team is responsible for “cleaning it up.” They do that, then I take a pass at it, making my corrections and suggestions (hand written on the pages), and then the editorial team implements my changes. If there is a weakness in my system it is at this stage. The editorial team does not make many changes on their own. I think they are uncomfortable editing the work of a classmates. The draft I get generally needs a lot of work. I need to strategize about how to improve this. (Suggestions? Leave them in the comments!) There is also an editorial team that is responsible for the entire book. They coordinate the editing of the full book at the end of the semester.

Images. The book requires many images—open and closed sets in $\mathbb{R}^n$, the torus, the Klein bottle, the Möbius strip, etc. I have yet to find a good drawing program that is easy to use, powerful enough to produce good images, and free (or cheap). I’ve tried many different pieces of software and have had mixed results with all of them. I’d be happy to hear suggestions for what to use.

Exams. I either give two midterms and a final exam or three midterms and no final exam.

Peer evaluations. I have the students evaluate each other (and themselves) at the end of the semester. I use this information to help me assign their class participation grades. I’ve found that sometimes students who are quiet in class will get rave reviews from their classmates (“she was always extremely helpful with editing,” “he was very good at explaining the concepts to me,” etc.). On the flip side, some of the more “active” members of class could be bad collaborators (“he never showed up for our editing sessions,” “her secretarial work was never done on time,” etc.). Here is an example of the evaluation sheet. (Excel file)

Final textbook. At the end of the semester, after the textbook has been edited multiple times, I give it a grade. The entire group gets the same grade on the book. Usually the grade is high, but occasionally (rarely) I’ll have a group that does not put in the required effort. When it is complete I quickly send it to our print center to have it printed and bound. The turn-around time is very short, so most students are able to pick it up before they leave for the semester. Typically the students are THRILLED to hold this book. They love to share it with their friends and parents. It is neat to see their pride. However, I used to have a few books weren’t picked up. So (and this also coincided with the recession and subsequent budget cuts) I now have them pay for the book (in advance)—which is usually \$5, or so. This way we only print the book for students who want it (usually 75% of the class).

Final thoughts. I love teaching this class and the students (for the most part) enjoy taking it. The material is very challenging and some of the weaker students struggle with it. But I think that a faster-paced traditional class would be even worse. I think that the students really learn the material using this approach. I like that the students spend time presenting mathematics at the board. This is missing from most other mathematics classes. And the emphasis on writing and revising is very valuable. Finally, I love seeing their faces light up when they pick up the finished book at the end of the semester.

Source files. Professors at other institutions have asked me for copies of the blank textbook. Unfortunately, because I was not the original creator of the book I don’t feel comfortable distributing it. However, there is a lot of good IBL content online. For example, look at the Journal of Inquiry-Based Learning in Mathematics and the links on their website.

What’s next? I’ve enjoyed this method of teaching so much that I’ll be teaching my Discrete Math/Intro to Proof class using this technique next semester. I’m very excited about it (and somewhat nervous about it too). I’m using these notes (by Dana Ernst) as a starting point. I’m planning to do things similarly, but slightly differently, in this class than in my topology class. But that’s a topic for another blog post.

Posted by: Dave Richeson | December 9, 2012

## The Pigpen Cipher in Latex

Recently my son and his friends have been enjoying sending secret messages back-and-forth using the pigpen cipher (also called the masonic cipher or Freemason’s cipher). It produces codes that look like:

The pigpen cipher is a simple substitution cipher—there is a 1-1 correspondence between these special symbols and letters of the alphabet. The correspondence is illustrated in the chart below. Each letter is replaced by the boundaries around the letter and a dot if there is one. So, for example,

I, being a lover of Latex, wondered if it was possible to create pigpen ciphers in Latex. It is! The cipher text text above was created using Oliver Corff’s Pigpen Cipher package. After downloading the package and including it in the Latex file you must simply type

{\pigpenfont MATH IS FUN}

to obtain

(Pigpen cipher key (picture) from Wikipedia)