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

I just uploaded an article “Circular reasoning: who first proved that  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…

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. 

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…

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…

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…

Online LaTeX editors

For the last 10+ years I’ve taught topology using a modified Moore method, also known as inquiry-based learning (IBL). The students are given the skeleton of a textbook; then they must prove all the theorems and solve all of the problems. They are forbidden from looking at outside sources. The class types up their work as…

Ancient number systems in XeTeX

I am teaching a history of mathematics class this semester. We are beginning with a brief discussion of ancient number systems: Egyptian, Babylonian, Mayan, Chinese, Incan, Greek, Roman, and Hindu-Arabic. As I was writing up the first homework assignment it occured to me that I should investigate whether these numbers could be typeset using LaTeX. It…

Mathematics departments at liberal arts colleges

I’m often curious about how other mathematics departments do things—how they structure their curriculum, run the Putnam Exam, handle research projects, etc. This invariably leads to a lot of web searching. So I decided to put together a collection of links to mathematics departments at schools like mine (a small liberal arts college). Because I…

Plato’s approximation of pi?

Today I came across an assertion that Plato used as an approximation of . Indeed, it is not a bad approximation: (although it is not within Archimedes’s bounds: ). Not only had I not seen this approximation before, I had not heard of any value of attributed to Plato. I investigated a little further and…

Puzzler: a squarable region from Leonardo da Vinci

It is famously impossible to square the circle. That is, given a circle, it is impossible, using only a compass and straightedge, to construct a square having the same area as the circle. I will let you read elsewhere about the exact rules behind compass and straightedge constructions. The punchline is that if you begin…

Angle trisection using origami

It is well known that it is impossible to trisect an arbitrary angle using only a compass and straightedge. However, as we will see in this post, it is possible to trisect an angle using origami. The technique shown here dates back to the 1970s and is due to Hisashi Abe. Assume, as in the…

An interesting multivariable calculus example

Earlier this semester in my Multivariable Calculus course we were discussing the second derivative test. Recall the pesky condition that if is a critical point and , then the test fails. A student emailed me after class and asked the following question. Suppose a function has a critical point at and . Moreover, suppose that…