Inspired by Chaim Goodman-Strauss’s recent video about symmetries, paper snowflakes, and paper dolls, I decided to post a few other paper-related videos. First is a video showing some cutting tricks for a Möbius strip. I show this to my topology class, then have them play around with Möbius strips—twisting them various numbers of times and…
Flash cards are a good idea
I recently came across an article by the mathematician Ethan Akin, whose work in topology and dynamical systems I admire greatly, called “In Defense of ‘Mindless Rote’“. In the article he defends the traditional education model of having students memorize mathematical facts and techniques. He begins with the following quote from Alfred North Whitehead’s Introduction to Mathematics….
DARPA’s 23 mathematical questions
DARPA (The Defense Advanced Research Projects Agency) recently released the DARPA Mathematical Challenges (Word document)—23 challenging mathematical problems “with the goal of dramatically revolutionizing mathematics and thereby strengthening DoD’s scientific and technological capabilities.” The titles of the challenges are: The Mathematics of the Brain The Dynamics of Networks Capture and Harness Stochasticity in Nature 21st Century Fluids…
Math quotes on Freakonomics blog
There’s a mathematical topic today on the New York Times’ Freakonomics blog. Their daily bleg (definition: using a blog to beg for information): What’s been said about math? Readers are invited to leave their favorite quotes about mathematics (or quotes by mathematicians) in the comments.
Kuratowski’s closure-complement theorem (solution)
Stop! This post contains spoilers. This page has the solution to the problem posed in yesterday’s post. We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations. In 1922 Kuratowski proved the following theorem. Theorem. At most 14 sets can…
Kuratowski’s closure-complement theorem
One of my favorite theorems in elementary topology is Kuratowski’s closure-complement theorem. First some notation. For any set let denote the complement of and denote the closure of . (Recall that and is the union of and all the limit points of ). Here’s the problem. Find a set so that we can construct as many…
Not so elementary my dear Doyle
I loved Sherlock Holmes mysteries when I was a child. A few years ago I learned the disappointing news that Sir Arthur Conan Doyle was a spiritualist who believed in séances. Because of this, his once close friend Harry Houdini became a rival. Their falling out began when Houdini joined the Doyles for an intimate séance,…
Technology of today’s students
Amherst College has over 1680 students. Of them, only 14 have desktop computers and 5 have landline telephones. Here is an interesting list of IT related facts about the college and their students. [via]
Königsberg today
In 1736 Leonhard Euler solved the now-famous bridges of Königsberg problem. It is often hailed as the birth of topology and graph theory. Although no graph appears in Euler’s paper, his argument is topological in spirit. (He proved another famous topological theorem a decade and a half later.) The problem is stated as follows. If a resident of Königsberg…
What is the difference between a theorem, a lemma, and a corollary?
I prepared the following handout for my Discrete Mathematics class (here’s a pdf version). Definition — a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true. Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In…
David Richeson: Division by Zero 