This is a call for help—or for suggestions, at least. I’m teaching a knot theory class next semester. I’m looking for good props to use in the class to make knots. I would like to be able to make knots such as the following (and have my students do so as well). I suppose the…
Tag: topology
Möbius bubble wrap
This week’s New York Times Magazine has an article called “The Year in Ideas.” One feature in the article is “Bubble Wrap That Never Ends,” by Vanessa Gregory. She writes about the popular Japanese keychain called Mugen Puchi Puchi. It has six small buttons on it, and pressing them simulates popping bubble wrap. The keychain…
Four color theorem applets
The four color theorem is a beloved result with a long and fascinating history. The theorem says that four colors suffice to color any map so that no two bordering regions are the same color. The conjecture was made in 1852 by Francis Guthrie. After many, many failed proofs, the conjecture was finally put to…
Lipson’s mathematical LEGO sculptures
Ξ at the the 360 blog just posted a neat LEGO fact: it is possible to snap together two 2×4 lego bricks in 24 different ways. Given six of these LEGOs it is possible to snap them together in 915,103,765 different ways! This inspired me to post a link to a cool website by Andrew Lipson….
Flatland and other videos about dimension
Not long ago I watched the DVD of Flatland staring Martin Sheen as the voice of Arthur Square. The movie is based on Edwin Abbott Abbott’s 1884 book of the same title. Flatland is a story of polygons living in a two dimensional world and A. Square’s discovery of the third dimension. It is also…
Cutting and folding paper
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…
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…
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…
Wobbly tables and the intermediate value theorem
Tomorrow I’ll be introducing the intermediate value theorem (IVT) to my calculus class. Recall the statement of the IVT: if is a continuous function on the interval and is between and , then there exists a value such that . In other words, achieves all of the intermediate values between and . This is a very underappreciated theorem…
David Richeson: Division by Zero 