Proof by Induction with a Difficult Base Case

I’m teaching Discrete Mathematics this semester. It is our “intro to proofs” class. One of the proof techniques the students learn is proof by induction. I told the class that usually the base case for induction proofs are easy and that most of the work occurs in the inductive step. Indeed, most of the proofs…

Proof Without Words: Gregory’s Theorem

Archimedes famously used inscribed and circumscribed polygons to approximate the circumference of a circle. He then repeatedly doubled the numbers of sides to get an approximation for π. In 1667, James Gregory did the same, but he used areas: He discovered the following beautiful double-recurrence relation that can be used to compute the areas of inscribed…

Möbius Band Ambigram

Almost 10 years ago I had some fun making ambigrams (a word or words that have some sort of symmetry—rotation, reflection, etc.) of my name. I posted some examples on this blog. Möbius bands have the surprising properties that they are one-sided and have only one edge. This inspired me to write the words MÖBIUS…

Rubik’s Cube Tri-Hexaflexagon

A few days ago I came across an animated gif of a Rubik’s Cube hexaflexagon kaleidocycle that somone made. I posted it on Twitter. I've got to make a Rubik's Cube hexaflexagon! pic.twitter.com/ZxdSkby1iv — Dave Richeson (@divbyzero) August 21, 2018 It got a lot of interest, so I thought I’d try making my own. Here’s the final…

I Heart Cardioids

Roll a circle around another circle of the same radius. A marked point on the first circle traces a curve called a cardioid. (In the figure below we rolled the orange circle around the red circle to draw the green cardioid.) This beautiful heart-shaped curve shows up in some of the most unexpected places. Grab…

Finite differences of polynomials

It is interesting watching my kids go through the school math curriculum. Since I’m a math professor, one would think that I would know all of the school-aged math. While that is mostly true, sometimes the teachers and textbooks use unfamiliar terminology for familiar mathematical ideas. (“Oh, ____ is just ___,” I’ve said multiple times.)…

Who was Pierre Wantzel? A translation crowdsourcing project

I would like to try an experiment. If you like math, history, and can read French—read on! I am interested in the so-called “problems of antiquity”—squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons. If you look in reference books, we now know that three of the four problems (all but…

The Division Symbol Goes Viral

A few days ago a Twitter user with the handle @Advil posted the following tweet: i just found out that the division symbol (÷) is just a blank fraction with dots replacing the numerator and denominator. oh my god. — abdul 🚀 (@Advil) September 11, 2017 As you can see, the tweet was widely “liked”…

Advice for College Students

I’m teaching a first-year seminar this semester. This isn’t a math course, although there will be some math in it. The title of my course is “Decisions, Decisions! Why We Make Bad Ones and How to Make Better Ones.” We will be using four texts, Writing Analytically, How to Think About Weird Things, Weapons of…