Every teacher has had the experience of writing a seemingly straightforward exam question only to realize when grading the exam that some of the students misunderstood the intent of the question. Oh, to be able to turn back the clock and to rewrite the question! That happened to me this semester, and the variety of answers…

## 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…

## Math Crafts: Salt Designs, Newton Snowflakes, Fractal Christmas Trees, Paper Pentagons, and Flip Books

I have had a crafty late fall and early winter. I’ve been good about posting my crafts on Twitter, but not so good at blogging about them. So, I’ve collected them all and will share them all here in one blog post. The Geometry of Salt I came across this neat pdf by Troy Jones…

## 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”…