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”…

## 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…

## The Math Behind a Reflected Double Rainbow

My friend Albert Sarvis posted this amazing photo that he took in the Grand Tetons—a double rainbow reflecting off of the water! After I got over the amazing artistic qualities of the photo, I started wondering about the math behind it. I know that if you point your arm directly away from the sun (so…

## Card Table Proof of the Pythagorean Theorem

We own a standard card table that we leave tucked away in the basement until the kids want to have a lemonade stand on the front sidewalk or we need the extra table space for a large Thanksgiving dinner. It is the standard kind with legs that fold underneath it so it is easy to store….

## Seventeen Facts about 17

I have encountered the number 17 several times in the last few weeks—enough times that it caught my attention. So I challenged myself to write a list of seventeen interesting things about the number 17. I tried to be as mathematical as possible. I wasn’t able to get seventeen facts on my own, so I turned…

## A Geometric Proof of Brooks’s Trisection?

[UPDATE: we have a proof! I included it at the end of the blog post.] Yesterday I was looking at a few methods of angle trisection. For instance, I made this applet showing how to use the “cycloid of Ceva” to trisect an angle. (It is based on Archimedes’s neusis [marked straightedge] construction.) I also found…

## Two More Impossible Cylinders

Earlier this year I wrote a couple blog posts about reverse engineering Sugihara’s impossible cylinder illusion. I then wrote it up more formally, and it has appeared in Math Horizons (pdf). The example I gave on my blog and in the article was a cylinder that looked like a circular cylinder but like a square cylinder in the…

## Make a Sugihara Circle/Square Optical Illusion Out of Paper

Yesterday I explained the mathematics behind Sugihara’s Circle/Square Optical Illusion, which appears in this video. Today I created a printable template from which you can make your own version of Sugihara’s object. Click the following image to download the pdf. Making the shape and seeing the illusion is easy. Cut out the figure at the top…