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…

Sugihara’s Circle/Square Optical Illusion

[Update: Check out my second post in which I provide a template so you can make your own Sugihara circle/square object out of paper.] Kokichi Sugihara created a video called Ambiguous Optical Illusion: Rectangles and Circles. In it he shows a variety of 3-dimensional objects that look like one shape when viewed from the front but look…

Measuring Tapes for Circles and Spheres

I’d like to thank Matt Parker for introducing me to diameter tapes (or D-tapes). These are measuring tapes used by foresters to measure the diameters of trees. The forester wraps the measuring tape around a tree as if measuring the circumference, but the scale on the tape is adjusted so that the measurement gives the diameter…

Zip-Apart Möbius Bands

I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what…

A Trig-free Proof of Crockett Johnson’s Theorem

I recently wrote a post about Crockett Johnson’s neusis construction of a regular heptagon. Johnson’s proof that the construction was correct required heavy trigonometry. I asked if there was a geometric proof that didn’t use trigonometry. My friend Dan Lawson came to the rescue—he posted the following lovely proof on Twitter. Thanks Dan!

A Geometry Theorem Looking for a Geometric Proof

[Update: Dan Lawson has proved the theorem without trigonometry. Thanks, Dan!] I spent a good chunk of last week reading about David Johnson Leisk (1906–1975), who is better known by his nom-de-plum Crockett Johnson. Johnson is most well known as the author of Harold and the Purple Crayon, a children’s book from 1955, and its sequels. Johnson was also the…

A Trisectrix from a Carpenter’s Square

UPDATE: The article is now published. Read it in Mathematics Magazine. Yesterday I posted an article to the arXiv, “A Trisectrix from a Carpenter’s Square.” Abstract: In 1928 Henry Scudder described how to use a carpenter’s square to trisect an angle. We use the ideas behind Scudder’s technique to define a trisectrix—a curve that can be…