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…

Good activity for an Introduction to Proofs class

I just read this post at Futility Closet. (Spoiler: Don’t click the link unless you want to know the punchline.) Perhaps the result is well known, but I hadn’t seen it before. The post made me think of a neat project for an “Introduction to proofs” class. I’ll have to save it for the next…

Editing a Very Poorly Written Proof

I’m teaching Discrete Math this semester. Discrete Math is our college’s “introduction to proofs” class. We spend a lot of time talking about and practicing proofwriting. In earlier blog posts I shared my “Nuts and Bolts of Writing Mathematics” and an “editing checklist” that I give to them. Yesterday, I gave them an example of…

Tangent lines to the sine function with rational slope

Today I was wondering the following thing (I won’t bore you with how I ended up with this question): Are there any rational values of for which the line is tangent to the graph of Clearly the answer is yes: But my gut feeling was that this was the only such After some head scratching,…

Gabriel’s paper horn

I just returned from the eleventh Gathering for Gardner. One of the many special things about this unusual conference is that the attendees are strongly encouraged to participate in the “gift exchange.” We were each asked to bring a physical exchange item (one for each of the 350 conference-goers) or to submit a written contribution….

Undergraduate Math Bloggers

I was interested seeing how undergraduate math students used blogs (and related platforms, like Tumblr). So I posted a call on Google+ and Twitter: I'm looking for mathematics blogs written by undergraduate students. Any recommendations? I'll retweet/repost them as they come in. — Dave Richeson (@divbyzero) February 17, 2014 I received quite a few links. I’m looking…

2013: the year of pi

A couple days ago I saw this tweet. This is the year of pi. Arctan2 + Arctan1 + Arctan0 + Arctan3 = pi. #pi @centerofmath @MathJesus1 @CutTheKnotMath @maanow @mathematicsprof — John Molokach (@mathemusician_) September 7, 2013 Pretty cool! Let’s see why Two terms are easy to deal with: and But why is One way to…