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…

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…

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…

Using a kayak to measure the perimeter of a lake

I’m on vacation this week on a lake in northern Michigan (hold up your right hand, palm toward you, point at the first knuckle of your middle finger—that’s where I am). Yesterday I paddled around the perimeter of the lake in a kayak. On a whim I brought my GPS-enabled phone. My route is shown…

Puzzler: a squarable region from Leonardo da Vinci

It is famously impossible to square the circle. That is, given a circle, it is impossible, using only a compass and straightedge, to construct a square having the same area as the circle. I will let you read elsewhere about the exact rules behind compass and straightedge constructions. The punchline is that if you begin…

Angle trisection using origami

It is well known that it is impossible to trisect an arbitrary angle using only a compass and straightedge. However, as we will see in this post, it is possible to trisect an angle using origami. The technique shown here dates back to the 1970s and is due to Hisashi Abe. Assume, as in the…

The Japanese theorem for nonconvex polygons

A couple of years ago I wrote blog posts about two beautiful theorems from geometry: the so-called Japanese theorem and Carnot’s theorem. Today I finished a draft of a web article that looks at both of these theorems in more detail. It contains all that you could want—connections between these theorems, generalizations of them, and consequence of…

Albrecht Dürer’s ruler and compass constructions

Albrecht Dürer (1471–1528) is a famous Renaissance artist. Mathematicians probably know him best for his work Melencolia I which contains a magic square, a mysterious polyhedron, a compass, etc. Today I was reading his book Underweysung der Messung mit dem Zirckel und Richtscheyt (The Painter’s Manual: A manual of measurement of lines, areas, and solids…

Lincoln and squaring the circle

I’d heard a long time ago that Abraham Lincoln was a largely self-taught man and that he read Euclid’s Elements on his own. Right now I’m reading Doris Kearns Goodwin’s Team of Rivals: The Political Genius of Abraham Lincoln, and from it I learned that not only did he read Euclid, he spent some time…

Three geometric theorems

Just for fun I thought I’d share a few interesting geometric theorems that I came across recently. Morley’s miracle In 1899 Frank Morley, a professor at Haverford, discovered the following remarkable theorem. The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. I’ve made a Geogebra…

Circle squaring limerick

I found this nice limerick on Charles Petzold’s blog: Said the man about town, ‘I have a flair For squaring the circle, I swear.’ But he found that the strain Was too great for his brain, So he’s gone back to circling the square. Petzold has a scan of the title page of E. H….

An Euler line geogebra applet

Lately I’ve been thinking a lot about Euler and his many contributions to mathematics. So, just for fun I decided to make a Geogeba applet showing the “Euler Line.” In 1763 Euler proved that three different centers of a triangle—the centroid, the orthocenter, and the circumcenter—are collinear. This line is called the Euler line. The centroid…