It is well known that we can make a cardioid by drawing straight lines inside a circle. Simply choose one point on the circle’s boundary to be the base point. Then, connect points on the circle to the points twice as far away from the base point (measured along the circumference). If there are n…
Tag: geometry
Tales of Impossibility: Now Published!
I’m very excited to announce that my new book, Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity (Princeton University Press, 2019), is now available! (OK. It was published about a month ago, but I am just now getting around to blogging about it.) Like my previous book, Euler’s Gem (Princeton University…
BraidTiles—A Mathematical Braid Puzzle
We can view braids mathematically as n strings hanging from a horizontal bar. Each piece of string runs downward and can cross neighboring strings. In the 1920s Emil Artin observed that braids of n strings form an algebraic group. To “multiply” two braids, we append the bottom of one braid with the top of another braid. The identity element in this group…
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…
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…
David Richeson: Division by Zero 