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…

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…

Circular reasoning: who first proved that C/d is a constant?

I just uploaded an article “Circular reasoning: who first proved that  is a constant?” to the arXiv. The abstract is below. It is on a topic that I’ve been thinking about and reading about off-and-on for the last year and a half. I’d be happy to hear people’s thoughts, reactions, and impressions. Abstract. We answer the…

Plato’s approximation of pi?

Today I came across an assertion that Plato used as an approximation of . Indeed, it is not a bad approximation: (although it is not within Archimedes’s bounds: ). Not only had I not seen this approximation before, I had not heard of any value of attributed to Plato. I investigated a little further and…

A pyramidologist’s value for pi

Recently I came across two theories about the design of Great Pyramid of Giza. If we construct a circle with the altitude of the pyramid as its radius, then the circumference of the circle is equal to the perimeter of the base of the pyramid. Said another way, if we build a hemisphere with the same…

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…

Irving Kaplansky’s “A Song about Pi”

Perhaps I should wait until mid-March to post this, but oh, well. Irving “Kap” Kaplansky (1917–2006), the mathematician and former head of MSRI, was also a pianist and songwriter. In 1973 he brought all of these interests together to pen a song called “A Song about Pi.” The tune is was inspired by the digits of…

An amazing paragraph from Euler’s Introductio

Today I’d like to share an amazing paragraph from Euler’s 1748 textbook Introductio in analysin infinitorum (Introduction to analysis of the infinite). This two–volume book is what Carl Boyer calls “The foremost textbook of modern times,” edging out, for example, Descartes’s Géométrie, Gauss’ Disquisitiones, and Newton’s Principia. Boyer writes that “Euler accomplished for analysis what Euclid…

Top ten transcendental numbers

Everyone loves a top ten list, and what’s better than a top ten list about numbers? (I’m reminded of David Letterman’s top ten numbers between one and ten from September 22, 1989.) So, on the heels of my previous posts about algebraic and transcendental numbers (here and here), here’s my list of the… Top Ten…

Trigonometric functions and rational multiples of pi

Recall that a real number is algebraic if it is the root of a polynomial with integer coefficients and that it is transcendental otherwise. For example is algebraic because it is a root of the polynomial , but is transcendental because it is not the root of any such equation. (On a recent blog post…

Happy tau day!

It is 6/28! Happy tau day—a day that is twice as fun a pi day! Let’s celebrate our new favorite mathematical constant:  Remember, as Bob Palais told us, is wrong! I’m a fan of tau.

More about the neat calculator trick

Yesterday I wrote about a neat calculator trick that I had just learned. We saw that if the calculator was set to degree mode, then times a high enough power of 10 is approximately . A commenter named Robert suggested looking at the difference between this approximation for and itself. He remarked that the error…