This is the solution to a problem I posed in my previous blog post. Please read that post first. [Update: I posted a solution to this problem two days ago. Yesterday Brent Yorgey pointed out that my analysis of “type 2” triangles was not quite correct. (See his comment below.) So I went back to…
Author: Dave Richeson
Counting triangles on a tin ceiling
This morning a student popped in to my office and said he had a math question for me, and that it wasn’t related to a class he was taking. He said that his apartment has a tin ceiling with square tiles. The tiles have X’s through them, as illustrated below: He asked: “Is there a…
Inspiration is for amateurs
I found this fantastic quote by the artist Chuck Close. It was his advice for young artists. However, I think that if you replace artist by researcher,* the same advice applies. I would certainly pass this advice along to young mathematicians: just start working, the ideas will come. Conversely, if you don’t put in the…
Music is math: ten songs about mathematics
Just for fun, here are ten songs about mathematics. Am I missing any good ones? Post them in the comments. 1. Finite Simple Group (of Order 2) by the Klein Four Group (lyrics). This excellent song was written and performed by graduate students at Northwestern University, where I did my graduate work. I think it…
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…
Beautiful theorems about dynamical systems on the plane
I was reading through some papers written by my Ph.D. advisor (John Franks) from the early 1990’s and was reminded of a few beautiful results about the dynamics of planar homeomorphisms. So I thought I’d share them here. For those of you who are not familiar with the terminology, a planar homeomorphism is a bijective…
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…
The Edmonton Eulers
A few years ago I found an alteration of the logo of the Edmonton Oilers (a Canadian hockey team in the NHL) in which “Oilers” was replaced with “Eulers.” I printed it and hung it outside my office door. Now I can’t find the original, but you can see a scanned copy on the left. I…
Google Translate now knows Latin
Yesterday Bruce Petrie (a graduate student studying the history of mathematics) and I were discussing Google Translate. While it is no substitute for a human translator, it is pretty good and getting better. In particular, it is perfect if you need a quick, approximate translation of a language that you do no know or don’t…
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…
David Richeson: Division by Zero 