How many squares does it take to express every whole number as the sum of squares? We saw that two...
Continue reading...medium
Koch Snowflake
Snowflakes are amazing creations of nature. They seem to have intricate detail no matter how closely you look at them....
Continue reading...Spherical Pythagorean Theorem
Did you know there is a version of the Pythagorean Theorem for right triangles on spheres? First, let’s define precisely what we...
Continue reading...Reuleaux Wheel
A Reuleaux Triangle is a plump triangle with rounded edges, formed in the following way: take the three points at the corners...
Continue reading...Arrow’s Impossibility Theorem
Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...
Continue reading...Hairy Ball Theorem
Another fun theorem from topology is the Hairy Ball Theorem. It states that given a ball with hairs all over it, it...
Continue reading...Products of Sums of Two Squares
Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the...
Continue reading...Thinned-Out Harmonic Series
You’re probably already aware that the harmonic series, which is the sum of the reciprocals of all natural numbers, diverges. In...
Continue reading...One, Two, Three, Pi
Here’s an interesting formula: Arctan(1) + Arctan(2) + Arctan(3) = Pi. (Everything’s in radians, of course). Presentation Suggestions:Challenge students to prove...
Continue reading...Kakeya Needle Problem
What is the smallest-area convex set in the plane inside which a needle (unit straight line segment) can be reversed...
Continue reading...