Beautiful Binomials

While leafing through the book Geometric Trilogy I – An Axiomatic Approach to Geometry, I came across two nice geometric depictions that are probably widely known but which I would love to have seen back in school. The first one is the geometric illustration of the algebraic fact that , the quadratic case of the binomial theorem. This can already be found in Book II of Euclid’s Elements. From this, it is easy to come up with a three dimensional version of the construction giving . Creating a good picture of this is somewhat tedious so I am very glad that I can use this opportunity to link to a wonderful blog where this and other wonderful mathematical illustrations and animations can be found: Hyrodium’s Graphical MathLand.

Let me close with a second figure inspired by the Geometric Trilogy books. It shows two different ways of (literally) seeing that the sum of the first odd numbers equals :