# The Power of Involutions

Time and again, there are moments when mathematics just feels magical.
For me, one example for this is given by generating functions (and that is why they can be found on this blog).
Today, I want to talk about another such example: involutions. We will look at how they are used to prove in one sentence that primes of the form $p=4n+1$ can be written as a sum of squares, in the proof of the wonderful Lindström-Gessel-Viennot lemma, and in the proof of Euler’s pentagonal number theorem.

# 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 $(a+b)^2 = a^2+2ab+b^2$, 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 $(a+b)^3=a^3+3a^2b+3ab^2+b^3$. 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.