# 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.

# How Algebra Sheds Light on Things

Egbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

# 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.

# How Hard is Your Maths?

In Brainiac: Science Abuse, there is a wonderful category called How Hard is Your Thing? (see [1]). Thaila Zucchi makes seemingly hard things undergo some of the ultimate stress tests: Can they stand the heat of thermite? The abrasion of an angle grinder? The impact of a ton of bricks?

There were several occasions in the history of mathematics when mathematicians had to answer to the question: how hard is your maths?

# The Ivory Tower

To me, mathematics is an ivory tower.
An ivory tower, vast, complex, and beautiful.
With its doors wide open.
At each level, helping hands are offered,