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 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.

# Examples

# A Beautiful Analytic Combination

Recently, I was looking for good motivating examples for complex analysis in several variables. There was already a short discussion of this question at MathOverflow. Some further searching led me to the book *Analytic Combinatorics in Several Variables* by Robin Pemantle and Mark C. Wilson. What is this all about and why did I fall in love immediately?

# 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 , 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.

# Permanent Determinant

The determinant and the permanent of a matrix are central characters in an endeavour to bring the powerful weapons of modern geometry to a battle in the epic war of computer science: the P vs. NP problem.

JM Landsberg has recently written a wonderful introduction to geometric complexity theory which is how the corresponding research field is called. This has inspired me to borrow some of it and write about the permanent and the determinant of a matrix.

# Euler and the Kids

Being on parental leave, somewhat naturally, gave me more time to spend in the kids’ room. After a chance encounter with Mr Pythagoras there some time ago, it was now time for a scheduled meeting with Mr Euler. The configuration in the image to this post is known to young parents as an Oball. To mathematicians, it is known as the truncated icosahedron. An interesting fact is that because it is constructed entirely from pentagons and hexagons, it has to have exactly 12 pentagons. Let’s see how Mr Euler can convince us of this necessity. Continue reading

# Pythagoras and the Kids

Yesterday, when playing with the kids, I found Mr. Pythagoras unexpectedly in their room. For a fraction of a second, I wondered why the Lego fence would fit perfectly across. Then, I saw that I had produced an instance of the 3-4-5 triangle.

# Neighbourly Polyhedra

In the plane, it is relatively easy to find four convex polygons which pairwise share an edge and are otherwise disjoint. Can you find five polygons in the plane with these properties? How about polyhedra? How many polyhedra can you find such that any pair of them share a face and are otherwise disjoint?

For me, this is an example where my three-dimensional imagination fails utterly. If you have never thought about this before, I suggest to try finding as many such polyhedra as possible (why not start with seven?) before reading on.

# The Groups of the Tetrahedron, the Octahedron, and the Icosahedron

After discussing the dihedral group, it is time to post my images of how Klein introduces the symmetry groups of the tetrahedron, the octahedron and the icosahedron.

By duality, this also handles the case of the dodecahedron (it is the dual of the icosahedron) and that of the cube (it is the dual of the octahedron) and thus all the Platonic solids are covered.

# The Dihedral Group

Snow flakes are one of nature’s beauties which are easy to appreciate even for the more mathematically or technically minded. Kenneth G. Libbrecht produces wonderful photographs of them; some are available online.

The symmetry of snow flakes is described by the dihedral group. This is one of the first groups described in Felix Klein’s book I have advertised before. Here is my illustration of how Klein described this group geometrically.