# 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.

# Lectures on the Icosahedron

Some weeks ago, I was looking for examples giving me a quick overview on how to control transparency in the raytracer POV-Ray. This took me to the website of David Dumas who has beautiful illustrations of limit sets. (A very good and accessible introduction to the beauty of the Kleinian groups behind this is given by the book “Indra’s Pearls” [1].) What took me as even more beautiful were the pages from old mathematics books he is using as a background.

# Believers in the Axiom of Choice

There was a time back at the university, when pretty much any question about mathematics I came up with led to the axiom of choice. This seemingly innocuous and at first sight quite reasonable axiom leads to pretty strange conclusions such as the Banach–Tarski paradox. Roughly speaking, it requires that for any collection of non-empty sets it should be possible to construct an element of the Cartesian product.

Here are two of the questions I had at the time:

# Galilean Moons

Galilean moons (image source)

From my childhood, I remember the Galilean moons as tiny spots in my small telescope.
Still this view was enough to stir interest and to excite my fantasy.

It is good to see that viewed close-up, they are even more spectacular.
All I really want to say with this is that the Wikipedia page on Galilean moons has spectacular images of them.

# Welcome

Every now and then, mathematics and the sciences allow us glimpses of a fascinating world. Here, I wish to share some of those glimpses which are particularly fascinating to me.

Stay tuned for my views on things I deem worth a look such as

Tilings,
Diatoms,
Virus capsides,