Discrete geometry, at the crossroads of disciplines

Discrete geometry is a field at the crossroads of several disciplines, such as geometry, arithmetic or combinatorics. To get an idea, let us replace the continuous Euclidean plane with a grid, made up of points with integer coordinates. The geometric objects we encounter there are, for example, polygons whose vertices lie on these points. Can we define a line, a circle? These questions arose with particular acuity from the 1950s onwards, when it became necessary to represent images on screens, discrete sets of pixels. Many problems in discrete geometry are still open and lead to exciting developments.

In science as well as in everyday life, we are led to encounter large numbers. The way to name them then depends, according to countries, on the choice of the scale used, long or short. The imagination of scientists, particularly mathematicians, being boundless, techniques for designating even larger numbers have been developed. This is the case, from Antiquity, with Archimedes' The Sand Reckoner. More recently, in the 20th century, John Conway and Donald Knuth developed new notations for numbers immensely larger than the number of elementary particles in the universe. And what is even more surprising, is that they are found in certain theories!