Experimental Mathematics Lab at DMATH in Luxembourg

The DMATH Experimental Mathematics Laboratory at the University of Luxembourg promotes experimental mathematics activities similar to those experiments in physics, chemistry or biology but in mathematics they can be carried out at an equally experimental level, often using computer methods. Applied to various types of mathematics (statistics, algebra, analysis, geometry) to visualize complex mathematical objects, use computers to work with examples that go beyond what is possible with pen and paper, find patterns in complex data, and much more. Creativity and imagination have no limits, because in the end, math is just fun.

Visualising systems of linear equations*

One can visualise sets of integral solutions of systems of linear equations (e.g. those representing magic squares or similar objects) with integral coefficients, and their reduction modulo an integer. For example, by colouring the solutions, where an entry in a solution (vector) which is 0 mod 2 could be represented by black square and 1 mod 2 by a white one. The aim of this project was to better understand systems of linear equations, whilst trying to produce beautiful images using rainbow spectrum, or also RGB (red-green-blue), CMY (cyan- magenta-yellow) systems, or Opacity, etc.
  • 2024, Visualising systems of linear equations, students: Ian BECK, Sevan CHRETIEN, Amelia GONZALEZ SAURA
* in collaboration with Gabor Wiese

Celebrating the 20th anniversary of our university
with Fourier series

One can create many two-dimensional "one-line drawings" with complex Fourier series. The aim of this project was to do this for some interesting logos, such as the Unilu university and the 'Roude Léiw' logo.
  • 2023, Celebrating the 20th anniversary of our university with Fourier series, students: Charles CHATTOU, Gil SCHROEDER

Visualising roots of algebraic numbers*

Visualising the distribution of roots of integral polynomials of degree $n$ with coefficients either $1$ or $-1$ (e.g. here for degree at most $24$) leads to see fascinating patterns. For instance, one can do a systematic study of the distribution of the roots, investigate similarity between the inner part of the picture and these fractal dragons, connected question of the proportion of irreducible, "zoom" into some of the interesting parts of the picture of zeros, etc.
  • 2023, Visualising roots of algebraic numbers II, students: Catia ALVES PIRES, Claudia MARICHAL
  • 2023, Visualising roots of algebraic numbers I, students: Salma BELMIR, Mikala EISEN, Melissa GENOUD
* in collaboration with Gabor Wiese

Lët'z box counting*

The goal was to illustrate the map of our country, Luxembourg, using box-counting method.
  • 2021, Lët'z box counting!, student: Kim DA CRUZ
* in collaboration with Lara Daw

Playing with Hadmard matrices

Have you ever asked yourself how the pictures from spacecrafts are send back to Earth and how the scientists can recover the original image free of any errors? The answear lies in the error correcting codes, which are one technique for building in the appropriate redundancy. In 1990, NASA used a matrix of the augmented Hadamard code, called the Reed–Muller code, for the Mariner 9 mission. Speaking of Hadamard matrices, there is still an open question, known as the Hadamard conjecture. Until now, no one was able to prove that Hadamard matrix of order $4n$ exists for every positive integer $n$. For example, recently studies showed that Hadamard matrices of order $764$ exist!
  • 2021, Behind the secrets of Hadamard matrices and their applications, students: Léa Mia MICARD, Eva RAGAZZINI, Lara SUYS

Solving polynomial equations over finite fields*

The goal was to implement, to explore and to illustrate various methods for solving polynomial equations in one variable, e.g. over the real numbers using Newton approximation, over finite fields using Berlekamp's algorithm, etc.
  • 2020, Solving polynomial equations, students: Kim DA CRUZ, Dylan MOTA, Clara POPESCU
* in collaboration with Gabor Wiese

Playing with knights (and queens)*

Among the famous chess inspired mathematical puzzles are those of the Knight's Tour and the Eight Queens. It is possible to make many variations of these puzzles on the usual board, but also on arbitrary chess boards, not necessarily square ones, of higher dimensions. The goal was to program these (and other) variations, illustrate them and make observations (for instance, on the (im)possibility for certain paths, the minimal length of paths, etc.).
  • 2020, Knights (and queens), students: Léa Mia MICARD, Eva RAGAZZINI, Lara SUYS
* in collaboration with Gabor Wiese

Magic objects of squares*

A magic square is a square of distinct (usually positive) integers such that the sum of each row, each column and each diagonal is the same. A magic square of squares is a magic square such that each of its entries is a square. It is an open problem to decide if a $3 \times 3$ magic square of squares exists. Our students tested and created interesting variations: bimagic squares, magic cubes, "congruence magic squares of squares" using modular arithmetic, magic stars, magic stairs, etc.
  • 2019, Magic squares of squares, students: Selma JUSOFOVIC, Manal KAAOUANE, Mirza MUHAREMOVIC, Gil MOES
  • 2020, Magic objects of squares in modular arithmetic, students: Sabrina CUNHA, Anne FISCH, Yannick VERBEELEN
  • 2020, Magic objects over the integers, students: Arjanita DINGU, Clara DUCHOSSOIS
* in collaboration with Gabor Wiese

Approximating integrals thanks to Monte-Carlo

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. The goal of this project was to vizualise and study several examples of situations where Monte Carlo estimation can be used.
  • 2018, Monte Carlo simulation, students: Philippe KARST, Bruni CARVALHO, Arno GEIMER

Exotic Delauny decomposition**

Given a set of points in the plane (resp. in Euclidean $3$-dimensional space, or in higher dimensions) one can define a "Delaunay triangulation" of their convex hull. More exotic types of Delaunay-like decompositions have been proved to exist, where the circles (resp. spheres) are replaced by special kinds of hyperboloids or paraboloids.
  • 2018, Triangulation de Delaunay, student: Guenda PALMIROTTA
** supervisor: Jean-Marc Schlenker