Folders

Paul Erdős solved many problems in all fields, but his primary passion is certainly number theory. He thus devoted many works to prime numbers. While most of his results, quite advanced, are difficult to popularize, some remain nevertheless accessible, at least in their formulation. This is the case, for example, in geometry, of problems of distances defined by a set of points, of the dissection of a square into squares of different sizes, or of the number of lines passing through given points.

Arab-Muslim mathematics hold particular importance for their heritage: the Latin translations that were made from them allowed the West to rediscover ancient texts but also and especially to benefit from the advances made by scholars from the lands of Islam. Sometimes, these Latin translations made it possible to preserve the core of certain Arabic texts whose original is today lost. However, one should not overlook the mathematical developments after what is often characterized as the "golden age" of Arabic science, which continue to be nourished by fascinating cultural exchanges.

The activity of a company is immersed in an environment that evolves regularly. It increasingly requires detailed studies around its technical or commercial operations as well as forecasts on the future of each of its activities. Mathematics is today called upon to better understand the nature of uncertainties that it is essential to take into account to anticipate the future. Everyone knew that finance, economics or computer science needed a wide range of high-performance mathematical tools, regularly expanded. Less well known is that many other sectors such as energy or even video games are demanding them.

Forgotten quadrilateral, little-known parallelogram, the rhombus is a figure almost ignored by geometry: its mathematical properties seem rather meager. Yet it retains a share of mystery, which this feature will help you unravel. First of all, the origin of the word "rhombus" remains enigmatic. Then, one may wonder whether every rhombus has an incircle, or is circumscribed by a circle (think about it for a moment, these are not such obvious questions!). Or whether one can construct polyhedra whose faces are rhombuses. Finally, just as there are "magic squares", there has recently existed a "magic rhombus"!

It's summer time, time for vacations and games, all sorts of games. • Video games? Math (vectors, matrices and even quaternions) is everywhere to manage the animation. • Card games? Shuffling well is not so simple, especially if the deck is infinite… • Program a mathematical game? Creativity is on display for all young talents who want to try their hand at it within the framework of the Bernard-Novelli prize. • Warming up your brain at the beach or on the banks of the Seine? The usual games and problems sections are always there, with the addition (mystery, of course) of two new pages of cryptarithms. Tangente wishes everyone a wonderful summer.

If the group is omnipresent in "pure and hard" mathematics, other domains have taken it up, aware of the advantages its presence brought: understanding of complex phenomena, new techniques, unification of ideas... not to mention the appeal that an abstract structure exerts on our thinking. In ethnology, Claude Lévi-Strauss used groups to model kinship relations in certain populations. In cryptology, the "clock arithmetic" is present for any encryption or decryption operation. Art is not left out, whether in music, literature, or painting. André Cadere's Round Wood Bars remain emblematic of the artistic representation of permutation groups.

In many mathematical fields, inequalities play an important role. This is the case for example in probability, with the Bienaymé‒Tchebychev inequality, or in all that relates to optimization. Many physical phenomena are also modeled by inequalities; thermodynamics gives us a striking example. In economic and social sciences, the study of inequalities is a central topic, whether they concern wealth, labor, housing, education…

Due to its properties, the tangent can be interpreted in various ways: line approximating a curve, trajectory of a light ray, geodesic of a particular space… Depending on the viewpoint adopted, the tangent under consideration then conveys a particular characteristic of the problem being studied. In physics, for example, we will use the focal properties of conics to explain a phenomenon. In economics, budget lines will help identify the achievable investment policies of a company. To every problem, its line, which very often turns out to be a tangent!

It is always a pleasure, in Tangente, to rediscover numbers, faithful companions of the mathematician's journey. We find them everywhere indeed: they are at the base of our numbering systems, we wonder about the distribution of prime numbers, we marvel at their arithmetic properties, as with the Chinese remainder theorem...

Do you think you know everything about conics? This feature aims to surprise you. Change your point of view – in the literal sense – and the familiar parabola or circle become unrecognizable. Abandon the 'as the crow flies' distance of the Euclidean plane for another metric, and conics transform into strange creatures you wouldn't have imagined. In the real world, architects and engineers have made conics their most faithful, and most discreet, allies. And in the sky, Kepler's three laws, the foundation on which Newton built universal gravitation, continue to govern the motion of celestial bodies. Conics, it seems, are everywhere.