Folders

Curves have always invaded everyday life. First lines and circles, of course, but also conics, spirals, helices, or others inspired by the line of a building, the layout of a track or the shape of a rope. Yet it is difficult to propose rigorous definitions. We had to wait for the birth of analysis to have a tool to study them. Arising from parametric or implicit equations, expressed in Cartesian coordinates or polar coordinates, they invariably fascinate scientists as well as lovers of beautiful geometry. Welcome to the land of curves!

Because we live in a three-dimensional universe, representing it has become a technological challenge. All the more so since volume is associated with the representation of motion. On our screens, the development of computer-aided design tools originally met the needs of an industry seeking to move beyond mockup prototyping, which was too costly in terms of time and resources. The volume illusion is achieved by studying the specifics of our visual system, which works from light signals collected by our two separated eyes. Adapted screens or glasses are needed to reproduce it, even if some think that one day this illusion may be obtained from a flat image. But 3D is not limited to the volume illusion. It is physically recreated with the rise of 3D printers, whose process, working layer by layer, or even line by line, is reminiscent of that of scanners.

China is far from being the only East Asian civilization to have achieved a high degree of mathematical depth. Japan and Korea drew from it to develop their own knowledge and practices. In Japan, the wasan of the Edo period is far from a pale copy of Chinese models: among other things, it features the intriguing practice of sangaku, which blends geometry, art and spirituality. Subsequently, the yōsan of the Meiji era notably addressed questions of method as well as linguistics in order to absorb English-language mathematics. As for Korea, it had mastered algebraic elimination methods far ahead of those in Europe, resulting in particular in magic squares that Euler rediscovered much later.

A mathematician, Nobel Prize in Physics! The 2020 Nobel Prize, awarded to Sir Roger Penrose, Andrea Ghez and Reinhard Genzel, is an opportunity to look back on the beautiful story of an exceptional scholar, passionate and enthusiastic, whose work in topology helped clarify our understanding of black holes. His training in pure mathematics concluded with a thesis in algebraic geometry. His ingenious discovery of impossible figures and aperiodic tilings made him known to the general public. The establishment of the first theorem on singularities in general relativity led to the justification of the existence of black holes. Sir Roger Penrose, generous with his time and energy, delivers in this issue of Tangente an exclusive interview.

It is part of the basics of mathematical analysis. The theorem (or lemma) of Rolle states that a differentiable function that takes two equal values has at least one 'peak' where its derivative is zero. But who was Michel Rolle? Did you know that he was a fierce opponent of differential calculus? This led to a memorable dispute with Pierre Varignon. With its extension the mean value theorem, Rolle's theorem admits multiple applications: in pure mathematics such as certain methods for solving equations, but also in various fields such as finance for performing certain interest calculations.

The Japanese votive tablets found at the entrance of certain temples have long interested mathematicians because they sometimes contain elegant geometric puzzles, where circles play an important role. Their resolution is easy when it requires no particular result. More often, one must appeal to classical theorems, such as the Pythagorean theorem. Finally, in some rare cases, the problem posed has resisted for centuries… One cannot imagine it: the last puzzle was only solved in 2020! Welcome to the world of sangaku.

Self-reference is at the source of well-known paradoxes, but also of a profound crisis in the foundations of logic and mathematics. At the heart of the famous Gödel incompleteness theorem, one can find a variant of the sentence "This proposition is false". Beyond thorny philosophical problems, self-reference allows us to play with infinity, both in art and in mathematics. Thus, certain sequences of numbers have symmetries that could never have been imagined without reflecting on the rich notion of self-similarity, which reveals, to our greatest pleasure, "self-descriptive" and even "fractal" sequences. Readers will discover the continuation of this ambitious two-part feature in the next issue (Tangente 192).

Self-reference is at the root of well-known paradoxes, but also of a profound crisis in the foundations of logic and mathematics. At the heart of the famous incompleteness theorem of Gödel, one can find a variant of the sentence "This proposition is false". Beyond thorny philosophical problems, self-reference allows us to play with infinity, both in art and in mathematics. Thus, certain sequences of numbers possess symmetries that one would never have been able to imagine without having reflected on the rich notion of self-similarity, which reveals, for our greatest pleasure, "self-descriptive" and even "fractal" sequences. Readers will discover the continuation of this ambitious two-part feature in the next issue (Tangente 192).

Beyond simply describing reality, statistics can also help, by modeling a situation using appropriate probabilistic formulas, to predict its plausible evolutions in the future. 'Mixture models' are a complex example that makes it possible to identify different classes within a population and... predict that of a new individual to, for example, offer them content or products that might interest them on provider platforms. In the field of health, forecasting models make it possible to improve patient care to the point of imagining personalized medicine. In sports, the analyses lead to dosing the quantity of training as well as possible that optimizes individual performance, and, in the collective context, progress goes as far as predicting the winner of the next FIFA World Cup from past performances!

A set is a collection of objects – the elements – between which there can exist all kinds of relations. At the elementary level, we can visualize sets using 'potatoids' connected by arrows to represent the relations that link them, but this naive representation quickly reaches its limits, in particular when the sets are infinite. Set theory mainly provides a framework for the rigorous formalization of all areas of mathematics and leads to mind-blowing proofs: more than a simple theory, it is a new approach to mathematics.