Knowledge
In-depth articles on mathematical knowledge and theories
Links and graphs: a two-way connection | Tangente
What do links and planar graphs have in common? They are connected by a relationship that associates a graph with any given link and, conversely, provides a simple way to encode—and therefore draw—any link using a graph.
Identifying knots with Gauss codes | Tangente
A visual encoding introduced by Gauss to make knots easier to recognize and manipulate leads to surprising developments in combinatorics, algebra and topology. Armed with paper, pencil and a few pieces of string, let's explore this world.
Classifying knots: topology and invariants | Tangente
Knots can be combined to form new ones—or, conversely, simplified. We can borrow the vocabulary of number theory and classify them rather like the chemical elements. What varied and unexpected facets knot theory has!
École polytechnique’s maths special | Tangente
La Jaune & La Rouge, the monthly magazine of the École polytechnique alumni and graduates association (AX), devotes its February 2023 feature section to mathematics.
Charles V's letter finally deciphered | Tangente
An encrypted letter from Charles V (1500–1558), Holy Roman Emperor, addressed in 1547 to Jean de Saint-Mauris, his ambassador to the king of France, and housed in the autograph collection of the Bibliothèque Stanislas in Nancy (Meurthe-et-Moselle), has just been not merely read but deciphered. Long passages in the letter are encoded.
AI improves the general algorithm for matrix multiplication
Every science student knows how to multiply two matrices. So why call it an "open problem"?
Sédimath: an initiative for promoting mathematics | Tangente
A new mathematics outreach initiative has just been launched: Sédimath, an online seminar organized by Aurélien Alvarez (ÉNS-Lyon) and Élise Raphael (Université de Genève, Switzerland).
Grenoble math center: local authorities pull out | Tangente
La Grange des Maths is an association founded in 2015 to renovate an old barn in Varces (Isère) and turn it into a place where people can meet and share mathematics.
Waring's problem: 250 years of research | Tangente
Since the 18th century, a famous problem—Waring's conjecture—has challenged mathematicians. Every integer N is a sum of at most four squares, or of at most nine cubes. Likewise, given an integer n, can every N be expressed as a sum of at most g(n) nth powers, and if so, what is the smallest possible value of g(n)?
When Euler gets it wrong
Even the greatest mathematicians sometimes make mistakes. It is only human! It even happened to one of mathematics’ legendary figures, the great Leonhard Euler, in connection with sums of powers.
Assigning a value… to a divergent series!
Beyond the "classical" convergence of numerical series, many methods—some particularly powerful—can accelerate the convergence of a series or even assign a value to the "sum of a divergent series." Here are a few notable examples.
The triumphs and tribulations of summing a series
Every convergent series can be assigned a sum—but what about divergent series? Mathematical orthodoxy holds that they cannot be assigned a value. Yet Leibniz and Euler suggested a few possible approaches. Poisson, Frobenius and Borel later crossed that forbidden line.
Converging to a number: is there only one way?
Traditional mathematics provides a precise definition of convergence for a numerical series, explored in the main body of this article. But this should not rule out less conventional approaches, discussed in the box and in several articles in this special issue.
Infinite sums: a matter of convention
Equality between two numbers poses no difficulty when both are defined by "finitary"* methods. But as soon as infinity enters the picture—represented by ellipses in formulas—the door opens to all manner of paradoxes. (* Finitism is an approach to mathematics that considers only finite objects.)
Catalan's problem
In 1844, Franco-Belgian mathematician Eugène Catalan published his famous conjecture in Crelle's Journal.
Spinoza and mathematical order in philosophy | Tangente
The Dutch philosopher Baruch Spinoza drew inspiration from mathematicians’ methods of inquiry and exposition in his "search for truth." Otherwise, he maintained, we prove nothing, merely supporting preconceived theses with plausible arguments.
François Plantade: Abel, a mathematical genius | Tangente
For Tangente, François Plantade looks back at the life of Niels Henrik Abel, a 19th-century Norwegian mathematician who remains little known despite his major influence on the mathematics of his day. He died at 26, and his work only began to gain recognition toward the end of his life.
From Sissa to RSA
What use is there in raising numbers to powers, except for the fun of uncovering some arithmetical property of the natural numbers? Unexpectedly, this ancient computational art lies at the heart of modern cryptography and secure data transmission.
The fifth operation
Although the four operations—addition, subtraction, multiplication and division—are studied from elementary school onward, students must wait until their third year of secondary school to discover a fifth: raising a number to a power.
Mechanical secrets of the 1642 Pascaline | Tangente
Electronic calculators, now built into our phones, have made us forget the mechanical marvels that came before them. At just 19, Blaise Pascal developed the first calculating machine and, after a hard struggle to perfect his invention, captivated his contemporaries.