Mathematical Themes
Explore the major mathematical themes: geometry, algebra, analysis, arithmetic, logic, and many other fascinating fields.
The Euclidean revolution
Some profound results, such as Pythagoras' theorem, predate any awareness of mathematics as a science. Rather, once theorized and proved, these results gave rise to this science.
A century of conjectures
Mathematics has always advanced through the curiosity of people seeking to answer new questions and pose new problems. Once solved, these give rise to others, creating an endless chain of advances in knowledge.
Toward open science | Tangente
Mathematical practice is undergoing profound change. Researchers are organizing, established ways of working are being questioned, and bringing knowledge to the general public has become a key concern. A new model—"open science"—is emerging.
The beginnings of collaborative research | Tangente
Until very recently, mathematicians worked and published alone. Now, thanks in part to the development of modern communication tools, it is not uncommon to find research papers co-authored by more than four people. At Tim Gowers's instigation, several websites have enabled collaborative research.
A spiral of digits | Tangente
An entertaining blend of arithmetic and graphics lets us display the successive terms of an arithmetic sequence visually. This calls for some explanation—and decoding!
The thousand-and-first tale | Tangente
Mise en abyme is a classic literary device that masterfully evokes an iterative process. One example is Olivier Salon's 2009 text Les Gens de légende, reproduced by kind permission of Éditions du Castor Astral.
Musical iterations | Tangente
The first iterative process in music gave rise to the Pythagorean scale. Since then, recursive processes have been widely used in musical composition, from Johann Sebastian Bach to contemporary music. Their use is limited only by composers’ imaginations!
Loops in programming | Tangente
The various types of loops are fundamental programming constructs. Although using them often comes naturally, they nevertheless raise a number of issues, such as whether the program will ever stop. Recursion offers useful solutions.
The online encyclopedia of integer sequences | Tangente
Founded in 1995 by British-American Neil Sloane, the OEIS (On-Line Encyclopedia of Integer Sequences) is a collaborative website where anyone can add a new way of generating a sequence of numbers, together with its known properties and the questions it raises. Nearly 400,000 sequences can be browsed there!
The Pythagorean scale
Pythagoras is often credited with the idea that everything is number. Less well known is that his view of mathematics was closely bound up with musical harmony—not only the "harmony of the spheres," but the actual construction of a musical scale using numbers.
Continued fractions,
Real numbers can be written without making an arbitrary choice of base. Euclid's algorithm provides a method that extends to all real numbers and leads to the notion of a continued fraction. In this notation, the golden ratio becomes the simplest irrational number to write!
Computational geometry: key challenges | Tangente
It is easy to overlook, but many geometric problems can be solved by iterative methods or, more broadly, algorithms. These questions are the subject of computational geometry! This is a vibrant, highly active field in which many seemingly elementary questions still await solutions.
The remarkably rich Prouhet–Thue–Morse sequence
The Morse word, or Prouhet–Thue–Morse sequence, is an easy-to-construct mathematical object with plenty of surprises in store. Join us in exploring a combinatorial sequence that would certainly deserve to be every bit as famous as Fibonacci's!
News from the Collatz conjecture
Syracuse... For some, the name evokes a classic song performed by the late Henri Salvador (1917–2008); for mathematics enthusiasts, however, it refers to the Collatz conjecture, whose statement is very simple but which remains unproved to this day. A recent result lends further support to the conjecture.
Fractals: the aesthetics of iteration
Repeating a geometric transformation at different scales produces fascinating figures whose aesthetic appeal is far from their only attraction. Let's (re)discover the most iconic fractals!
Iterative methods for solving equations
Sequences, including recursively defined sequences, arise in many areas of mathematics. This is true of numerical methods for solving equations, which are in fact iterative methods—hence the presence of sequences.
Heron's algorithm
Heron of Alexandria is known for a famous formula that gives the area of a triangle without requiring its height. He also devised highly sophisticated mechanisms and an extraordinarily efficient recursive method for approximating the square root of a positive number.
Blaise Pascal takes on induction
Although specialists still debate the origins of proof by induction, they often agree that Blaise Pascal's Traité du triangle arithmétique (Treatise on the Arithmetical Triangle) was the first work to set it out explicitly. A guided tour of a landmark in the history of proof.
Linear recurrences and an epidemic of mathematical talent
Recurrence sequences are a common feature of games and problems. They offer a good opportunity to discuss the linear and affine recurrences that often arise in them.
The many faces of mathematical induction
Mathematical induction is both a classic and an indispensable tool, and its principle is easily stated.