Knowledge
In-depth articles on mathematical knowledge and theories
Percentages, for better or worse
Since antiquity, data have been compared by scaling the total to 100. The introduction of Arabic numerals gave percentages an important role, offering a clear, quick view of a wealth of statistical data... provided, of course, that one understands how they work and can count!
The "happy ending" problem
In 1933, Hungarian mathematician Eszter Klein posed an apparently elementary geometry problem to a group of young colleagues. It would lead to the birth of a new field of mathematics, discrete geometry, and... a marriage.
Solar System stability: the KAM theorem | Tangente
Jürgen Moser belongs to the long line of mathematicians who have tackled the problem of the Solar System's stability. He is the "M" in the KAM theorem's trinity—Kolmogorov, Arnold and Moser—which links number theory and celestial mechanics.
Approximating a function and following its curve
Assuming that a function is a polynomial yields useful approximation formulas. A complicated function can thus be replaced by a polynomial, simplifying most calculations. More surprisingly, interpolation lies at the heart of a secret-sharing technique.
Sphere packing:
We live in extraordinary times! Not a month goes by without a new mathematical breakthrough. The latest? A young researcher has solved the problem of the optimal packing of identical spheres in dimensions 8 and 24.
Between algebra and analysis: an indispensable world
Polynomials belong to both algebra and analysis, which can lead to all kinds of confusion! This dual nature offers a simple way to explain the subtle differences between variables, unknowns and indeterminates.
Sandpile geometry: surfaces and skeleton | Tangente
There are a thousand ways to keep your mind active while soaking up the sun on the beach. Let's set aside those—not so foolish after all—that involve not tanning at all. Instead, let's borrow our young nephew's bucket, spade and sieve, and embark on the great geometric adventure of sandpiles!
Shadows revealed: mathematics and vision | Tangente
As summer begins and temperatures soar, we all seek out a little spot sheltered from the sun. Let us not forget that drawing shadows is where perspective begins! Seeing an object's shadow already means recognizing it to some extent—and therefore knowing how to draw it in 3D.
Fascinating prime numbers
A look at the latest theorems about prime numbers, which continue to inspire just as much as ever.
The inexhaustible prime number theorem
Prime numbers are an endless source of mathematical surprises: they are infinite in number yet rare, and attempts to count them bring transcendental functions into play, such as the Riemann zeta function, which at first glance seem far removed from arithmetic.
Solving equations: Rolle's theorem to the rescue
The mean value theorem guarantees that very concrete compounding problems do indeed have a solution, and that applying the standard governing equations does indeed lead to that solution. A dream for any economist!
Rolle: a cascade of theorems!
Rolle's theorem, like the celebrated mean value theorem, has been dropped from the secondary-school curriculum. Yet these two major results in analysis are of far more than historical importance. They can be used to solve a great many mathematical problems!
Heine and the rigorous study of continuous functions | Tangente
Eduard Heine is not among the best-known mathematicians, even though his name is attached to a theorem on the uniform continuity of continuous functions on a closed interval.
The scant remainder of Euclidean division
When we divide a by n using Euclidean division, we obtain a remainder. This remainder can take only a limited range of values: there are just n of them. This new perspective can simplify many calculations and cast a whole host of problems in a different light!
Drawing a line on a computer
What could be simpler than drawing a line between two points? All you need is a ruler and a pencil! But how do you do it on a computer screen? How does graphics software manage it? The task is to turn certain points on the screen black. But which ones?
The projective line: a fruitful new perspective
The line is the simplest geometric figure imaginable. The variety of situations encountered in geometry may seem more complicated, and yet… might there be a special perspective from which the plane could be understood as a line? That is precisely what projective geometry is about!
Straight lines in curved spaces: geodesics
If we restrict ourselves to a surface, the shortest path from one point to another is called a geodesic. This concept leads "straight" to non-Euclidean geometries and applications in navigation and cartography.
Tangents and asymptotes
From yesterday's "touching line" to today's tangent, from "vanishing quantities" to asymptotes, the story has been a long mathematical epic. Here are a few glorious episodes from this geometric quest to "approximate curves with straight lines."
Caustic envelopes
Lines whose direction varies continuously may reveal the curve to which they are all tangent. Such curves, enveloped by straight lines, appear in optics as caustics. Their properties give rise to geometric construction methods.
Harmonic pencils of lines
How can we express simply that several lines are concurrent? Despite appearances, pure geometry is not the best tool for the job! Introducing a coordinate system and a few equations may prove more useful.