Mathematical Themes
Explore the major mathematical themes: geometry, algebra, analysis, arithmetic, logic, and many other fascinating fields.
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.
Surfaces… made of straight lines!
A plane is generated by straight lines, and one might think it is the only surface that can be constructed in this way. But that is not so! Surfaces generated by straight lines even have a name: ruled surfaces.
Miraculous collinearity theorems!
A ruler, a compass, a sharp pencil—and the adventure has already begun! It can come as quite a surprise when three points turn out to lie on the same line. Plane geometry abounds in wondrous collinearity theorems.
Lines that have a name (2)
Some lines have a name, and some mathematicians have their own line.
Euclid: an introduction to the straight line
Written more than two thousand years ago, Euclid's Elements is a foundational text of classical geometry. The concept of the straight line is central to it. Let's meet this scholar of ancient Greece and take a closer look at his view of the straight line.
These lines that bear a name (1)
Some lines carry a mathematician's name. They are generally lines associated with the triangle, linked to certain notable points.
The topological line
The topology of the real line underpins the concept of a limit and hence, among other things, those of continuity and differentiability. It deals with notions of proximity between points that arise from the order relation.
Equations of a line
What defines a line, and in what geometric setting? Each possible answer leads to a representation from which the corresponding equations follow.
Descartes and Cartesian coordinates | Tangente
To locate points in a plane, Cartesian coordinates are generally used. Descartes is credited with inventing this method, so much so that it bears his name. Wrongly?
Geometry or numbers? The number line
It is easy to picture numbers as points on an oriented line with an origin. Yet this construction must be carried out with some rigor—especially if we want to represent infinitesimal or infinite numbers!
Without straightedge, without compass: Mascheroni | Tangente
Every straightedge-and-compass construction can be carried out with compass alone. This rather extraordinary result comes as a surprise.
The number line: definitions versus intuition
In the 19th century, mathematicians sought rigorous definitions of the quantities that make up the continuum of the line. How did they rigorously characterize the real continuum?
Straight lines and optical illusions — geometry | Tangente
For a mathematician, a line is a point in motion. For an artist, however, it is what defines the outlines of things. Without lines, there could be no shapes…
Antoine Pevsner: homage to the straight line
In 1956, the sculptor Antoine Pevsner declared: "The most important feature of my work today […], the principle on which all my sculpture is now built, is the exclusive use of straight lines."
The semantics of droite | Tangente
A good droite is not necessarily mathematical: you find it in politics just as in boxing…
The definition of the straight line through the ages
You thought you knew what a straight line is? Read on, and you may well start to doubt it…
Mathematical vs astrological semantics | Tangente
Horoscope advice can seem sensible. Why is that? While advice in objective areas consists solely of tautologies, subjective advice draws on a multiplicity of meanings, allowing clients always to interpret it as they wish.
Astrological calculations: how consistent are they?
The notion of houses, which originated with the Babylonians, who mainly observed the movements of the stars in the sky as seen from Earth, is an early form of astrology.
How to calculate cube roots by hand | Tangente
Have you mastered the square root algorithm? Impress your friends with cube root extraction!
Roots in the complex plane
The idea of a root extends readily to the field of complex numbers. In fact, it is even more at home there than among the real numbers, since this extension gives it a geometric interpretation. Welcome to the fascinating mathematical world of cyclotomy!