Mathematical Themes
Explore the major mathematical themes: geometry, algebra, analysis, arithmetic, logic, and many other fascinating fields.
France at the forefront of music research | Tangente
It took many a political battle, but today France can pride itself on having world-class centers dedicated to music research.
Oumupo: The workshop of potential music | Tangente
In 1960, François Le Lionnais and Raymond Queneau founded a collective dedicated to renewing literary forms: Oulipo (the Workshop of Potential Literature). From the outset, Le Lionnais considered extending this approach to other forms of expression.
When mathematics enters musical composition | Tangente
The deep connections between mathematics and music in the process of composition grew richer during the twentieth century.
Atonal music: a rearguard debate | Tangente
In 2013, Karol Beffa, a composer and improvising pianist who held a chair at the Collège de France, organized a lecture. Drawing on examples, he set out to demonstrate the continued existence of the tonal—or neo-tonal—system and the failings of an atonal system that he considered dogmatic, inaccessible to the ear and obsolete.
Jean Bourgain, exceptional Fields medallist | Tangente
The prodigious Belgian mathematician Jean Bourgain (1954–2018) worked in many fields of mathematics, forging wholly unexpected links between seemingly separate areas. His outstanding results earned him numerous awards.
Instruments in the service of music
Marking time, giving music its rhythm by keeping the beat: these are the functions of an object that accompanies many musicians—the metronome. Originally mechanical, it later became electronic, no doubt to keep in tune and move with the times…
Acoustics and computer music: Helmholtz | Tangente
Thanks to the work of Joseph Sauveur and later Hermann von Helmholtz on sound phenomena, we can analyze a sound in terms of its pitch, intensity and timbre. This is the analysis used in computer music, which is so popular with contemporary composers.
Astor Piazzolla and tango nuevo:
The freedom championed by the rise of contemporary music led Astor Piazzolla to break free from the stifling conventions of classical tango, both harmonically and rhythmically, and create a new form. This evolution in musical structure can be analyzed.
A free ticket to the stars: Newton | Tangente
How far we have come since Newton's famous apple! Recently discovered remarkable properties of the gravitational field raise the prospect of space missions requiring almost no energy—not only to the Moon, but to other planets as well…
A century of mathematical music | Tangente
Since 1945, contemporary "art" music has become so diverse that any overview is now impossible. Yet technology and the introduction of some form of chance or randomness into composition remain ubiquitous.
An invariant under central projection
The need to model visual perception gave rise to a new geometry. Renaissance painters felt compelled to study it closely in order to depict depth. Lengths, angles: nothing seemed to be preserved, apart from a curious relation linking four collinear points…
A cross-ratio from another world
During the 19th century, the search for quantities invariant under a particular group of transformations became the main focus of the various branches of geometry. The cross-ratio, a fundamental invariant of projective geometry, also appears in non-Euclidean geometries and their models.
A geometric tool of unparalleled power
The harmonic range, like the more general notion of cross-ratio, has proved essential to geometric reasoning, particularly when dealing with cocyclicity—the property of points in the plane lying on the same circle—or pencils of lines.
Statistics and epidemiology: clusters | Tangente
An article in Science et pseudo-sciences explains the statistical methods used in epidemiological studies, using cases of "babies born without arms" as an example
Arithmetic: Were things really better before? | Tangente
The latest report from the Directorate of Evaluation, Forecasting and Performance warns of declining achievement among pupils at the end of CM2.
Pick up your pens: maths writing competition | Tangente
The publication of Daniel Bouix's article "Coloring the Tiles of the Plane" gives us an opportunity to remind readers that the 2019 Best Article Award competition is now under way. Entrants have until September 30 to submit their work.
Breakthrough in quasi-linear multiplication | Tangente
Multiplying two integers dates back several thousand years. In 2019, at last, the algorithm was improved!
First black hole photograph: Katie Bouman | Tangente
The first photograph of a black hole was produced using an algorithm tasked with assembling all the data collected by a network of telescopes.
Karen Uhlenbeck: first woman to win Abel Prize | Tangente
Following Ingrid Daubechies and Claire Voisin, another female mathematician has been honored this year.
The theorems of Menelaus and Ceva
Menelaus's and Ceva's theorems, two classics of plane geometry, are similar in form. This resemblance becomes clearer when the notion of cross-ratio is introduced.