A crucial link between mathematics and music is nicely illustrated by the Tonnetz, a geometric diagram representing the harmonic relationships between the notes of the musical scale. An early version of the Tonnetz appeared in Leonhard Euler’s book Tentamen novae theoriae musicae (A new theory of music), published in 1739 [TM274 or search for “thatsmaths” at irishtimes.com]. A modern … Continue reading Music and Maths are Inextricably Intertwined.
Tag: Euler
Euler’s Identity: the Most Beautiful Equation in Mathematics
A recent entry in the visitor’s book of the James Joyce Tower & Museum in Sandycove, Dublin perplexed the Friends of Joyce’s Tower, the volunteers who run the museum. The entry, reproduced as the boxed equation in the photograph, seemed as impenetrable as a passage of Finnegans Wake and quite beyond decryption. When invited to … Continue reading Euler’s Identity: the Most Beautiful Equation in Mathematics
The Golden Key to Riemann’s Hypothesis
The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that … Continue reading The Golden Key to Riemann’s Hypothesis
The Dual in the Crown
The beginning of topology is often traced to Euler's solution of a puzzle, the Bridges of Königsberg. The problem posed was to follow a path through the Prussian city that crossed all seven bridges exactly once. Euler proved that the problem has no solution. He drastically simplifying it by replacing the geographical context by a … Continue reading The Dual in the Crown
The Arithmetic Triangle is Analytical too
Pascal's triangle is one of the most famous of all mathematical diagrams. It is simple to construct and rich in mathematical patterns. There is always a chance of finding something never seen before, and the discovery of new patterns is very satisfying. Not too long ago, Harlan Brothers found Euler's number $latex {e}&fg=000000$ in the … Continue reading The Arithmetic Triangle is Analytical too
Letters to a German Princess: Euler’s Blockbuster Lives On
The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his Opera Omnia began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of … Continue reading Letters to a German Princess: Euler’s Blockbuster Lives On
Euler’s Journey to Saint Petersburg
It all began with an invitation to Leonhard Euler to accept a chair of mathematics at the new Imperial Academy of Science in the city founded by Peter the Great. Euler’s journey from Basel to Saint Petersburg was a highly influential factor for the development of the mathematical sciences. The journey is described in detail … Continue reading Euler’s Journey to Saint Petersburg
Number Partitions: Euler’s Astonishing Insight
In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term partitio numerorum. Many of Euler's results in number theory involved divergent … Continue reading Number Partitions: Euler’s Astonishing Insight
Goldbach’s Conjecture: if it’s Unprovable, it must be True
The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of … Continue reading Goldbach’s Conjecture: if it’s Unprovable, it must be True
Euler’s Product: the Golden Key
The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general zeta function, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics. … Continue reading Euler’s Product: the Golden Key
Euler: a mathematician without equal and an overall nice guy
Mathematicians are an odd bunch. Isaac Newton was decidedly unpleasant, secretive and resentful while Carl Friedrich Gauss, according to several biographies, was cold and austere, more likely to criticize than to praise. It is frequently claimed that a disproportionate number of mathematicians exhibit signs of autism and have significant difficulties with social interaction and everyday … Continue reading Euler: a mathematician without equal and an overall nice guy
The Basel Problem: Euler’s Bravura Performance
The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to $latex {\ln 2}&fg=000000$. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers, $latex \displaystyle … Continue reading The Basel Problem: Euler’s Bravura Performance
Euler’s “Degree of Agreeableness” for Musical Chords
The links between music and mathematics stretch back to Pythagoras and many leading mathematicians have studied the theory of music. Music and mathematics were pillars of the Quadrivium, the four-fold way that formed the basis of higher education for thousands of years. Music was a central theme for Johannes Kepler in his Harmonices Mundi – … Continue reading Euler’s “Degree of Agreeableness” for Musical Chords
Euler and the Fountains of Sanssouci
When Frederick the Great was crowned King of Prussia in 1740 he immediately revived the Berlin Academy of Sciences and invited scholars from throughout Europe to Berlin. The most luminous of these was Leonhard Euler, who arrived at the academy in 1741. Euler was an outstanding genius, brilliant in both mathematics and physics. Yet, a … Continue reading Euler and the Fountains of Sanssouci
Brun’s Constant and the Pentium Bug
Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler's conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges: $latex \displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty &fg=000000$ Obviously, this could not happen if there were … Continue reading Brun’s Constant and the Pentium Bug
The Bridges of Paris
Leonhard Euler considered a problem known as The Seven Bridges of Königsberg. It involves a walk around the city now known as Kaliningrad, in the Russian exclave between Poland and Lithuania. Since Kaliningrad is out of the way for most of us, let's have a look closer to home, at the bridges of Paris. [TM073: … Continue reading The Bridges of Paris
Clothoids Drive Us Round the Bend
The article in this week’s That’s Maths column in the Irish Times ( TM043 ) is about the mathematical curves called clothoids, used in the design of motorways. * * * Next time you travel on a motorway, take heed of the graceful curves and elegant dips and crests of the road. Every twist and … Continue reading Clothoids Drive Us Round the Bend
Experiment and Proof
Many mathematicians spend their time proving results. The (very old) joke is that they are machines for turning coffee into theorems. A theorem is a statement that has been shown, by a sequence of irrefutable steps, to follow logically from a set of fundamental assumptions known as axioms. These axioms themselves may be self-evident, or … Continue reading Experiment and Proof
Euler’s Gem
This week, That’s Maths in The Irish Times ( TM032 ) is about Euler's Polyhedron Formula and its consequences. Euler's Polyhedron Formula The highlight of the thirteenth and final book of Euclid's Elements was the proof that there are just five “Platonic solids”. Recall that a regular polygon is a plane figure with all sides … Continue reading Euler’s Gem
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