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We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be located by giving its Cartesian coordinates 
Continue reading ‘Space-Filling Curves, Part I: “I see it, but I don’t believe it”’
Poincare’s hyperbolic disk model.
Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.
Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under forty years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, is the crowning achievement of their careers [TM235 or search for “thatsmaths” at irishtimes.com].
Clockwise from top left: Maryna Viazovska, James Maynard, June Huh and Hugo Duminil-Copin. Image Credits: Mattero Fieni, Ryan Cowan, Lance Murphy.
The congress, which ran from 6th to 14th July, was originally to take place in St Petersburg. When events made that impossible, the action shifted to Helsinki and the conference presentations were moved online. The International Mathematical Union generously allowed participants to register at no cost.
Goldbach’s Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude.
The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to Leonhard Euler. The truth of the conjecture for all even numbers up to four million million million (
Continue reading ‘Goldbach’s Conjecture and Goldbach’s Variation’
Schematic diagram of 
Cardinals and Ordinals
The cardinal number of a set is an indicator of the size of the set. It depends only on the elements of the set. Sets with the same cardinal number — or cardinality — are said to be equinumerate or (with unfortunate terminology) to be the same size. For finite sets there are no problems. Two sets, each having the same number
Atmospheric motions are chaotic: a minute perturbation can lead to major changes in the subsequent evolution of the flow. How do we know this? There is just one atmosphere and, if we perturb it, we can never know how it might have evolved if left alone.
We know, from simple nonlinear models that exhibit chaos, that the flow is very sensitive to the starting conditions. We can run “identical twin” experiments, where the initial conditions for two runs are almost identical, and watch how the two solutions diverge. This — and an abundance of other evidence — leads us to the conclusion that the atmosphere behaves in a similar way.
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 
Continue reading ‘The Arithmetic Triangle is Analytical too’
In just three weeks the largest global mathematical get-together will be under way. The opening ceremony of the 2022 International Congress of Mathematicians (ICM) opens on Wednesday 6 July and continues for nine days. Prior to the ICM, the International Mathematical Union (IMU) will host its 19th General Assembly in Helsinki on 3–4 July [TM234 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘ICM 2022 — Plans Disrupted but not Derailed’
{Left: Swinging spring (three d.o.f.). Right: the Wilberforce spring (two d.o.f.).
The swinging spring, or elastic pendulum, exhibits some fascinating dynamics. The bob is free to swing like a spherical pendulum, but also to bounce up and down due to the stretching action of the spring. The behaviour of the swinging spring has been described in a previous post on this blog [Reference 1 below].
In some recent posts, here and here we discussed the extension of the concept of parity (Odd v. Even) from the integers to the rational numbers. We found that it is natural to consider three parity classes, determined by the parities of the numerator and denominator of a rational number 
odd and odd. even and odd. odd and even.or, in symbolic form,
Here, 
Fairy Lights on the Farey Tree. Parity types are coloured as follows: Even: Blue; Odd: Green; None: Red.
The rational numbers 
Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined [TM233 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Image Processing Emerges from the Shadows’
In last week’s post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found — even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal density in
We define an extension of parity from the integers to the rational numbers. Three parity classes are found — even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure.
Continue reading ‘Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes’
Henri Poincaré described a beautiful geometric model with some intriguing properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid’s Elements. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute zero on the boundary, and that lengths varied in proportion to the temperature.
Euclid flourished about fifty years after Aristotle and was certainly familiar with Aristotle’s Logic. Euclid’s organization of the work of earlier geometers was truly innovative. His results depended upon basic assumptions, called axioms and “common notions”. There are in total 23 definitions, five axioms and five common notions in The Elements. The axioms, or postulates, are specific assumptions that may be considered as self-evident, for example “the whole is greater than the part” [TM232 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘The Whole is Greater than the Part — Or is it?’
Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican City use euro coins so, from 2015, there have been 23 countries, each with its own national coin designs. For the €1 and €2 coins, there are 23 distinct national patterns; for the smaller denominations, there are many more. Thus, there is a wide variety of designs in circulation.
National designs of Finland, France, Germany, Ireland and Netherlands.
[This is a condensed version of an article [5] in Mathematics Today]
A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon’s Theorem states that `The area of a tangent sweep of a curve is equal to the area of its tangent cluster’. We shall illustrate how this theorem can help to solve a range of integration problems.
Continue reading ‘Mamikon’s Visual Calculus and Hamilton’s Hodograph’
Abraham Robinson (1918-1974) and his book, first published in 1966.
A few weeks ago, I wrote about Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two.
Continue reading ‘Infinitesimals: vanishingly small but not quite zero’
To introduce the problem in the title, we begin with a quotation from the Foreword, written by Branko Grünbaum, to the book by Alexander Soifer (2009): The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators:
If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors?
About 70 years ago it was shown that the least number of colours needed for such a colouring is one of 4, 5, 6 and 7. But which of these is the correct number? Despite efforts by many very clever people, some of whom had solved problems that appeared to be much harder, no advance has been made to narrow the gap
A by-election for the Seanad Éireann Dublin University constituency, arising from the election of Ivana Bacik to Dáil Éireann, is in progress. There are seventeen candidates, eight men and nine women. Examining the ballot paper, I immediately noticed an imbalance: the top three candidates, and seven of the top ten, are men. The last six candidates listed are all women. Is there a conspiracy, or could such a lopsided distribution be a matter of pure chance?
To avoid bias, the names on the ballot paper are always listed in alphabetical order. We may assume that the name of a randomly chosen candidate is equally likely to appear at any of the positions on the list; with 17 candidates, there about 6% chance for each of the 17 positions; the distribution for a single candidate is uniform. However, when several candidates are grouped, the distribution is more complicated [TM231 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘The Improbability Principle and the Seanad Election’
Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In the 1800s, Bolzano, Cauchy and Weierstrass developed the 
Continue reading ‘Hyperreals and Nonstandard Analysis’
Lewis Fry Richardson in 1931.
One hundred years ago, a remarkable book was published by Cambridge University Press. It was a commercial flop: although the print run was just 750 copies, it was still in print thirty years later. Yet, it held the key to forecasting the weather by scientific means. The book, Weather Prediction by Numerical Process, was written by Lewis Fry Richardson, a brilliant, eccentric mathematician. He described in detail how the mathematical equations that govern the evolution of the atmosphere could be solved by numerical means to deduce future weather conditions from a set of observations [TM230 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘A Prescient Vision of Modern Weather Forecasting’
A game of Wordle solved in 3 guesses (a birdie).
Hula hoops were all the rage in 1958. Yo-yos, popular before World War II, were relaunched in the 1960s. Rubik’s Cube, invented in 1974, quickly became a global craze. Sudoku, which had been around for years, was wildly popular when it started to appear in American and European newspapers in 2004.
Henri Poincaré
Where does new mathematics come from? The great French mathematician Henri Poincaré, a brilliant expositor of the scientific method, described how he grappled for months with an arcane problem in function theory. Exasperated by lack of progress, he went on vacation and forgot about the problem. But, as he was boarding a bus in Caen, the answer came to him in a flash. He was later able to return to his office and complete a proof of the result [TM229 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Sources and Scenes of Mathematical Inspiration’
Ecliptic plane [Wikimedia Commons].
In his scientific best-seller, A Brief History of Time, Stephen Hawking remarked that every equation he included would halve sales of the book, so he put only one in it, Einstein’s equation relating mass and energy, E = mc2. This cynical view is a disservice to science; we should realize that, far from being inimical, equations are our friends [TM228 or search for “thatsmaths” at irishtimes.com].
We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers.
The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set
The set ![{\mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{g \in \mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
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 letters [TM227 or search for “thatsmaths” at irishtimes.com].
For several years, starting in 1760, Euler wrote a series of letters to Friederike Charlotte, Princess of Brandenburg-Schwedt, a niece of Frederick the Great of Prussia. The collection of 234 letters was first published in French, the language of the nobility, as Lettres à une Princesse d’Allemagne. This remarkably successful popularisation of science appeared in many editions, in several languages, and was widely read. Subtitled “On various subjects in physics and philosophy”, the first two of three volumes were published in 1768 by the Imperial Academy of Sciences in St. Petersburg, with the support of the empress, Catherine II.
Continue reading ‘Letters to a German Princess: Euler’s Blockbuster Lives On’
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 in a full-length biography of Euler by Ronald Calinger (2016). The account below is heavily dependent on Calinger’s book.
Drawing based on a map of Europe from about 1740 (from Calinger, 2016, pg. 39). Euler’s route from Basel to Saint Petersburg is marked by the heavy dashed line.
What are mathematicians really like? What are the characteristics or traits of personality typical amongst them? Mathematicians are rarely the heroes of novels, so we have little to learn from literature. A few films have featured mathematicians, but most give little insight into the personalities of their subjects [TM226 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Some Characteristics of the Mathematical Psyche’
It is a simple matter to post a paper on arXiv.org claiming to prove Goldbach’s Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored.
Mathematicians establish their claims and convince their colleagues by submitting their work to peer-reviewed journals. The work is then critically scrutinized and evaluated by mathematicians familiar with the relevant field, and is either accepted for publication, sent back for correction or revision or flatly rejected.
Continue reading ‘De Branges’s Proof of the Bieberbach Conjecture’
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 series. He was courageous in manipulating these but had remarkable insight and, almost invariably, his findings, although not rigorously established, were valid.
Partitions
In number theory, a partition of a positive integer
Continue reading ‘Number Partitions: Euler’s Astonishing Insight’
Swiss postage stamp, issued in 1994 for the International Congress of Mathematicians in Zurich, featuring Jakob Bernoulli and illustrating his “golden theorem”.
Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli’s great work, Ars Conjectandi, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, and it was the first limit theorem in probability, and the first attempt to apply probability outside the realm of games of chance [TM225 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Bernoulli’s Golden Theorem and the Law of Large Numbers’
In pure set-theoretic terms, the set of even positive numbers is the same size, or cardinality, as the set of all natural numbers: both are infinite countable sets that can be put in one-to-one correspondence through the mapping 
every second number is even and, intuitively, we feel that there are half as many even numbers as natural numbers. In particular, our intuition tells us that if 
Continue reading ‘Set Density: are even numbers more numerous than odd ones?’
In addition to some beautiful photos and maps and descriptions of upland challenges in Ireland and abroad, the November issue of The Summit, the Mountain Views Quarterly Newsletter for hikers and hillwalkers, describes a method to find the length of a walk based on ideas originating with the French naturalist and mathematician George-Louis Leclerc, Comte de Buffon [TM224 or search for “thatsmaths” at irishtimes.com].
[Image from November issue of The Summit, the Mountain Views Quarterly Newsletter.]
An object is chiral if it differs from its mirror image. The favourite example is a hand: our right hands are reflections of our left ones. The two hands cannot be superimposed. The term chiral comes from 
According to Wikipedia, it was William Thomson, aka Lord Kelvin, who wrote:
“I call any geometrical figure, or group of points, ‘chiral‘, and say that it has chirality if its image in a plane mirror … cannot be brought to coincide with itself.”
A remarkable French natural philosopher and mathematician who lived in the early eighteenth century, Émilie Du Châtalet, is generally remembered for her translation of Isaac Newton’s Principia Mathematica, but her work was much more than a simple translation: she added an extensive commentary in which she included new developments in mechanics, the most important being her formulation of the principle of conservation of energy [TM223 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Émilie Du Châtelet and the Conservation of Energy’
The power set of the set {x,y,z}, containing all its subsets, has 2^3=8 elements. Image from Wikimedia Commons.
In 1891, Georg Cantor published a seminal paper, U”ber eine elementare Frage der Mannigfaltigkeitslehren — On an elementary question of the theory of manifolds — in which his “diagonal argument” first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater than that of the natural numbers. But his theorem is much more general, and it implies that the set of cardinals is without limit: there is no greatest order of infinity.
Continue reading ‘Cantor’s Theorem and the Unending Hierarchy of Infinities’
Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic assumptions axioms and the consequent results theorems. But can things be done the other way around, using theorems to prove axioms? This is a central question of reverse mathematics [TM222 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Topsy-turvy Maths: Proving Axioms from Theorems’
Let 
Open any mathematical journal and read the first sentence of a paper chosen at random. You will probably find something along the following lines: “Let X be a Banach space”. That is fine if you know what a Banach space is, but meaningless if you don’t.
Continue reading ‘How to Write a Convincing Mathematical Paper’
Edna St Vincent Millay’s sonnet “Euclid alone has looked on beauty bare” evokes the ethereal, otherworldly quality of mathematics. Scandalous behaviour is not usually associated with mathematicians, but they are human: pride, overblown ego and thirst for fame have led to skulduggery, plagiarism and even murder. Some of the more egregious scandals are reviewed here [TM221 or search for “thatsmaths” at irishtimes.com].
French postage stamp issued in 1984.
Spiral of Theodorus [image Wikimedia Commons].
The square-root spiral is attributed to Theodorus, a tutor of Plato. It comprises a sequence of right-angled triangles, placed edge to edge, all having a common point and having hypotenuse lengths equal to the roots of the natural numbers.
The spiral is built from right-angled triangles. At the centre is an isosceles triangle of unit side and hypotenuse 
Rene Descartes
There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and methods, there are also strong overlaps and interdependencies. Several attempts have been made to construct a grand unified theory that embraces the entire field of maths [TM220 or search for “thatsmaths” at irishtimes.com].
We are all familiar with Pascal’s Triangle, also known as the Arithmetic Triangle (AT). Each entry in the AT is the sum of the two closest entries in the row above it. The 
M
In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured with high accuracy, while distances to these bodies are much more difficult to find and can be determined only indirectly.
Continue reading ‘Embedding: Reconstructing Solutions from a Delay Map’
Image credit: NASA.
The climate is changing, and we need to know what changes to expect and how soon to expect them. Earth system models, which simulate all relevant components of the Earth system, are the primary means of anticipating future changes of our climate [TM219 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Earth System Models simulate the changing climate’
Abstract: Continuity is defined relative to a topology. For two distinct topological spaces 
Continue reading ‘The Signum Function may be Continuous’
On a cold December night in 1976, a group of mathematicians assembled in a room in Trinity College Dublin for the inaugural meeting of the Irish Mathematical Society (IMS). Most European countries already had such societies, several going back hundreds of years, and it was felt that the establishment of an Irish society to promote the subject, foster research and support teaching of mathematics was timely [TM218 or search for “thatsmaths” at irishtimes.com].
The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.
Rounding error and formula error as functions of step size 
is crucial: if is too large, the estimate of the derivative is poor, due to truncation error; if is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if is very small, they produce zero values.
Where it can be applied, complex step differentiation provides a stable and accurate method for computing 
Continue reading ‘Real Derivatives from Imaginary Increments’
![\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BZ%7D%5Bi%5D+%5Cequiv+%5C%7B+m+%2B+i+n+%3A+m%2C+n+%5Cin+%5Cmathbb%7BZ%7D+%5C%7D+%5C%2C.+&bg=ffffff&fg=000000&s=0&c=20201002)
ThatsMaths 