Author: thatsmaths

Space-Filling Curves, Part II: Computing the Limit Function

The Approximating Functions It is simple to define a mapping from the unit interval $latex {I := [0,1]}&fg=000000$ into the unit square $latex {Q:=[0,1]\times[0,1]}&fg=000000$. Georg Cantor found a one-to-one map from $latex {I}&fg=000000$ onto $latex {Q}&fg=000000$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor's map was not continuous, but … Continue reading Space-Filling Curves, Part II: Computing the Limit Function →

Space-Filling Curves, Part I: “I see it, but I don’t believe it”

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 … Continue reading Space-Filling Curves, Part I: “I see it, but I don’t believe it” →

Poincare’s Square and Unbounded Gomoku

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 … Continue reading Poincare’s Square and Unbounded Gomoku →

Goldbach’s Conjecture and Goldbach’s Variation

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 … Continue reading Goldbach’s Conjecture and Goldbach’s Variation →

ICM 2022 — Plans Disrupted but not Derailed

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 … Continue reading ICM 2022 — Plans Disrupted but not Derailed →

Fairy Lights on the Farey Tree

The rational numbers $latex {\mathbb{Q}}&fg=000000$ are dense in the real numbers $latex {\mathbb{R}}&fg=000000$. The cardinality of rational numbers in the interval $latex {(0,1)}&fg=000000$ is $latex {\boldsymbol{\aleph}_0}&fg=000000$. We cannot list them in ascending order, because there is no least rational number greater than $latex {0}&fg=000000$. However, there are several ways of enumerating the rational numbers. The … Continue reading Fairy Lights on the Farey Tree →

Image Processing Emerges from the Shadows

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 … Continue reading Image Processing Emerges from the Shadows →

Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes

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 … Continue reading Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes →

Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

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. The natural numbers $latex {\mathbb{N}}&fg=000000$ split nicely into two subsets, the odd and even numbers $latex \displaystyle … Continue reading Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes →

A Finite but Unbounded Universe

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 … Continue reading A Finite but Unbounded Universe →

The Whole is Greater than the Part — Or is it?

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, … Continue reading The Whole is Greater than the Part — Or is it? →

Mamikon’s Visual Calculus and Hamilton’s Hodograph

[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 … Continue reading Mamikon’s Visual Calculus and Hamilton’s Hodograph →

Infinitesimals: vanishingly small but not quite zero

A few weeks ago, I wrote about  Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two. We know that 2.999… is equal to three. But many people have a sneaking suspicion that there is “something” between the number with all those 9’s after the 2 and the number 3, that is not … Continue reading Infinitesimals: vanishingly small but not quite zero →

The Improbability Principle and the Seanad Election

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 … Continue reading The Improbability Principle and the Seanad Election →

Hyperreals and Nonstandard Analysis

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 … Continue reading Hyperreals and Nonstandard Analysis →

A Prescient Vision of Modern Weather Forecasting

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 … Continue reading A Prescient Vision of Modern Weather Forecasting →

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 →

Some Characteristics of the Mathematical Psyche

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]. Absentmindedness Sweeping … Continue reading Some Characteristics of the Mathematical Psyche →

De Branges’s Proof of the Bieberbach Conjecture

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 … Continue reading De Branges’s Proof of the Bieberbach Conjecture →

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 →

Bernoulli’s Golden Theorem and the Law of Large Numbers

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, … Continue reading Bernoulli’s Golden Theorem and the Law of Large Numbers →

Set Density: are even numbers more numerous than odd ones?

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 $latex {n \rightarrow 2n}&fg=000000$. This was known to Galileo. However, with the usual ordering, $latex \displaystyle \mathbb{N} … Continue reading Set Density: are even numbers more numerous than odd ones? →

Buffon’s Noodle and the Mathematics of Hillwalking  

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 … Continue reading Buffon’s Noodle and the Mathematics of Hillwalking   →

Émilie Du Châtelet and the Conservation of Energy

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 … Continue reading Émilie Du Châtelet and the Conservation of Energy →

Cantor’s Theorem and the Unending Hierarchy of Infinities

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 … Continue reading Cantor’s Theorem and the Unending Hierarchy of Infinities →

Mathematical Scandals and Scoundrels

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 … Continue reading Mathematical Scandals and Scoundrels →

A Grand Unification of Mathematics

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 … Continue reading A Grand Unification of Mathematics →

The Spine of Pascal’s Triangle

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 $latex {k}&fg=000000$-th entry in row $latex {n}&fg=000000$ is the binomial coefficient $latex {\binom{n}{k}}&fg=000000$ (read $latex {n}&fg=000000$-choose-$latex {k}&fg=000000$), the number of ways of … Continue reading The Spine of Pascal’s Triangle →

Embedding: Reconstructing Solutions from a Delay Map

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 … Continue reading Embedding: Reconstructing Solutions from a Delay Map →

Earth System Models simulate the changing climate

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]. A Holistic View Over the past century, our … Continue reading Earth System Models simulate the changing climate →

The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces $latex {(X,\mathcal{O}_1)}&fg=000000$ and $latex {(X,\mathcal{O}_2)}&fg=000000$ having the same underlying set $latex {X}&fg=000000$ but different families of open sets, a function may be continuous in one but discontinuous in the other. The signum function is defined on the real line as follows: $latex … Continue reading The Signum Function may be Continuous →