ThatsMaths

Chiral and Achiral Knots

Published November 25, 2021 Occasional Leave a Comment
Tags: Topology

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 ${\chi\epsilon\rho\iota}$ , Greek for hand. If chirality is absent, we have an achiral object.

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.”

Image from Lehninger Principles of Biochemistry, 2008: Nelson, Lehninger; and Cox. © W H Freeman.

Chirality is important in many sciences, most notably in chemistry. In mathematics, chirality is the property of a curve, surface or solid such that it is not identical to its mirror image. The term is found in topology and, in particular, in knot theory.

A helix may be left-handed or right-handed (see Figure). The DNA molecule is formed of right-handed helices, although it is not uncommon to see diagrams that use the left-handed versions. There are also left-handed and right-handed versions of the Möbius band.

Chiral and Achiral Knots

A knot is achiral if it can be continuously deformed into its mirror image. The unknot and the figure-of-eight knot are achiral. The unknot is essentially a continuously distorted circle (technically, via an ambient isotopy).

Transformation of a figure-of-eight knot into its mirror image.

The Figure above illustrates how a figure-of-eight knot can be continuously transformed into its mirror image, demonstrating that it is achiral.

The trefoil knot is chiral: there are left-handed and right-handed trefoils (see Figure below) and it is impossible to transform one into another by a continuous deformation.

These two trefoil knots are not equivalent; the trefoil is chiral [image from Wikimedia Commons].

A Knotty Puzzle

Imagine a bug confined to move within a thin flexible pipe closed by joining its ends. Assume that the pipe is in deep space. The bug can measure speed and acceleration; this is its only source of data.

Can the bug deduce the shape of the pipe and determine whether or not it is knotted? Comments to thatsmaths@gmail.com (Subject: Bug in pipe), preferably before the end of November 2021.

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Émilie Du Châtelet and the Conservation of Energy

Published November 18, 2021 Irish Times Leave a Comment
Tags: History, Mechanics

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].

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Cantor’s Theorem and the Unending Hierarchy of Infinities

Published November 11, 2021 Occasional Leave a Comment
Tags: Cantor, Set Theory

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.

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Topsy-turvy Maths: Proving Axioms from Theorems

Published November 4, 2021 Irish Times Leave a Comment
Tags: Logic, Philosophy

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].

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How to Write a Convincing Mathematical Paper

Published October 28, 2021 Occasional Leave a Comment

Let ${X}$ be a Banach Space

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.

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Mathematical Scandals and Scoundrels

Published October 21, 2021 Irish Times Leave a Comment
Tags: History

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.

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The Square Root Spiral of Theodorus

Published October 14, 2021 Occasional Leave a Comment
Tags: Analysis, Geometry

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 ${\sqrt{2}}$ . Another triangle, with sides ${1}$ and ${\sqrt{2}}$ and hypotenuse ${\sqrt{3}}$ is stacked upon the first. This process continues, giving hypotenuse lengths ${\sqrt{n}}$ for all ${n}$ .

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The Spine of Pascal’s Triangle

Published September 30, 2021 Occasional Leave a Comment
Tags: Arithmetic, Number Theory

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 ${k}$ -th entry in row ${n}$ is the binomial coefficient ${\binom{n}{k}}$ (read ${n}$ -choose- ${k}$ ), the number of ways of selecting ${k}$ elements from a set of ${n}$ distinct elements.

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Embedding: Reconstructing Solutions from a Delay Map

Published September 23, 2021 Occasional Leave a Comment
Tags: Mechanics, Topology

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’

Earth System Models simulate the changing climate

Published September 16, 2021 Irish Times Leave a Comment
Tags: biology, Climate Modelling, Geophysics

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].

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The Signum Function may be Continuous

Published September 9, 2021 Occasional Leave a Comment
Tags: Topology

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces ${(X,\mathcal{O}_1)}$ and ${(X,\mathcal{O}_2)}$ having the same underlying set ${X}$ but different families of open sets, a function may be continuous in one but discontinuous in the other. Continue reading ‘The Signum Function may be Continuous’

The Social Side of Mathematics

Published September 2, 2021 Irish Times Leave a Comment
Tags: Education, Ireland

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].

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Real Derivatives from Imaginary Increments

Published August 26, 2021 Occasional Leave a Comment
Tags: Numerical Analysis

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 ${h}$ [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size ${h}$ is crucial: if ${h}$ is too large, the estimate of the derivative is poor, due to truncation error; if ${h}$ is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if ${h}$ is very small, they produce zero values.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing ${f^\prime(x)}$ .

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Changing Views on the Age of the Earth

Published August 19, 2021 Irish Times Leave a Comment
Tags: Geophysics, History, Ireland

[Image credit: NASA]

In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at irishtimes.com].

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Carnival of Mathematics

Published August 12, 2021 Occasional 1 Comment

The Aperiodical is described on its `About’ page as “a meeting-place for people who already know they like maths and would like to know more”. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of thatsmaths.com to host CoM.
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Phantom traffic-jams are all too real

Published August 5, 2021 Irish Times Leave a Comment
Tags: modelling

Driving along the motorway on a busy day, you see brake-lights ahead and slow down until the flow grinds to a halt. The traffic stutters forward for five minutes or so until, mysteriously, the way ahead is clear again. But, before long, you arrive at the back of another stagnant queue. Hold-ups like this, with no apparent cause, are known as phantom traffic jams and you may experience several such delays on a journey of a few hours [TM216 or search for “thatsmaths” at irishtimes.com].

Traffic jams can have many causes [Image © Susanneiles.com. JPEG]

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Simple Models of Atmospheric Vortices

Published July 29, 2021 Occasional Leave a Comment
Tags: Fluid Dynamics, Geophysics

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

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Finding Fixed Points

Published July 22, 2021 Occasional Leave a Comment
Tags: Algebra

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group ${E(n)}$ is the group of isometries of ${n}$ -dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane ${\mathbb{E}^2}$ , we have the group ${E(2)}$ , comprising all combinations of translations, rotations and reflections of the plane.

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All Numbers Great and Small

Published July 15, 2021 Irish Times Leave a Comment
Tags: Number Theory, Physics

Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we do not know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our number system represent physical space and time? [TM215 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘All Numbers Great and Small’

Approximating the Circumference of an Ellipse

Published July 8, 2021 Occasional Leave a Comment

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference ${C}$ to diameter ${D}$ the same for all? Searching in Euclid’s Elements, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that ${C / D}$ is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

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Kalman Filters: from the Moon to the Motorway

Published July 1, 2021 Irish Times Leave a Comment
Tags: Algorithms

Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie in the early 1960s, when the space race triggered by the Russian launch of Sputnik I was raging [TM214 or search for “thatsmaths” at irishtimes.com].

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Gauss Predicts the Orbit of Ceres

Published June 24, 2021 Occasional Leave a Comment
Tags: Astronomy, Gauss

Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].

On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without success to locate it. Without accurate knowledge of its orbit, the search seemed hopeless. How could its trajectory be determined from a few observations made from the Earth, which itself was moving around the Sun?

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Seeing beyond the Horizon

Published June 17, 2021 Irish Times Leave a Comment
Tags: Geometry, Geophysics

From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around AD 1020, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within 1% of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous [TM213 or search for “thatsmaths” at irishtimes.com].

Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).

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Al Biruni and the Size of the Earth

Published June 10, 2021 Occasional Leave a Comment
Tags: Geometry, History, Trigonometry

Abu Rayhan al-Biruni (AD 973–1048)

Al Biruni at Persian Scholars Pavilion in Vienna.

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius ${a}$ . He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

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The Simple Arithmetic Triangle is full of Surprises

Published June 3, 2021 Irish Times Leave a Comment
Tags: Arithmetic, Number Theory

Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before [TM212 or search for “thatsmaths” at irishtimes.com].

Pascal’s triangle as found in Zhu Shiji’s treatise The Precious Mirror of the Four Elements (1303).

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Hanoi Graphs and Sierpinski’s Triangle

Published May 27, 2021 Occasional Leave a Comment
Tags: Algorithms, Fractals, Games

The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

Only one disk can be moved at a time.
No disk can be placed upon a smaller one.

Tower of Hanoi [image Wikimedia Commons].

Continue reading ‘Hanoi Graphs and Sierpinski’s Triangle’

Multi-faceted aspects of Euclid’s Elements

Published May 20, 2021 Irish Times Leave a Comment
Tags: Geometry

A truncated octahedron within the coronavirus [image from Cosico et al, 2020].

Euclid’s Elements was the first major work to organise mathematics as an axiomatic system. Starting from a set of clearly-stated and self-evident truths called axioms, a large collection of theorems is constructed by logical reasoning. For some, the Elements is a magnificent triumph of human thought; for others, it is a tedious tome, painfully prolix and patently pointless [TM211 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Multi-faceted aspects of Euclid’s Elements’

A Model for Elliptic Geometry

Published May 13, 2021 Occasional Leave a Comment
Tags: Geometry, Spherical Trigonometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

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Improving Weather Forecasts by Reducing Precision

Published May 6, 2021 Irish Times Leave a Comment
Tags: Numerical Weather Prediction

Weather forecasting relies on supercomputers, used to solve the mathematical equations that describe atmospheric flow. The accuracy of the forecasts is constrained by available computing power. Processor speeds have not increased much in recent years and speed-ups are achieved by running many processes in parallel. Energy costs have risen rapidly: there is a multimillion Euro annual power bill to run a supercomputer, which may consume something like 10 megawatts [TM210 or search for “thatsmaths” at irishtimes.com].

The characteristic butterfly pattern for solutions of Lorenz’s equations [Image credit: source unknown].

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Can You Believe Your Eyes?

Published April 29, 2021 Occasional Leave a Comment
Tags: Numerical Analysis

Scene from John Ford’s Stagecoach (1939).

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

Continue reading ‘Can You Believe Your Eyes?’

The Size of Things

Published April 22, 2021 Occasional Leave a Comment
Tags: Analysis, Set Theory

In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

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Entropy and the Relentless Drift from Order to Chaos

Published April 15, 2021 Irish Times Leave a Comment
Tags: History, Physics

In a famous lecture in 1959, scientist and author C P Snow spoke of a gulf of comprehension between science and the humanities, which had become split into “two cultures”. Many people in each group had a lack of appreciation of the concerns of the other group, causing grave misunderstandings and making the world’s problems more difficult to solve. Snow compared ignorance of the Second Law of Thermodynamics to ignorance of Shakespeare [TM209 or search for “thatsmaths” at irishtimes.com].

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Circles, polygons and the Kepler-Bouwkamp constant

Published April 8, 2021 Occasional Leave a Comment
Tags: Analysis

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is ${\cos \pi/3 = 0.5}$ . More generally, for an N-gon the ratio is easily shown to be ${\cos \pi/N}$ . Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the Mysterium Cosmographicum here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established.

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Was Space Weather the cause of the Titanic Disaster?

Published April 1, 2021 Occasional Leave a Comment
Tags: Geophysics

Space weather, first studied in the 1950’s, has grown in importance with recent technological advances. It concerns the influence on the Earth’s magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of the Global Positioning System, causing serious navigation problems [TM208 or search for “thatsmaths” at irishtimes.com].

Solar disturbances disrupt the Earth’s magnetic field [Image: ESA].

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The Dimension of a Point that isn’t there

Published March 25, 2021 Occasional Leave a Comment
Tags: Topology

A slice of Swiss cheese has one-dimensional holes;
a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

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Making the Best of Waiting in Line

Published March 18, 2021 Irish Times Leave a Comment
Tags: Applied Maths

Queueing system with several queues, one for each serving point [Wikimedia Commons].

Queueing is a bore and waiting to be served is one of life’s unavoidable irritants. Whether we are hanging onto a phone, waiting for response from a web server or seeking a medical procedure, we have little choice but to join the queue and wait. It may surprise readers that there is a well-developed mathematical theory of queues. It covers several stages of the process, from patterns of arrival, through moving gradually towards the front, being served and departing [TM207 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Making the Best of Waiting in Line’

Differential Forms and Stokes’ Theorem

Published March 11, 2021 Occasional Leave a Comment
Tags: Analysis

Elie Cartan (1869–1951).

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in ${n}$ dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

Continue reading ‘Differential Forms and Stokes’ Theorem’

Goldbach’s Conjecture: if it’s Unprovable, it must be True

Published March 4, 2021 Irish Times 1 Comment
Tags: Euler, Logic, Number Theory

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 logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’

Mamikon’s Theorem and the area under a cycloid arch

Published February 25, 2021 Occasional Leave a Comment
Tags: Analysis, Geometry

The cycloid, the locus of a point on the rim of a rolling disk.

The Cycloid

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

$\displaystyle x = r (\theta - \sin\theta)\,, \qquad y = r (1 - \cos\theta ) \ \ \ \ \ (1)$

where ${\theta}$ is the angle through which the disk has rotated. The centre of the disk is at ${(x_0,y_0) = (r\theta, r)}$ .

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Machine Learning and Climate Change Prediction

Published February 18, 2021 Irish Times Leave a Comment
Tags: Climate Modelling

Current climate prediction models are markedly defective, even in reproducing the changes that have already occurred. Given the great importance of climate change, we must identify the causes of model errors and reduce the uncertainty of climate predictions [TM205 or search for “thatsmaths” at irishtimes.com].

Schematic diagram of some key physical processes in the climate system.

Continue reading ‘Machine Learning and Climate Change Prediction’

Apples and Lemons in a Doughnut

Published February 11, 2021 Occasional Leave a Comment

A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let ${r}$ be the radius of the circle and ${R}$ the distance from the axis to the centre of the circle, with ${R>r}$ .

Generating a ring torus by rotating a circle of radius ${r}$ about an axis at distance ${R>r}$ from its centre.

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Complexity: are easily-checked problems also easily solved?

Published February 4, 2021 Irish Times Leave a Comment
Tags: Algorithms, Computer Science

From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a cake (output). For a computer algorithm, the inputs are the known numerical quantities and the output is the required solution [TM204 or search for “thatsmaths” at irishtimes.com].

Al Khwarizmi, Persian polymath (c. 780 – 850) [image, courtesy of Prof. Irfan Shahid].

Continue reading ‘Complexity: are easily-checked problems also easily solved?’

Euler’s Product: the Golden Key

Published January 28, 2021 Occasional Leave a Comment
Tags: Analysis, Euler

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.

Euler treated the Taylor series for ${\sin x}$ as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

$\displaystyle \frac{\sin x}{x} = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{(2n+1)!} = \prod_{n=1}^{\infty} \left( 1 - \frac{x^{2} }{(n\pi)^2} \right) \nonumber \ \ \ \ \ (1)$

This enabled him to deduce the remarkable result

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \left( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots \right) = \frac{\pi^2}{6}$

which he described as an unexpected and elegant formula.

Continue reading ‘Euler’s Product: the Golden Key’

Euler: a mathematician without equal and an overall nice guy

Published January 21, 2021 Irish Times 1 Comment
Tags: Euler

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 communication [TM203 or search for “thatsmaths” at irishtimes.com].

It is true that some of the greatest fit this stereotype, but the incomparable Leonhard Euler is a refreshing counter-example. He was described by his contemporaries as a generous man, kind and loving to his 13 children and maintaining his good-natured disposition even after he became completely blind. He is comforting proof that a neurotic personality is not essential for mathematical prowess.

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The Basel Problem: Euler’s Bravura Performance

Published January 14, 2021 Occasional Leave a Comment
Tags: Analysis, Euler

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 ${\ln 2}$ . The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \ ?$

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

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That’s Maths II: A Ton of Wonders

by Peter Lynch has just appeared.
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We are living at the bottom of an ocean

Published January 7, 2021 Irish Times Leave a Comment
Tags: Geophysics

Anyone who lives by the sea is familiar with the regular ebb and flow of the tides. But we all live at the bottom of an ocean of air. The atmosphere, like the ocean, is a fluid envelop surrounding the Earth, and is subject to the influence of the Sun and Moon. While sea tides have been known for more than two thousand years, the discovery of tides in the atmosphere had to await the invention of the barometer [TM202 or search for “thatsmaths” at irishtimes.com].

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Derangements and Continued Fractions for e

Published December 31, 2020 Occasional Leave a Comment
Tags: Combinatorics, Number Theory

We show in this post that an elegant continued fraction for ${e}$ can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position.

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Arrangements and Derangements

Published December 24, 2020 Irish Times Leave a Comment
Tags: Combinatorics, Number Theory

Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.

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On what Weekday is Christmas? Use the Doomsday Rule

Published December 17, 2020 Irish Times Leave a Comment
Tags: Algorithms

An old nursery rhyme begins “Monday’s child is fair of face / Tuesday’s child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born without the aid of computers or calendars? Is there an algorithm – a recipe or rule – giving the answer? [TM201 or search for “thatsmaths” at irishtimes.com].

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