Archive Page 2

The Simple Arithmetic Triangle is full of Surprises

Published June 3, 2021 Irish Times Leave a Comment
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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
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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
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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
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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
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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].

Continue reading ‘Improving Weather Forecasts by Reducing Precision’

Can You Believe Your Eyes?

Published April 29, 2021 Occasional Leave a Comment
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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
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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.

Continue reading ‘The Size of Things’

Entropy and the Relentless Drift from Order to Chaos

Published April 15, 2021 Irish Times Leave a Comment
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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
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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. 

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’

Was Space Weather the cause of the Titanic Disaster?

Published April 1, 2021 Occasional Leave a Comment
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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].
Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’

The Dimension of a Point that isn’t there

Published March 25, 2021 Occasional Leave a Comment
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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.

Continue reading ‘The Dimension of a Point that isn’t there’

Making the Best of Waiting in Line

Published March 18, 2021 Irish Times Leave a Comment
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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
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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
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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
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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)}.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

>>  Review in The Irish Times  <<

* * * * *

 

Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

Machine Learning and Climate Change Prediction

Published February 18, 2021 Irish Times Leave a Comment
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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.

Continue reading ‘Apples and Lemons in a Doughnut’

Complexity: are easily-checked problems also easily solved?

Published February 4, 2021 Irish Times Leave a Comment
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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
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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
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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.

Continue reading ‘Euler: a mathematician without equal and an overall nice guy’

The Basel Problem: Euler’s Bravura Performance

Published January 14, 2021 Occasional Leave a Comment
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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.

* * * * *

That’s Maths II: A Ton of Wonders

by Peter Lynch has just appeared.
Full details and links to suppliers at
http://logicpress.ie/2020-3/

* * * * *

Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

We are living at the bottom of an ocean

Published January 7, 2021 Irish Times Leave a Comment
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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].

Continue reading ‘We are living at the bottom of an ocean’

Derangements and Continued Fractions for e

Published December 31, 2020 Occasional Leave a Comment
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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.

Continue reading ‘Derangements and Continued Fractions for e’

Arrangements and Derangements

Published December 24, 2020 Irish Times Leave a Comment
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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.

Continue reading ‘Arrangements and Derangements’

On what Weekday is Christmas? Use the Doomsday Rule

Published December 17, 2020 Irish Times Leave a Comment
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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].

Continue reading ‘On what Weekday is Christmas? Use the Doomsday Rule’

Will RH be Proved by a Physicist?

Published December 10, 2020 Occasional Leave a Comment
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The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, {\Re(s) = 1/2}. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:

The non-trivial zeros of {\zeta(s)} are the
eigenvalues of a hermitian operator.

Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

Continue reading ‘Will RH be Proved by a Physicist?’

Decorating Christmas Trees with the Four Colour Theorem

Published December 3, 2020 Irish Times Leave a Comment
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When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let’s suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Decorating Christmas Trees with the Four Colour Theorem’

Laczkovich Squares the Circle

Published November 26, 2020 Occasional Leave a Comment
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The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

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Ireland’s Mapping Grid in Harmony with GPS

Published November 19, 2020 Irish Times Leave a Comment
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The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential  [TM199 or search for “thatsmaths” at irishtimes.com].

Transverse Mercator projection with central meridian at Greenwich.

Continue reading ‘Ireland’s Mapping Grid in Harmony with GPS’

Aleph, Beth, Continuum

Published November 12, 2020 Occasional Leave a Comment
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Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that not all infinite sets are created equal. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

Continue reading ‘Aleph, Beth, Continuum’

Weather Forecasts get Better and Better

Published November 5, 2020 Irish Times Leave a Comment
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Weather forecasts are getting better. Fifty years ago, predictions beyond one day ahead were of dubious utility. Now, forecasts out to a week ahead are generally reliable  [TM198 or search for “thatsmaths” at irishtimes.com].

Anomaly correlation of ECMWF 500 hPa height forecasts over three decades [Image from ECMWF].

Careful measurements of forecast accuracy have shown that the range for a fixed level of skill has been increasing by one day every decade. Thus, today’s one-week forecasts are about as good as a typical three-day forecast was in 1980. How has this happened? And will this remarkable progress continue?

Continue reading ‘Weather Forecasts get Better and Better’

The p-Adic Numbers (Part 2)

Published October 29, 2020 Occasional Leave a Comment
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Kurt Hensel (1861-1941)

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.

Continue reading ‘The p-Adic Numbers (Part 2)’

The p-Adic Numbers (Part I)

Published October 22, 2020 Occasional Leave a Comment
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Image from Cover of Katok, 2007.

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers {\mathbb{N}}, and ratios of these, the positive rational numbers {\mathbb{Q}^{+}}. It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of completion gives us the real numbers \mathbb{R}, which include rationals, irrationals like {\sqrt{2}} and transcendental numbers like {\pi}.

Continue reading ‘The p-Adic Numbers (Part I)’

Terence Tao to deliver the Hamilton Lecture

Published October 15, 2020 Irish Times Leave a Comment
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Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter, we have 5, 16, 8, 4, 2 and 1. From then on, the value cycles from 1 to 4 to 2 and back to 1 again, forever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1–4–2–1 cycle  [TM197 or search for “thatsmaths” at irishtimes.com].

Fields Medalist Professor Terence Tao.

Continue reading ‘Terence Tao to deliver the Hamilton Lecture’

From Impossible Shapes to the Nobel Prize

Published October 8, 2020 Occasional 1 Comment
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Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.

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Mathematics and the Nature of Physical Reality

Published October 1, 2020 Irish Times 1 Comment
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Applied mathematics is the use of maths to address questions and solve problems outside maths itself. Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood mathematically? [TM196, or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Mathematics and the Nature of Physical Reality’

Doughnuts and Dumplings are Distinct: Homopoty-101

Published September 24, 2020 Occasional Leave a Comment
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As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana.

Continue reading ‘Doughnuts and Dumplings are Distinct: Homopoty-101’

Will mathematicians be replaced by computers?

Published September 17, 2020 Irish Times Leave a Comment
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There are ongoing rapid advances in the power and versatility of AI or artificial intelligence. Computers are now producing results in several fields that are far beyond human capability. The trend is unstoppable, and is having profound effects in many areas of our lives. Will mathematicians be replaced by computers?  [TM195 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Will mathematicians be replaced by computers?’

TeX: A Boon for Mathematicians

Published September 10, 2020 Occasional Leave a Comment

Donald E Knuth, designer of the \TeX mathematical typesetting system.

Mathematicians owe a great debt of gratitude to Donald Knuth. A renowned American computer scientist and mathematician, Knuth is an emeritus professor at Stanford University. He is author of many books, including the multi-volume work, The Art of Computer Programming.

Knuth is the author of the powerful and versatile mathematical typesetting system called TeX. The original version, designed and written by Knuth, was released in 1978.

TeX is a powerful system for typesetting mathematical formulae. It is ideal both for simple mathematical notes with few formulas and for more complex documents and books involving subtle and sophisticated mathematical typography. TeX is used by almost all research mathematicians. It is also popular in computer science, engineering, physics, statistics, and and many other sciences.

Continue reading ‘TeX: A Boon for Mathematicians’

Suitable Names for Large Numbers

Published September 3, 2020 Irish Times Leave a Comment
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One year ago, there were just two centibillionaires, Jeff Bezos and Bill Gates. Recently, Facebook’s Mark Zuckerberg has joined the Amazon and Microsoft founders. Elon Musk, CEO of Tesla and SpaceX, is tipped to be next to join this exclusive club [TM194 or search for “thatsmaths” at irishtimes.com].

Shot from “A Suitable Boy” with Maan Kapoor (Ishaan Khatter), Mrs. Mahesh Kapoor (Geeta Agarwal) and Bhaskar (Yusuf Akhtar), covered in colours during the Holi festival [image from Instagram.  See also here].

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Jung’s Theorem: Enclosing a Set of Points

Published August 27, 2020 Occasional Leave a Comment
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Let us imagine that we have a finite set {P} of points in the plane {\mathbb{R}^2} (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle?  The answer is given by Jung’s Theorem.


Left: a set P of points in the real plane. Right: The span s is the maximum distance between two points of P.

Continue reading ‘Jung’s Theorem: Enclosing a Set of Points’

Is There Anyone Out There? The Drake Equation gives a Clue

Published August 20, 2020 Irish Times Leave a Comment
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The Drake Equation is a formula for the number of developed civilizations in our galaxy, the Milky Way. This number is determined by seven factors. Some are known with good accuracy but the values of most are quite uncertain. It is a simple equation comprising seven terms multiplied together [TM193 or search for “thatsmaths” at irishtimes.com].

Drake-Equation-Plaque-NRAO

A plaque commemorating the first appearance of the Drake Equation at a conference at the National Radio Astronomy Observatory, Green Bank, West Virginia in 1961.

Continue reading ‘Is There Anyone Out There? The Drake Equation gives a Clue’

Think of a Number: What are the Odds that it is Even?

Published August 13, 2020 Occasional Leave a Comment
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Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?

EulerProbDist

Probability distribution {P(n)=1/(\zeta(s)n^s)} for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).

Continue reading ‘Think of a Number: What are the Odds that it is Even?’

Resolution of Paradox: a Gateway to Mathematical Progress

Published August 6, 2020 Occasional Leave a Comment

Weyl-49-Cover

Hermann Weyl (1885-1955)

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the science of the infinite”  [TM192 or search for “thatsmaths” at irishtimes.com].

Ever since there has been time to wonder, humankind has been intrigued by the enigma of infinity. The idea has perplexed philosophers and mystified mathematicians for millennia. The Greeks were contemplating infinity from the time of Pythagoras. Head-on confrontation of infinity led to contradictory conclusions; the paradoxes of Zeno are amongst the most noted of these.

Continue reading ‘Resolution of Paradox: a Gateway to Mathematical Progress’

Berry’s Paradox and Gödel’s Incompleteness Theorem

Published July 30, 2020 Occasional Leave a Comment
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Chaitin-Boolos

Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form

THE SMALLEST NATURAL NUMBER THAT CANNOT BE
DEFINED IN FEWER THAN TWENTY WORDS.

Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’

Does Numerical Integration Reflect the Truth?

Published July 23, 2020 Occasional Leave a Comment
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Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:

“Does the numerical solution resemble the true solution of the equation?”

The answer is: “Not necessarily”.

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.

CFL-Criterion-1

Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).

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Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Published July 16, 2020 Irish Times Leave a Comment
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Lanczos240In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

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Buridan’s Ass

Published July 9, 2020 Occasional Leave a Comment
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Jean-Buridan

Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

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The Ever-growing Goals of Googology

Published July 2, 2020 Irish Times Leave a Comment
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In 1920, a kindergarten class was asked to describe the biggest number that they could imagine. One child proposed to “write down digits until you get tired”. A more concrete idea was to write a one followed by 100 zeros. This number, which scientists would express as ten to the power 100, was given the name “googol” by its inventor [TM190; or search for “thatsmaths” at irishtimes.com ].

OneGoogol

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The Online Encyclopedia of Integer Sequences

Published June 25, 2020 Occasional Leave a Comment
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Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.

OEIS-Homepage

The Home page of OEIS: https://oeis.org

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