As you pass through the main entrance of Trinity College, the iconic campanile stands before you, flanked, in pleasing symmetry, by two life-size statues. On the right, on a granite plinth is the historian and essayist William Lecky. On the left, George Salmon (1819–1904) sits on a limestone platform. Salmon was a distinguished mathematician and … Continue reading George Salmon, Mathematician & Theologian
Category: Irish Times
Articles in Irish Times
An English Lady with a Certain Taste
One hundred years ago, an English lady, Dr Muriel Bristol, amazed some leading statisticians by proving that she could determine by taste the order in which the constituents are poured in a cup of tea. One of the statisticians was Ronald Fisher. The other was William Roach, who was to marry Dr Bristol shortly afterwards. … Continue reading An English Lady with a Certain Taste
The Brief and Tragic Life of Évariste Galois
On the morning of 30 May 1832 a young man stood twenty-five paces from his friend. Both men fired, but only one pistol was loaded. Évariste Galois, a twenty year old mathematical genius, fell to the ground. The cause of Galois's death is veiled in mystery and speculation. Whether both men loved the same woman … Continue reading The Brief and Tragic Life of Évariste Galois
Stokes’s 200th Birthday Anniversary
Next Tuesday, the 30th of August, is the 200th anniversary of the birth of George Gabriel Stokes. This extended blog post is to mark that occasion. See also an article in The Irish Times. Whether we are designing aircraft, modelling blood flow, studying propulsion, lubrication or the dynamics of swimming, constructing wind turbines or forecasting … Continue reading Stokes’s 200th Birthday Anniversary
Algorithms: Recipes for Success
The impact of computing on society is ever-increasing. Web-based commerce continues to grow and artificial intelligence now pervades our lives. To make wise choices, we need to understand how computers operate and how we can deploy them most constructively. Listen to any computer scientist and soon you will hear the word “algorithm” [TM168 or search for … Continue reading Algorithms: Recipes for Success
Learning Maths without even Trying
Children have an almost limitless capacity to absorb knowledge if it is presented in an appealing and entertaining manner. Mathematics can be daunting, but it is possible to convey key ideas visually so that they are instantly accessible. Visiting Explorium recently, I saw such a visual display demonstrating the theorem of Pythagoras, which, according to … Continue reading Learning Maths without even Trying
What did the Romans ever do for Maths?
The ancient Romans developed many new techniques for engineering and architecture. The citizens of Rome enjoyed fountains, public baths, central heating, underground sewage systems and public toilets. All right, but apart from sanitation, medicine, education, irrigation, roads and aqueducts, what did the Romans ever do for maths? [TM166 or search for “thatsmaths” at irishtimes.com]. It might … Continue reading What did the Romans ever do for Maths?
Simple Curves that Perplex Mathematicians and Inspire Artists
The preoccupations of mathematicians can seem curious and strange to normal people. They sometimes expend great energy proving results that appear glaringly obvious. One such result is called the Jordan Curve Theorem. We all know that a circle has an inside and an outside, and that this property also holds for a much larger collection … Continue reading Simple Curves that Perplex Mathematicians and Inspire Artists
Spin-off Effects of the Turning Earth
On the rotating Earth, a moving object deviates from a straight line, being deflected to the right in the northern hemisphere and to the left in the southern hemisphere. The deflecting force is named after a nineteenth century French engineer, Gaspard-Gustave de Coriolis [TM164 or search for “thatsmaths” at irishtimes.com]. Coriolis was interested in the dynamics of machines, … Continue reading Spin-off Effects of the Turning Earth
The Rise and Rise of Women in Mathematics
The influential collection of biographical essays by Eric Temple Bell, Men of Mathematics, was published in 1937. It covered the lives of about forty mathematicians, from ancient times to the beginning of the twentieth century. The book inspired many boys to become mathematicians. However, it seems unlikely that it inspired many girls: the only woman … Continue reading The Rise and Rise of Women in Mathematics
Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph
The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of … Continue reading Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph
Closing the Gap between Prime Numbers
Occasionally, a major mathematical discovery comes from an individual working in isolation, and this gives rise to great surprise. Such an advance was announced by Yitang Zhang six years ago. [TM161 or search for “thatsmaths” at irishtimes.com]. After completing his doctorate at Purdue in 1991, Zhang had great difficulty finding an academic position and worked at various … Continue reading Closing the Gap between Prime Numbers
A Pioneer of Climate Modelling and Prediction
Today we benefit greatly from accurate weather forecasts. These are the outcome of a long struggle to advance the science of meteorology. One of the major contributors to that advancement was Norman A. Phillips, who died in mid-March, aged 95. Phillips was the first person to show, using a simple computer model, that mathematical simulation … Continue reading A Pioneer of Climate Modelling and Prediction
Joseph Fourier and the Greenhouse Effect
Jean-Baptiste Joseph Fourier, French mathematician and physicist, was born in Auxerre 251 years ago today. He is best known for the mathematical techniques that he developed in his analytical theory of heat transfer. Over the past two centuries, his methods have evolved into a major subject, harmonic analysis, with widespread applications in number theory, signal … Continue reading Joseph Fourier and the Greenhouse Effect
Hokusai’s Great Wave and Roguish Behaviour
Hokusai's woodcut “The Great Wave off Kanagawa”. “The Great Wave off Kanagawa”, one of the most iconic works of Japanese art, shows a huge breaking wave with foam thrusting forward at its crest, towering over three fishing boats, with Mt Fuji in the background [TM158 or search for “thatsmaths” at irishtimes.com]. This woodcut, produced by … Continue reading Hokusai’s Great Wave and Roguish Behaviour
Multiple Discoveries of the Thue-Morse Sequence
It is common practice in science to name important advances after the first discoverer or inventor. However, this process often goes awry. A humorous principle called Stigler's Law holds that no scientific result is named after its original discoverer. This law was formulated by Professor Stephen Stigler of the University of Chicago in his publication … Continue reading Multiple Discoveries of the Thue-Morse Sequence
Rambling and Reckoning
A walk on the beach, in the hills or along a river bank provides great opportunities for mathematical reflection. How high is the mountain? How many grains of sand are on the beach? How much water is flowing in the river? [TM156 or search for “thatsmaths” at irishtimes.com]. Daily average flow (cubic metres per second) at … Continue reading Rambling and Reckoning
Discoveries by Amateurs and Distractions by Cranks
Do amateurs ever solve outstanding mathematical problems? Professional mathematicians are aware that almost every new idea they have about a mathematical problem has already occurred to others. Any really new idea must have some feature that explains why no one has thought of it before [TM155 or search for “thatsmaths” at irishtimes.com]. It is both difficult and … Continue reading Discoveries by Amateurs and Distractions by Cranks
Trappist-1 & the Age of Aquarius
The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book Harmonices Mundi, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com]. Kepler's … Continue reading Trappist-1 & the Age of Aquarius
Consider a Spherical Christmas Tree
A minor seasonal challenge is how to distribute the fairy lights evenly around the tree, with no large gaps or local clusters. Since the lights are strung on a wire, we are not free to place them individually but must weave them around the branches, attempting to achieve a pleasing arrangement. Optimization problems like this … Continue reading Consider a Spherical Christmas Tree
Random Numbers Plucked from the Atmosphere
Randomness is a slippery concept, defying precise definition. A simple example of a random series is provided by repeatedly tossing a coin. Assigning “1” for heads and “0” for tails, we generate a random sequence of binary digits or bits. Ten tosses might produce a sequence such as 1001110100. Continuing thus, we can generate a … Continue reading Random Numbers Plucked from the Atmosphere
The “Napoleon of Crime” and The Laws of Thought
A fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents … Continue reading The “Napoleon of Crime” and The Laws of Thought
Johannes Kepler and the Song of the Earth
Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological … Continue reading Johannes Kepler and the Song of the Earth
Who Uses Maths? Almost Everyone!
In the midst of Maths Week Ireland, many students may be asking “What use is mathematics and what purpose is served by studying it?” Mathematicians often stress the inherent beauty and intellectual charm of the subject, but that is unlikely to persuade many people, who demand to know how mathematics can be of use and … Continue reading Who Uses Maths? Almost Everyone!
The Many Modern Uses of Quaternions
The story of William Rowan Hamilton's discovery of new four-dimensional numbers called quaternions is familiar. The solution of a problem that had bothered him for years occurred to him in a flash of insight as he walked along the Royal Canal in Dublin. But this Eureka moment did not arise spontaneously: it was the result … Continue reading The Many Modern Uses of Quaternions
Tom Lehrer: Comical Musical Mathematical Genius
Tom Lehrer, mathematician, singer, songwriter and satirist, was born in New York ninety years ago. He was active in public performance for about 25 years from 1945 to 1970. He is most renowned for his hilarious satirical songs, many of which he recorded and which are available today on YouTube [see TM147, or search for “thatsmaths” … Continue reading Tom Lehrer: Comical Musical Mathematical Genius
Face Recognition
As you pass through an airport, you are photographed several times by security systems. Face recognition systems can identify you by comparing your digital image to faces stored in a database. This form of identification is gaining popularity, allowing you to access online banking without a PIN or password. [see TM146, or search for “thatsmaths” at irishtimes.com]. … Continue reading Face Recognition
The Miraculous Spiral on Booterstown Strand
We all know what a spiral looks like. Or do we? Ask your friends to describe one and they will probably trace out the form of a winding staircase. But that is actually a helix, a curve in three-dimensional space. A spiral is confined to a plane – it is a flat curve. In general … Continue reading The Miraculous Spiral on Booterstown Strand
Tides: a Tug-of-War between Earth, Moon and Sun
All who set a sail, cast a hook or take a dip have a keen interest in the water level, and the regular ebb and flow of the tides. At most places the tidal variations are semi-diurnal, with high and low water twice each day [see TM144, or search for “thatsmaths” at irishtimes.com]. Equilibrium Tides In the … Continue reading Tides: a Tug-of-War between Earth, Moon and Sun
The Empty Set is Nothing to Worry About
Today's article is about nothing: nothing at all, as encapsulated in the number zero and the empty set. It took humanity millennia to move beyond the counting numbers. Zero emerged in several civilizations, first as a place-holder to denote a space or gap between digits, and later as a true number, which could be manipulated … Continue reading The Empty Set is Nothing to Worry About
Trigonometric Comfort Blankets on Hilltops
On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country [see TM142, or search for “thatsmaths” at irishtimes.com]. The pillars are about … Continue reading Trigonometric Comfort Blankets on Hilltops
Optical Refinements at the Parthenon
The Parthenon is a masterpiece of symmetry and proportion. This temple to the Goddess Athena was built with pure white marble quarried at Pentelikon, about 20km from Athens. It was erected without mortar or cement, the stones being carved to great accuracy and locked together by iron clamps. The building and sculptures were completed in … Continue reading Optical Refinements at the Parthenon
Leopold Bloom’s Arithmetical Adventures
As Bloomsday approaches, we reflect on James Joyce and mathematics. Joyce entered UCD in September 1898. His examination marks are recorded in the archives of the National University of Ireland, and summarized in a table in Richard Ellmann's biography of Joyce (reproduced below) [TM140 or search for “thatsmaths” at irishtimes.com]. The marks fluctuate widely, suggesting some lack of … Continue reading Leopold Bloom’s Arithmetical Adventures
Mathematics at the Science Museum
The new Winton Gallery at London's Science Museum in South Kensington holds a permanent display on the history of mathematics over the past 400 years. The exhibition shows how mathematics has underpinned astronomy, navigation and surveying in the past, and how it continues to pervade the modern world [see TM139, or search for “thatsmaths” at … Continue reading Mathematics at the Science Museum
Stan Ulam, a mathematician who figured how to initiate fusion
Stanislaw Ulam, born in Poland in 1909, was a key member of the remarkable Lvov School of Mathematics, which flourished in that city between the two world wars. Ulam studied mathematics at the Lvov Polytechnic Institute, getting his PhD in 1933. His original research was in abstract mathematics, but he later became interested in a … Continue reading Stan Ulam, a mathematician who figured how to initiate fusion
Geodesics on the Spheroidal Earth-II
Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as … Continue reading Geodesics on the Spheroidal Earth-II
Fourier’s Wonderful Idea – II
Solving PDEs by a Roundabout Route Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only … Continue reading Fourier’s Wonderful Idea – II
Cubic Skulduggery & Intrigue
Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields … Continue reading Cubic Skulduggery & Intrigue
Reducing R-naught to stem the spread of Epidemics
We are reminded each year to get vaccinated against the influenza virus. The severity of the annual outbreak is not known with certainty in advance, but a major pandemic is bound to occur sooner or later. Mathematical models play an indispensable role in understanding and managing infectious diseases. Models vary in sophistication from the simple … Continue reading Reducing R-naught to stem the spread of Epidemics
Galileo’s Book of Nature
In 1971, astronaut David Scott, standing on the Moon, dropped a hammer and a feather and found that both reached the surface at the same time. This popular experiment during the Apollo 15 mission was a dramatic demonstration of a prediction made by Galileo three centuries earlier. Galileo was born in Pisa on 15 February … Continue reading Galileo’s Book of Nature
Staying Put or Going with the Flow
The atmospheric temperature at a fixed spot may change in two ways. First, heat sources or sinks may increase or decrease the thermal energy; for example, sunshine may warm the air or radiation at night may cool it. Second, warmer or cooler air may be transported to the spot by the air flow in a … Continue reading Staying Put or Going with the Flow
The Heart of Mathematics
At five litres per minute the average human heart pumps nearly 200 megalitres of blood through the body in a lifetime. Heart disease causes 40 percent of deaths in the EU and costs hundreds of billions of Euros every year. Mathematics can help to improve our knowledge of heart disease and our understanding of cardiac … Continue reading The Heart of Mathematics
Energy Cascades in Van Gogh’s Starry Night
"Big whirls have little whirls that feed on their velocity, And little whirls have lesser whirls, and so on to viscosity." We are all familiar with the measurement of speed, the distance travelled in a given time. Allowing for the direction as well as the magnitude of movement, we get velocity, a vector quantity. In … Continue reading Energy Cascades in Van Gogh’s Starry Night
Darker Mornings, Brighter Evenings
Today is the winter solstice, the shortest day of the year. We might expect that the latest sunrise and earliest sunset also occur today. In fact, the earliest sunset, the darkest day of the year, was on 13 December, over a week ago, and the latest sunrise is still more than a week away. This … Continue reading Darker Mornings, Brighter Evenings
The Star of Bethlehem … or was it a Planet?
People of old were more aware than we are of the night sky and took a keen interest in unusual happenings above them. The configuration of the stars was believed to be linked to human affairs and many astronomical phenomena were interpreted as signs of good or evil in the offing. The Three Wise Men … Continue reading The Star of Bethlehem … or was it a Planet?
Slingshot Orbit to Asteroid Bennu
The Voyager 1 and Voyager 2 spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The Voyager spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as slingshots, are essential … Continue reading Slingshot Orbit to Asteroid Bennu
Modular Arithmetic: from Clock Time to High Tech
You may never have heard of modular arithmetic, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders [TM126 or search for “thatsmaths” at irishtimes.com]. When reckoning hours, we count up to twelve and start again … Continue reading Modular Arithmetic: from Clock Time to High Tech
Learning Maths has never been Easier
Maths is hard: many people find it inscrutable and have negative attitudes towards maths. They may have bad memories of school maths or have been told they lack mathematical talents. This is unfortunate: we all have the capacity to apply reasoning and logic and we can all do maths. Given the vital role mathematics plays … Continue reading Learning Maths has never been Easier
From Sailing on a Rhumb to Flying on a Geodesic
If you fly 14,500 km due westward from New York you will come to Beijing. The two cities are on the fortieth parallel of latitude. However, by flying a great circle route over the Arctic, you can reach Beijing in 11,000 km, saving 3,500 km and much time and aviation fuel. [TM124 or search for … Continue reading From Sailing on a Rhumb to Flying on a Geodesic
Andrey Markov’s Brilliant Ideas are still a Driving Force
Imagine examining the first 20,000 letters of a book, counting frequencies and studying patterns. This is precisely what Andrey Markov did when he analyzed the text of Alexander Pushkin's verse novel Eugene Onegin. This work comprises almost 400 stanzas of iambic tetrameter and is a classic of Russian literature. Markov studied the way vowels and … Continue reading Andrey Markov’s Brilliant Ideas are still a Driving Force
ThatsMaths 