Given a function $latex {f(x)}&fg=000000$ of a real variable, we often have to find the values of $latex {x}&fg=000000$ for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton's method or as the Newton-Raphson method. It usually produces highly … Continue reading Zeroing in on Zeros
George Salmon, Mathematician & Theologian
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
Spiralling Primes
The prime numbers have presented mathematicians with some of their most challenging problems. They continue to play a central role in number theory, and many key questions remain unsolved. Order and Chaos The primes have many intriguing properties. In his article ``The first 50 million prime numbers'', Don Zagier noted two contradictory characteristics of the … Continue reading Spiralling Primes
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
ToplDice is Markovian
Many problems in probability are solved by assuming independence of separate experiments. When we toss a coin, it is assumed that the outcome does not depend on the results of previous tosses. Similarly, each cast of a die is assumed to be independent of previous casts. However, this assumption is frequently invalid. Draw a card … Continue reading ToplDice is Markovian
The curious behaviour of the Wilberforce Spring.
The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to stretch up and down and to twist about its axis. In equilibrium, the spring hangs down with the pull of gravity … Continue reading The curious behaviour of the Wilberforce Spring.
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
Billiards & Ballyards
In (mathematical) billiards, the ball travels in a straight line between impacts with the boundary, when it changes suddenly and discontinuously We can approximate the hard-edged, flat-bedded billiard by a smooth sloping surface, that we call a ``ballyard''. Then the continuous dynamics of the ballyard approach the motions on a billiard. Elliptical Billiards We idealize … Continue reading Billiards & Ballyards
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
Boxes and Loops
We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions. Libration and Rotation of a Pendulum The simple pendulum, with one degree of freedom, provides a … Continue reading Boxes and Loops
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?
Cumbersome Calculations in Ancient Rome
“Typus Arithmeticae” is a woodcut from the book Margarita Philosophica by Gregor Reisch of Freiburg, published in 1503. In the centre of the figure stands Arithmetica, the muse of mathematics. She is watching a competition between the Roman mathematician Boethius and the great Pythagoras. Boethius is crunching out a calculation using Hindu-Arabic numerals, while Pythagoras … Continue reading Cumbersome Calculations in Ancient Rome
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
Bernard Bolzano, a Voice Crying in the Wilderness
Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but … Continue reading Bernard Bolzano, a Voice Crying in the Wilderness
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
Symplectic Geometry
For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum ``physics is geometry'' is a guiding principle of modern theoretical physics. Einstein's General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of … Continue reading Symplectic Geometry
Chase and Escape: Pursuit Problems
From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved … Continue reading Chase and Escape: Pursuit Problems
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
Bouncing Billiard Balls Produce Pi
There are many ways of evaluating $latex {\pi}&fg=000000$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method. Historical Methods Archimedes used inscribed and circumscribed polygons to deduce that $latex \displaystyle \textstyle{3\frac{10}{71} < \pi < 3\frac{10}{70}} &fg=000000$ giving roughly … Continue reading Bouncing Billiard Balls Produce Pi
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
K3 implies the Inverse Square Law.
Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler's Astronomia Nova. The first law (K1) describes the orbit of a planet as an ellipse with the Sun at one focus. The … Continue reading K3 implies the Inverse Square Law.
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
Massive Collaboration in Maths: the Polymath Project
Sometimes proofs of long-outstanding problems emerge without prior warning. In the 1990s, Andrew Wiles proved Fermat's Last Theorem. More recently, Yitang Zhang announced a key result on bounded gaps in the prime numbers. Both Wiles and Zhang had worked for years in isolation, keeping abreast of developments but carrying out intensive research programs unaided by … Continue reading Massive Collaboration in Maths: the Polymath Project
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
A Chirping Elliptic Rocker
Sitting at the breakfast table, I noticed that a small cereal bowl placed within another larger one was rocking, and that the period became shorter as the amplitude died down. What was going on? The handles of the smaller bowl appeared to be elliptical in cross-section, so I considered how a rigid body shaped … Continue reading A Chirping Elliptic Rocker
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
The Kill-zone: How to Dodge a Sniper’s Bullet
Under mild simplifying assumptions, a projectile follows a parabolic trajectory. This results from Newton's law of motion. Thus, for a fixed energy, there is an accessible region around the firing point comprising all the points that can be reached. We will derive a mathematical description for this kill-zone (the term kill-zone, used for dramatic effect, … Continue reading The Kill-zone: How to Dodge a Sniper’s Bullet
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
Don’t be Phased by Waveform Distortions
For many years there has been an ongoing debate about the importance of phase changes in music. Some people claim that we cannot hear the effects of phase errors, others claim that we can. Who is right? The figure below shows a waveform of a perfect fifth, with components in the ratio $latex {3 : … Continue reading Don’t be Phased by Waveform Distortions
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
Folding Maps: A Simple but Unsolved Problem
Paper-folding is a recurring theme in mathematics. The art of origami is much-loved by many who also enjoy recreational maths. One particular folding problem is remarkably easy to state, but the solution remains elusive: Given a map with M × N panels, how many different ways can it be folded? Each panel is considered to … Continue reading Folding Maps: A Simple but Unsolved Problem
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
Our Dearest Problems
A Colloquium on Recreational Mathematics took place in Lisbon this week. The meeting, RMC-VI (G4GEurope), a great success, was organised by the Ludus Association, with support from several other agencies: MUHNAC, ULisboa, CMAF-IO, CIUHCT, CEMAPRE, and FCT. It was the third meeting integrated in the Gathering for Gardner movement, which celebrates the great populariser of … Continue reading Our Dearest Problems
From a Wide Wake to the Width of the World
The finite angular width of a ship's turbulent wake at the horizon enables the Earth's radius to be estimated. By ignoring evidence, Flat-Earthers remain secure in their delusions. The rest of us benefit greatly from accurate geodesy. Satellite communications, GPS navigation, large-scale surveying and cartography all require precise knowledge of the shape and form of the … Continue reading From a Wide Wake to the Width of the World
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
Really, 0.999999… is equal to 1. Surreally, this is not so!
The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory. In school we learn that … Continue reading Really, 0.999999… is equal to 1. Surreally, this is not so!
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
Gaussian Curvature: the Theorema Egregium
One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his ``Disquisitiones generales circa superficies curvas'', or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by … Continue reading Gaussian Curvature: the Theorema Egregium
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
The 3 : 2 Resonance between Neptune and Pluto
For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto. The Harmony of the Spheres Pythagoras based his musical analysis on two ratios: the octave 2 : 1 and the perfect fifth 3 : 2. He … Continue reading The 3 : 2 Resonance between Neptune and Pluto
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 Two Envelopes Fallacy
During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed … Continue reading The Two Envelopes Fallacy
Gravitational Waves & Ringing Teacups
Newton's law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide. Energy and angular momentum are carried … Continue reading Gravitational Waves & Ringing Teacups
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
Listing the Rational Numbers III: The Calkin-Wilf Tree
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce … Continue reading Listing the Rational Numbers III: The Calkin-Wilf Tree
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
Saving Daylight with Hip-hop Time: a Modest Proposal
At 2:00 AM on Sunday 28 October the clocks throughout Europe will be set back one hour, reverting to Standard Time. In many countries, the clocks are put forward one hour in Spring and set back to Standard Time in the Autumn. Daylight saving time gives brighter evenings in Summer. In Summer, the mornings are … Continue reading Saving Daylight with Hip-hop Time: a Modest Proposal
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!
ThatsMaths 