That’s Maths in The Irish Times this week (TM059, or Search for “thatsmaths” at irishtimes.com) is about data compression and its uses in modern technology. The arrival of mobile phones was followed rapidly by "txtese", an abbreviation of language to enable messages to be written and transmitted rapidly using SMS (Short Message Service). The simplest … Continue reading Information Theory
New Curves for Old: Inversion
Special Curves A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space $latex {\mathbb{R}^2}&fg=000000$ so that it generates a plane curve as it moves. … Continue reading New Curves for Old: Inversion
The Year of George Boole
This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen's College Cork. Mathematician and logician George Boole died just 150 years ago, on 8 December 1864, following a drenching as he was walking between his home and Queen's … Continue reading The Year of George Boole
Falling Bodies [2]: Philae
The ESA Rosetta Mission, launched in March 2004, rendezvoused with comet 67P/C-G in August 2014. The lander Philae touched down on the comet on 12 November and came to rest after bouncing twice (the harpoon tethers and cold gas retro-jet failed to fire). Rosetta was in orbit around the comet and, after detatchment, the lander … Continue reading Falling Bodies [2]: Philae
Earth’s Shape and Spin Won’t Make You Thin
Using a simple pendulum we can determine the shape of the Earth. That amazing story is told in this week’s That’s Maths column in The Irish Times (TM057 or search for “thatsmaths” at http://www.irishtimes.com ). Many of us struggle to lose weight, or at least to keep our weight within a manageable range. There is no … Continue reading Earth’s Shape and Spin Won’t Make You Thin
Falling Bodies [1]: Sky-diving
Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle's theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a … Continue reading Falling Bodies [1]: Sky-diving
El Niño likely this Winter
This week’s That’s Maths column in The Irish Times (TM056 or search for “thatsmaths” at irishtimes.com) is about El Niño and the ENSO phenomenon. In 1997-98, abnormally high ocean temperatures off South America caused a collapse of the anchovy fisheries. Anchovies are a vital link in the food-chain and shortages can bring great hardship. Weather … Continue reading El Niño likely this Winter
Light Weight (*)
Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a … Continue reading Light Weight (*)
Waring’s Problem & Lagrange’s Four-Square Theorem
$latex \displaystyle \mathrm{num}\ = \square+\square+\square+\square &fg=000000$ Introduction We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved … Continue reading Waring’s Problem & Lagrange’s Four-Square Theorem
Old Octonions may rule the World
This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton. On this day in 1843, the great Irish mathematician William Rowan Hamilton discovered a new kind of numbers called quaternions. Each quaternion has … Continue reading Old Octonions may rule the World
Triangular Numbers: EYPHKA
The maths teacher was at his wits' end. To get some respite, he set the class a task: Add up the first one hundred numbers. “That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later … Continue reading Triangular Numbers: EYPHKA
Algebra in the Golden Age
This week’s That’s Maths column in The Irish Times (TM054, or search for “thatsmaths” at irishtimes.com) is about the emergence of algebra in the Golden Age of Islam. The Chester Beatty Library in Dublin has several thousand Arabic manuscripts, many on mathematics and science. "The ink of a scholar is holier than the blood of … Continue reading Algebra in the Golden Age
Curves with Singularities
Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it … Continue reading Curves with Singularities
How Big was the Bomb?
By a brilliant application of dimensional analysis, G.I.Taylor estimated the explosive energy of the first atomic blast, the Trinity Test (see this week’s That’s Maths column in The Irish Times, TM053, or search for “thatsmaths” at irishtimes.com). Physicists, engineers and applied mathematicians have an arsenal of problem-solving techniques. Computers crunch out numerical solutions in short … Continue reading How Big was the Bomb?
Cartoon Curves
The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves
The Biggest Harp in Ireland
This week’s That’s Maths column in The Irish Times (TM052, or search for “thatsmaths” at irishtimes.com) is about "Samuel Beckett Playing Bridge in Dublin". Mathematics Models Nature The life of Pythagoras is shrouded in myth and legend. He was born on the island of Samos and travelled to Egypt, Mesopotamia and possibly India before arriving … Continue reading The Biggest Harp in Ireland
Temperamental Tuning
Every pure musical tone has a frequency, the number of oscillations per second in the sound wave. Doubling the frequency corresponds to moving up one octave. A musical note consists of a base frequency or pitch, called the fundamental together with a series of harmonics, or oscillations whose frequencies are whole-number multiples of the fundamental … Continue reading Temperamental Tuning
Biomathematics: the New Frontier
Mathematics is coming to Life in a Big Way. This week's That’s Maths in The Irish Times (TM051, or Search for “thatsmaths” at irishtimes.com) is about the increasing importance of mathematics in the biological sciences. Once upon a time biology meant zoology and botany, the study of animals and plants. The invention of the microscope … Continue reading Biomathematics: the New Frontier
Do you remember Venn?
Do you recall coming across those diagrams with overlapping circles that were popularised in the 'sixties', in conjunction with the “New Maths”. They were originally introduced around 1880 by John Venn, and now bear his name. John Venn Venn was a logician and philosopher, born in Hull, Yorkshire in 1834. He studied at Cambridge University, … Continue reading Do you remember Venn?
“Come See the Spinning Globe”
That’s Maths in The Irish Times this week (TM050, or Search for “thatsmaths” at irishtimes.com) is about how a simple pendulum can demonstrate the rotation of the Earth. Spectators gathered in Paris in March 1851 were astonished to witness visible evidence of the Earth's rotation. With a simple apparatus comprising a heavy ball swinging on … Continue reading “Come See the Spinning Globe”
Degrees of Infinity
Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no … Continue reading Degrees of Infinity
Invention or Discovery?
Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture: Conjecture: Definitions are invented. Theorems are discovered. The goal is to prove … Continue reading Invention or Discovery?
Digital Dentistry
That’s Maths in The Irish Times this week (TM049, or Search for “thatsmaths” at irishtimes.com) is about applications of computer aided design and computer aided manufacture to making dental crowns. Next time you spot a kid immersed in a video game, pause before uttering a condemnatory remark. The spin-offs from computer gaming are benefiting us … Continue reading Digital Dentistry
Gauss’s Great Triangle and the Shape of Space
In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a … Continue reading Gauss’s Great Triangle and the Shape of Space
Beauty is the First Test
This week, That’s Maths in The Irish Times (TM048: Search for "thatsmaths" at irishtimes.com) is about the beauty of mathematics. “Beauty is truth, truth beauty” wrote John Keats in his Ode on a Grecian Urn. Mathematical theorems have an enduring truth and the truly great ones are profoundly beautiful. Proofs of theorems can be technically … Continue reading Beauty is the First Test
Balancing a Pencil
Does quantum mechanics matter at everyday scales? It would be very surprising if quantum effects were to be manifest in a macroscopic system. This has been claimed for the problem of balancing a pencil on its tip. But the behaviour of a tipping pencil can be explained in purely classical terms. Modelling a balanced pencil … Continue reading Balancing a Pencil
When did Hammurabi reign?
The consequences of the Earth’s changing climate may be very grave. It is essential to understand past climate change so that we can anticipate future changes. This week, That’s Maths in The Irish Times ( TM047 ) is about the chronology of the Middle East. Surprisingly, this has important implications for our understanding of climate … Continue reading When did Hammurabi reign?
Biscuits, Books, Coins and Cards: Massive Hangovers
Have you ever tried to build a high stack of coins? In theory it's fine: as long as the centre of mass of the coins above each level remains over the next coin, the stack should stand. But as the height grows, it becomes increasingly trickier to avoid collapse. In theory it is possible to … Continue reading Biscuits, Books, Coins and Cards: Massive Hangovers
Sunflowers and Fibonacci: Models of Efficiency
The article in this week’s That’s Maths column in The Irish Times ( TM046 ) is about the maths behind the efficient packing of sunflowers and many other plants Strolling along Baggot Street in Dublin recently, I noticed a plaque at the entrance to the Ibec head office. It showed a circular pattern of dots, … Continue reading Sunflowers and Fibonacci: Models of Efficiency
The High-Power Hypar
Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space --- the hyperbolic paraboloid, or hypar --- pops up in unexpected … Continue reading The High-Power Hypar
The Chaos Game
The term "Chaos Game" was coined by Michael Barnsley [1], who developed this ingenious technique for generating mathematical objects called fractals. We have discussed a particular fractal set on this blog: see Cantor's Ternary Set. The Chaos Game is a simple algorithm that identifies one point in the plane at each stage. The sets of … Continue reading The Chaos Game
The Future of Society: Prosperity or Collapse?
The article in this week’s That’s Maths column in the Irish Times ( TM045 ) is about a mathematical model to simulate the future of society. Our extravagant lifestyle is draining the Earth's natural resources. Population is climbing and climate change looms ever larger. Is the collapse of society imminent? The historical precedents are ominous. Many civilizations … Continue reading The Future of Society: Prosperity or Collapse?
Predator-Prey Models
Next week's post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here. The Lotka-Volterra Model Many ecological process can be modelled by simple systems of equations. An early example of this is the predator-prey model, developed independently by American mathematician Alfred Lotka … Continue reading Predator-Prey Models
The Faraday of Statistics
This week, That’s Maths in The Irish Times ( TM044 ) is about the originator of Students t-distribution. In October 2012 a plaque was unveiled at St Patrick's National School, Blackrock, to commemorate William Sealy Gosset, who had lived nearby for 22 years. Sir Ronald Fisher, a giant among statisticians, called Gosset “The Faraday of … Continue reading The Faraday of Statistics
Breaking Weather Records
In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each … Continue reading Breaking Weather Records
Clothoids Drive Us Round the Bend
The article in this week’s That’s Maths column in the Irish Times ( TM043 ) is about the mathematical curves called clothoids, used in the design of motorways. * * * Next time you travel on a motorway, take heed of the graceful curves and elegant dips and crests of the road. Every twist and … Continue reading Clothoids Drive Us Round the Bend
Rollercoaster Loops
We all know the feeling when a car takes a corner too fast and we are thrown outward by the centrifugal force. This effect is deliberately exploited, and accentuated, in designing rollercoasters: rapid twists and turns, surges and plunges thrill the willing riders. Many modern rollercoasters have vertical loops that take the trains through 360 … Continue reading Rollercoaster Loops
The Predictive Power of Maths
This week, That’s Maths in The Irish Times ( TM042 ) is about the remarkable capacity of mathematics to anticipate physical phenomena that have not yet been observed. The mathematical equations that express the laws of physics describe phenomena seen in the real world. But they also allow us to anticipate completely new phenomena. * … Continue reading The Predictive Power of Maths
Solar System Perturbations
Remarkable progress in understanding the dynamics of the planets has been possible thanks to their relatively small masses and the overwhelming dominance of the Sun. The figure below shows the relative masses of the Sun, planets and some natural satellites, taking the mass of Earth to be unity. The Sun is one million times more … Continue reading Solar System Perturbations
The Unity of Mathematics
This week, That’s Maths in The Irish Times ( TM041 ) is about an ambitious program to unify mathematics. Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have … Continue reading The Unity of Mathematics
The Langlands Program
An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects. Substantial strides have been made in the … Continue reading The Langlands Program
Simulating the Future Climate
The Earth's climate is changing, and the consequences may be very grave. This week, That’s Maths in The Irish Times ( TM040 ) is about computer models for simulating and predicting the future climate. Liffey Bursts its Banks: St. Stephen's Green Flooded Again The above is an improbable but not entirely impossible future headline. Sea … Continue reading Simulating the Future Climate
The Prime Number Theorem
God may not play dice with the Universe, but something strange is going on with the prime numbers [Paul Erdös, paraphrasing Albert Einstein] The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural … Continue reading The Prime Number Theorem
Euclid in Technicolor
The article in this week’s That’s Maths column in the Irish Times ( TM039 ) is about Oliver Byrne's amazing technicolor Elements of Euclid, recently re-published by Taschen. Oliver Byrne (1810–1890), a Victorian civil engineer, was a prolific writer on science. He published more than twenty books on mathematics and several more on mechanics. The … Continue reading Euclid in Technicolor
Speed Cubing & Group Theory
The article in this week’s That’s Maths column in the Irish Times ( TM038 ) is about Rubik's Cube and the Group Theory that underlies its solution. The Rubik's cube craze ran through the world like wildfire in the 1980s. This simple mechanical puzzle is made from small pieces, called “cubies”, in a 3x3x3 structure … Continue reading Speed Cubing & Group Theory
French Curves and Bézier Splines
A French curve is a template, normally plastic, used for manually drawing smooth curves. These simple drafting instruments provided innocent if puerile merriment to generations of engineering students, but they have now been rendered obsolete by computer aided design (CAD) packages, which enable us to construct complicated curves and surfaces using mathematical functions called Bézier … Continue reading French Curves and Bézier Splines
Bézout’s Theorem
Two lines in a plane intersect at one point, a line cuts a circle at two points, a cubic (an S-shaped curve) crosses the x-axis three times and two ellipses, one tall and one squat, intersect in four places. In fact, these four statements may or may not be true. For example, two parallel lines … Continue reading Bézout’s Theorem
Pythagorean triples
The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It can be written as an equation, a2 + b2 = c2, where c is the length of the hypotenuse, and a and b are the lengths of … Continue reading Pythagorean triples
Robots & Biology
The article in this week’s That’s Maths column in the Irish Times ( TM037 ) is about connections between robotics and biological systems via mechanics. The application of mathematics in biology is a flourishing research field. Most living organisms are far too complex to be modelled in their entirety, but great progress is under way … Continue reading Robots & Biology
White Holes in the Kitchen Sink
A tidal bore is a wall of water about a metre high travelling rapidly upstream as the tide floods in. It occurs where the tidal range is large and the estuary is funnel-shaped (see previous post on this blog). The nearest river to Ireland where bores can be regularly seen is the Severn, where favourable … Continue reading White Holes in the Kitchen Sink
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