The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. But it is not obvious how to construct a list that is sure to contain every rational number precisely once. In a previous post we described the Farey Sequences. Here we examine another, related, approach. The Stern-Brocot Tree We … Continue reading Listing the Rational Numbers II: The Stern-Brocot Tree
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
Listing the Rational Numbers: I. Farey Sequences
We know, thanks to Georg Cantor, that the rational numbers --- ratios of integers --- are countable: they can be put into one-to-one correspondence with the natural numbers. How can we make a list that includes all rationals? For the present, let us consider rationals in the interval $latex {[0,1]}&fg=000000$. It would be nice if … Continue reading Listing the Rational Numbers: I. Farey Sequences
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
A Trapezoidal Prism on the Serpentine
Walking in Hyde Park recently, I spied what appeared to be a huge red pyramid in the middle of the Serpentine. On closer approach, and with a changing angle of view, it became clear that it was prismatic in shape, composed of numerous barrels in red, blue and purple. An isoceles trapezoidal prism A prism … Continue reading A Trapezoidal Prism on the Serpentine
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
A Zero-Order Front
Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions. Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed. Life-cycle … Continue reading A Zero-Order Front
The Flight of the Bumble Bee
Alice and Bob, initially a distance l apart, walk towards each other, each at a speed w. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed f until Alice and Bob meet. How far does the bumble bee fly? There … Continue reading The Flight of the Bumble Bee
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
Euler’s “Degree of Agreeableness” for Musical Chords
The links between music and mathematics stretch back to Pythagoras and many leading mathematicians have studied the theory of music. Music and mathematics were pillars of the Quadrivium, the four-fold way that formed the basis of higher education for thousands of years. Music was a central theme for Johannes Kepler in his Harmonices Mundi – … Continue reading Euler’s “Degree of Agreeableness” for Musical Chords
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
Grandi’s Series: A Second Look
In an earlier post, we discussed Grandi's series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series $latex \displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots &fg=000000$ This is a divergent series: the sequence of partial sums is $latex {\{ 1, … Continue reading Grandi’s Series: A Second Look
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
Grandi’s Series: Divergent but Summable
Is the Light On or Off? Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states ``on'' and … Continue reading Grandi’s Series: Divergent but Summable
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
Numbers with Nines
What proportion of all numbers less than a given size N have a 9 in their decimal expansion? A naive argument would be that, since 9 is one of ten distinct digits, the answer must be about 10%. But this is not "remotely close" to the true answer. Counting the Nines It is a simple … Continue reading Numbers with Nines
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
“Dividends and Divisors Ever Diminishing”
Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week's ThatsMaths post] In Ithaca, the narrator takes every … Continue reading “Dividends and Divisors Ever Diminishing”
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
Motifs: Molecules of Music
Motif: A short musical unit, usually just few notes, used again and again. A recurrent short phrase that is developed in the course of a composition. A motif in music is a small group of notes encapsulating an idea or theme. It often contains the essence of the composition. For example, the opening four notes … Continue reading Motifs: Molecules of Music
A Glowing Geometric Proof that Root-2 is Irrational
It was a great shock to the Pythagoreans to discover that the diagonal of a unit square could not be expressed as a ratio of whole numbers. This discovery represented a fundamental fracture between the mathematical domains of Arithmetic and Geometry: since the Greeks recognized only whole numbers and ratios of whole numbers, the result … Continue reading A Glowing Geometric Proof that Root-2 is Irrational
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
Marden’s Marvel
Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden's Theorem, published in 1945, is one such -- delightful -- result. For centuries, mathematicians have struggled to find roots of polynomials like p(x) ≡ xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + … a1 x … Continue reading Marden’s Marvel
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
Waves Packed in Envelopes
In this article we take a look at group velocity and at the extraction of the envelope of a wave packet using the ideas of the Hilbert transform. Interference of two waves A single sinusoidal wave is infinite in extent and periodic in space and time. When waves interact, the dynamics are more interesting. The … Continue reading Waves Packed in Envelopes
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
Geodesics on the Spheroidal Earth – I
Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not? The General Equation for Geodesics Open a typical text … Continue reading Geodesics on the Spheroidal Earth – I
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
Fourier’s Wonderful Idea – I
Breaking Complex Objects into Simple Pieces ``In a memorable session of the French Academy on the 21st of December 1807, the mathematician and engineer Joseph Fourier announced a thesis which inaugurated a new chapter in the history of mathematics. The claim of Fourier appeared to the older members of the Academy, including the great analyst … Continue reading Fourier’s Wonderful Idea – I
Sophus Lie
“It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002). Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no … Continue reading Sophus Lie
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
Subtract 0 and divide by 1
We all know that division by zero is a prohibited operation, and that ratios that reduce to ``zero divided by zero'' are indeterminate. We probably also recall proving in elementary calculus class that $latex \displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1 &fg=000000$ This is an essential step in deriving an expression for the derivative of … Continue reading Subtract 0 and divide by 1
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
The Evolute: Envelope of Normals
Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute. Suppose the curve $latex {\gamma}&fg=000000$ is specified in parametric form $latex {(x(t), y(t))}&fg=000000$ for $latex {t \in [0,1]}&fg=000000$. The centre of curvature $latex {\Gamma = (X, Y)}&fg=000000$ at a particular point … Continue reading The Evolute: Envelope of Normals
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
Hardy’s Apology
Godfrey Harold Hardy's memoir, A Mathematician's Apology, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of … Continue reading Hardy’s Apology
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
Kaprekar’s Number 6174
The Indian mathematician D. R. Kaprekar spent many happy hours during his youth solving mathematical puzzles. He graduated from Fergusson College in Pune in 1929 and became a mathematical teacher at a school in Devlali, north-east of Mumbai. Kaprekar is remembered today for a range of curious mathematical patterns that he discovered. The best known … Continue reading Kaprekar’s Number 6174
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
Moebiquity: Ubiquity and Versitility of the Möbius Band
The Möbius strip or Möbius band, with one side and one edge, has been a source of fascination since its discovery in 1858, independently by August Möbius and Johann Listing. It is easily formed from a strip of paper by giving it a half-twist before joining the ends. The Möbius band may be characterised in … Continue reading Moebiquity: Ubiquity and Versitility of the Möbius Band
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
Doughnuts and Tonnetze
The circle of fifths is a remarkably useful diagram for the analysis of music. It shows the twelve notes of the chromatic scale arranged in a circle, with notes that are harmonically related (like C and G) being close together and notes that are discordant (like C and C♯) more distant from each other. The … Continue reading Doughnuts and Tonnetze
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
Vanishing Hyperballs
We all know that the area of a disk --- the interior of a circle --- is $latex {\pi r^2}&fg=000000$ where $latex {r}&fg=000000$ is the radius. Some of us may also remember that the volume of a ball --- the interior of a sphere --- is $latex {\frac{4}{3}\pi r^3}&fg=000000$. The unit disk and ball have … Continue reading Vanishing Hyperballs
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?
Disentangling Loops with an Ambient Isotopy
The surface in the left panel above has two linked loops. In the right hand panel, the loops are unlinked. Is it possible to continuously distort the left-hand surface so as to unlink the loops and produce the right-hand figure? This seems impossible, but intuition is not always reliable. Knot Theory A knot is an … Continue reading Disentangling Loops with an Ambient Isotopy
A Symbol for Global Circulation
The recycling symbol consisting of three bent arrows is found on bottles, cartons and packaging of all kinds. It originated in 1970 when the Chicago-based Container Corporation of America (CCA) held a competition for the design of a symbol suitable for printing on cartons, to encourage recycling and re-use of packaging materials. The competition for … Continue reading A Symbol for Global Circulation
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
More on Moduli
We wrote last week on modular arithmetic, the arithmetic of remainders. Here we will examine a few other aspects of this huge subject. Modular arithmetic was advanced by Gauss in his Disquisitiones Arithmeticae. In this system, number wrap around when they reach a point known as the modulus. Numbers that differ by a multiple of … Continue reading More on Moduli
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
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