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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 already bright before most of us wake up but, in Winter, the mornings would be too dark unless we reverted to Standard Time.

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 value to them. [TM149, or search for “thatsmaths” at].


In reality, mathematics is essential in numerous contexts: the diversity is remarkable, and you may be surprised how maths plays a vital role in the everyday work of so many people.

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Listing the Rational Numbers II: The Stern-Brocot Tree

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.

Mediant-red Continue reading ‘Listing the Rational Numbers II: The Stern-Brocot Tree’

The Many Modern Uses of Quaternions

Hamiltons-Bridge-PlaqueThe 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 of years of intense effort. The great French mathematician Henri Poincaré also described how sudden inspiration occurs unexpectedly, but always following a period of concentrated research [TM148, or search for “thatsmaths” at].

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.


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” at].

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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.


Changing perspective on approach to the Mastaba

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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].


Jimmy Wales, co-founder of Wikipedia, answering a question. Face detection indicated by squares.

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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.

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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?


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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 terms, a spiral is formed by a point moving around a fixed centre while its distance increases or decreases as it revolves [see TM145, or search for “thatsmaths” at].


The spiral sandbank on Booterstown strand (satellite image digitally enhanced by Andrew Lynch).

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Euler’s “Degree of Agreeableness” for Musical Chords

Euler-10_Swiss_Franc_banknoteThe 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 – Harmony of the World, and René Descartes’ first work was a compendium of music.

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].


Animation of tide prediction machine, showing outputs for New York (semi-diurnal tides) and Kuril Islands (diurnal tides) [Source: American Mathematical Society (see below)].

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

\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots

This is a divergent series: the sequence of partial sums is {\{ 1, 0, 1, 0, 1, 0, \dots \}}, which obviously does not converge, but alternates between {0} and {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 like any other. [see TM143, or search for “thatsmaths” at].


A selection of images of zero (google images).

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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 “off” by {1} and {0}, the sequence of states over the first minute is {\{ 1, 0, 1, 0, 1, 0, \dots \}}. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.


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].


Trig pillar on summit of Croaghan Moira, Wicklow [Image from

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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.

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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 just 15 years, between 447 and 432 BC. [TM141 or search for “thatsmaths” at].


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“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]


Joyce in Zurich: did he meet Zermelo?

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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].


Joyce’s examination marks [archives of the National University of Ireland].

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 of Beethoven’s Fifth Symphony express a musical idea that is repeated throughout the symphony. 


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A Glowing Geometric Proof that Root-2 is Irrational

Tennenbaum-00It 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 meant that there was no number to describe the diagonal of a unit square.

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].


Central Display at the Science Museum

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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.


Cubic with roots at x=1, x=2 and x=3.

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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 wide range of applications. He once joked that he was “a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points” [TM138 or search for “thatsmaths” at].


Operation Castle, Bikini Atoll, 1954

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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.


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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 astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at].


Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.

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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?


A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

Continue reading ‘Geodesics on the Spheroidal Earth – I’

Fourier’s Wonderful Idea – II

Solving PDEs by a Roundabout Route


Joseph Fourier (1768-1830)

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 in frequency  [TM136 or search for “thatsmaths” at].

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 Lagrange, entirely incredible.”



Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

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 (1842-1899)

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 hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

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Cubic Skulduggery & Intrigue


Solution of a cubic equation, usually called Cardano’s formula.

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 a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at].

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

\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1

This is an essential step in deriving an expression for the derivative of {\sin x}.


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Reducing R-naught to stem the spread of Epidemics

Vaccine-1We 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 SIR model with just three variables to highly complex simulation models with millions of variables [TM134 or search for “thatsmaths” at]. 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.


Sin t (blue) and its evolute (red).

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 1564, just 454 years ago today [TM133 or search for “thatsmaths” at].


Image: NASA

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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 Hardy’s opinions are difficult to support and some of his predictions have turned out to be utterly wrong, the book is still well worth reading.


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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 process called advection. Normally, the two mechanisms act together, sometimes negating and sometimes reinforcing each other. What is true for temperature is also true for other quantities: pressure, density, humidity and even the flow velocity itself. This last effect may be described by saying that “the wind blows the wind” [TM132 or search for “thatsmaths” at].


Hurricane Ophelia approaching Ireland, 16 October 2017, 1200Z. Image from

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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 process for three digit numbers converging to 495 [Wikimedia Commons].

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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 malfunction [TM131 or search for “thatsmaths” at].


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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.


Möbius band in 3-space and a flat representation in 2-space.

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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 the flow of a viscous fluid, such as treacle pouring off a spoon, the velocity is smooth and steady. Such flow is called laminar, and variations of velocity from place to place are small. By contrast, the motion of the atmosphere, a fluid with low viscosity, can be irregular and rapidly fluctuating. We experience this when out and about on a gusty day. Such chaotic fluid flow is called turbulence, and this topic continues to challenge the most brilliant scientists [TM130 or search for “thatsmaths” at].


Vincent Van Gogh’s Starry Night.

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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 Tonnetz diagram (note that the arrangement here is inverted relative to that used in the text.  It appears that there is no rigid standard, and several arrangements are in use) [Image from WikimediaCommons].

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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 curious behaviour is due to the unsteady path of the Earth around the Sun. Our clocks, which run regularly at what is called mean time, move in and out of synchronization with solar time [TM129 or search for “thatsmaths” at].


Sunrise in Newgrange on winter solstice [image from

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Vanishing Hyperballs


Spherical ball contained within a cubic region
[Image from ].

We all know that the area of a disk — the interior of a circle — is {\pi r^2} where {r} is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is {\frac{4}{3}\pi r^3}.

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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 or Magi were astrologers, experts in celestial matters, and would have drawn inferences from what they observed in the sky [TM128 or search for “thatsmaths” at].

Star-of-Bethlehem-1 Continue reading ‘The Star of Bethlehem … or was it a Planet?’

Disentangling Loops with an Ambient Isotopy


Can one of these shapes be continuously distorted to produce the other?

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. 

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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.


Original (Moebius) and a variation (3-twist) of the universal recycling symbol.

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 for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.


Artist’s impression of OSIRIS-REx orbiting Bennu [Photo Credit: NASA]

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