The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed. The Reynolds number is defined as Re = VL/ν where V is the flow speed, L the length … Continue reading Life’s a Drag Crisis
Mathematics Solving Crimes
What use is maths? Why should we learn it? A forensic scientist could answer that virtually all the mathematics we learn at school is used to solve crimes. Forensic science considers physical evidence relating to criminal activity and practitioners need competence in mathematics as well as in the physical, chemical and biological sciences [TM080: search … Continue reading Mathematics Solving Crimes
Numbering the Family Tree
The availability of large historical data sets online has spurred interest in genealogy and family history. Anyone who has assembled information knows how important it is to organize it systematically. A simple family tree showing the direct ancestors of Wanda One is shown here: This has just three generations but, as more people are added, … Continue reading Numbering the Family Tree
Melencolia: An Enigma for Half a Millennium
Albrecht Dürer, master painter and engraver of the German Renaissance, made his Melencolia I in 1514, just over five centuries ago. It is one of the most brilliant engravings of all time, and amongst the most intensively debated works of art [TM079; or search for “thatsmaths” at irishtimes.com ]. The winged figure, Melancholy, sits in a … Continue reading Melencolia: An Enigma for Half a Millennium
Mowing the Lawn in Spirals
Like a circle in a spiral / Like a wheel within a wheel / Never ending or beginning / On an ever-spinning reel. The Windmills Of Your Mind Broadly speaking, a spiral curve originates at a central point and gets further away (or closer) as it revolves around the point. Spirals abound in nature, being … Continue reading Mowing the Lawn in Spirals
A Few Wild Functions
Sine Function: $latex {\mathbf{y=\sin x}}&fg=000000$ The function $latex {y=\sin x}&fg=000000$ is beautifully behaved, oscillating regularly along the entire real line $latex {\mathbb{R}}&fg=000000$ (it is also well-behaved for complex $latex {x}&fg=000000$ but we won't consider that here). Chirp Function: $latex {\mathbf{y=\sin x^2}}&fg=000000$ Now $latex {y=\sin x^2}&fg=000000$ is also well-behaved: its oscillations become more rapid as $latex … Continue reading A Few Wild Functions
It’s a Small – Networked – World
Networks are everywhere in the modern world. They may be physical constructs, like the transport system or power grid, or more abstract entities like family trees or the World Wide Web. A network is a collection of nodes linked together, like cities connected by roads or people genetically related to each other. Such a system … Continue reading It’s a Small – Networked – World
Which Way did the Bicycle Go?
``A bicycle, certainly, but not the bicycle," said Holmes. In Conan-Doyle's short story The Adventure of the Priory School Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal … Continue reading Which Way did the Bicycle Go?
New Tricks: No Clicks
The quality of music recordings on compact discs or CDs is excellent. In the age of vinyl records, irritating clicks resulting from surface scratches were almost impossible to avoid. Modern recording media are largely free from this shortcoming. But this is curious: there are many reasons why CD music can be contaminated: dirt on the … Continue reading New Tricks: No Clicks
Hamming’s Smart Error-correcting Codes
In the late 1940s, Richard Hamming, working at Bell Labs, was exasperated with the high level of errors occurring in the electro-mechanical computing equipment he was using. Punched card machines were constantly misreading, forcing him to restart his programs. He decided to do something about it. This was when error-correcting codes were invented. A simple … Continue reading Hamming’s Smart Error-correcting Codes
The Ubiquitous Cycloid
Puzzle: However fast a train is travelling, part of it is moving backwards. Which part? For the answer, see the end of this post. Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would … Continue reading The Ubiquitous Cycloid
Holbein’s Anamorphic Skull
Hans Holbein the Younger, court painter during the reign of Henry VIII, produced some spectacular works. Amongst the most celebrated is a double portrait of Jean de Dinteville, French Ambassador to Henry's court, and Georges de Selve, Bishop of Lavaur. Painted by Holbein in 1533, the picture, known as The Ambassadors, hangs in the National … Continue reading Holbein’s Anamorphic Skull
James Joseph Sylvester
James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at irishtimes.com ] Sylvester's mathematical talents became evident at … Continue reading James Joseph Sylvester
Thomas Harriot: Mathematician, Astronomer and Navigator
Sir Walter Raleigh, adventurer, explorer and privateer, was among most colourful characters of Tudor times. He acquired extensive estates in Waterford and Cork, including Molana Abbey near Youghal, which he gave to his friend and advisor, the brilliant mathematician and astronomer Thomas Harriot. Raleigh needed an excellent navigator on his transatlantic voyages, and he brought … Continue reading Thomas Harriot: Mathematician, Astronomer and Navigator
The Great American Eclipse
Just two years from now, on Monday, August 21, 2017, the Moon's shadow will sweep across the United States at a speed of over 2,000 km/hr. The Great American Eclipse of 2017 will generate a frenzy of activity. [TM074: search for “thatsmaths” at irishtimes.com ]. Solar eclipses are not especially rare, but this one is of … Continue reading The Great American Eclipse
Buffon was no Buffoon
The Buffon Needle method of estimating $latex {\pi}&fg=000000$ is hopelessly inefficient. With one million throws of the needle we might expect to get an approximation accurate to about three digits. The idea is more of philosophical than of practical interest. Buffon never envisaged it as a means of computing $latex {\pi}&fg=000000$. Buffon and his Sticks … Continue reading Buffon was no Buffoon
The Bridges of Paris
Leonhard Euler considered a problem known as The Seven Bridges of Königsberg. It involves a walk around the city now known as Kaliningrad, in the Russian exclave between Poland and Lithuania. Since Kaliningrad is out of the way for most of us, let's have a look closer to home, at the bridges of Paris. [TM073: … Continue reading The Bridges of Paris
Who Needs EirCode?
The idea of using two numbers to identify a position on the Earth's surface is very old. The Greek astronomer Hipparchus (190–120 BC) was the first to specify location using latitude and longitude. However, while latitude could be measured relatively easily, the accurate determination of longitude was more difficult, especially for sailors out of site … Continue reading Who Needs EirCode?
Bent Coins: What are the Odds?
If we toss a `fair' coin, one for which heads and tails are equally likely, a large number of times, we expect approximately equal numbers of heads and tails. But what is `approximate' here? How large a deviation from equal values might raise suspicion that the coin is biased? Surely, 12 heads and 8 tails … Continue reading Bent Coins: What are the Odds?
RT60 and Acoustic Excellence
This week’s That’s Maths column (TM072) [search for “thatsmaths” at irishtimes.com] is about architectural acoustics, and about the remarkable work of Wallace Clement Sabine. Attending a mathematical seminar in UCD recently, I could understand hardly a word. The problem lay not with the arcane mathematics but with the poor acoustics of the room. The lecturer … Continue reading RT60 and Acoustic Excellence
Fun and Games on a Honeycombed Rhomboard.
Hex is an amusing game for two players, using a board or sheet of paper divided into hexagonal cells like a honeycomb. The playing board is rhomboidal in shape with an equal number of hexagons along each edge. Players take turns placing a counter or stone on a single cell of the board. One uses … Continue reading Fun and Games on a Honeycombed Rhomboard.
Pluto’s Unruly Family
An astrodynamical miracle is happening in the sky above. Our ability to launch a space-probe from the revolving Earth to reach a moving target billions of kilometres away almost ten years later with pinpoint accuracy is truly astounding. "New Horizons" promises to enhance our knowledge of the solar system and it may help us to … Continue reading Pluto’s Unruly Family
Increasingly Abstract Algebra
In the seventeenth century, the algebraic approach to geometry proved to be enormously fruitful. When René Descartes (1596-1650) developed coordinate geometry, the study of equations (algebra) and shapes (geometry) became inextricably interlinked. The move towards greater abstraction can make mathematics appear more abstruse and impenetrable, but it brings greater clarity and power, and can lead … Continue reading Increasingly Abstract Algebra
Emmy Noether’s beautiful theorem
The number of women who have excelled in mathematics is lamentably small. Many reasons may be given, foremost being that the rules of society well into the twentieth century debarred women from any leading role in mathematics and indeed in science. But a handful of women broke through the gender barrier and made major contributions. … Continue reading Emmy Noether’s beautiful theorem
Game Theory & Nash Equilibrium
Game theory deals with mathematical models of situations involving conflict, cooperation and competition. Such situations are central in the social and behavioural sciences. Game Theory is a framework for making rational decisions in many fields: economics, political science, psychology, computer science and biology. It is also used in industry, for decisions on manufacturing, distribution, consumption, … Continue reading Game Theory & Nash Equilibrium
The Tragic Demise of a Beautiful Mind
John Nash, who was the subject of the book and film A Beautiful Mind, won the Abel Prize recently. But his journey home from the award ceremony in Norway ended in tragedy [see this week’s That’s Maths column (TM069): search for “thatsmaths” at irishtimes.com]. We learn at school how to solve polynomial equations of first … Continue reading The Tragic Demise of a Beautiful Mind
Maps on the Web
In a nutshell: In web maps, geographical coordinates are projected as if the Earth were a perfect sphere. The results are great for general use but not for high-precision applications. Mercator's Projection Cartographers have devised numerous projections, each having advantages and drawbacks. No one projection is ideal. Mercator's projection is based on a cylinder tangent … Continue reading Maps on the Web
Mercator’s Marvellous Map
Try to wrap a football in aluminium foil and you will discover that you have to crumple up the foil to make it fit snugly to the ball. In the same way, it is impossible to represent the curved surface of the Earth on a flat plane without some distortion. [See this week’s That’s Maths … Continue reading Mercator’s Marvellous Map
Eccentric Pizza Slices
Suppose six friends visit a pizzeria and have enough cash for just one big pizza. They need to divide it fairly into six equal pieces. That is simple: cut the pizza in the usual way into six equal sectors. But suppose there is meat in the centre of the pizza and some of the friends … Continue reading Eccentric Pizza Slices
Modelling the Markets
Mathematics now plays a fundamental role in modelling market movements [see this week’s That’s Maths column (TM067) or search for “thatsmaths” at irishtimes.com]. The state of the stock market displayed on a trader's screen is history. Big changes can occur in the fraction of a second that it takes for information to reach the screen. … Continue reading Modelling the Markets
Brouwer’s Fixed-Point Theorem
A climber sets out at 8 a.m. from sea-level, reaching his goal, a 2,000 metre peak, ten hours later. He camps at the summit and starts his return the next morning at 8 a.m. After a leisurely descent, he is back at sea-level ten hours later. Is there some time of day at which his … Continue reading Brouwer’s Fixed-Point Theorem
Tap-tap-tap the Cosine Button
Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not … Continue reading Tap-tap-tap the Cosine Button
For Good Comms, Leaky Cables are Best
A counter-intuitive result of Oliver Heaviside showed how telegraph cables should be designed [see this week’s That’s Maths column (TM066) or search for “thatsmaths” at irishtimes.com]. Robert Halpin In Wicklow town an obelisk commemorates Robert Halpin, a Master Mariner born at the nearby Bridge Tavern. Halpin, one of the more important mariners of the nineteenth … Continue reading For Good Comms, Leaky Cables are Best
The Hodograph
The Hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy. Hodographs are valuable in fluid dynamics, astronomy and meteorology. The idea of a hodograph is very simple. … Continue reading The Hodograph
Mode-S: Aircraft Data improves Weather Forecasts
A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com]. It has often happened that an instrument designed for one purpose has proved invaluable for another. Galileo observed the regular swinging of a pendulum. Christiaan Huygens derived a mathematical … Continue reading Mode-S: Aircraft Data improves Weather Forecasts
Golden Moments
Suppose a circle is divided by two radii and the two arcs a and b are in the golden ratio: b / a = ( a + b ) / b = φ ≈ 1.618 Then the smaller angle formed by the radii is called the golden angle. It is equal to about 137.5° or … Continue reading Golden Moments
You Can Do Maths
Bragging about mathematical ineptitude is not cool. There is nothing admirable about ignorance and incompetence. Moreover, everyone thinks mathematically all the time, even if they are not aware of it. Can we all do maths? Yes, we can! [See this week’s That’s Maths column (TM064) or search for “thatsmaths” at irishtimes.com]. We use simple arithmetic … Continue reading You Can Do Maths
A King of Infinite Space: Euclid I.
O God, I could be bounded in a nutshell, and count myself a king of infinite space ... [Hamlet] The Elements – far and away the most successful textbook ever written – is not just a great mathematics book. It is a great book. There is nothing personal in the book, nothing to give any clue as … Continue reading A King of Infinite Space: Euclid I.
Café Mathematics in Lvov
For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at irishtimes.com). Zygmunt Janeszewski, who … Continue reading Café Mathematics in Lvov
The Birth of Functional Analysis
Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. … Continue reading The Birth of Functional Analysis
MGP: Tracing our Mathematical Ancestry
There is great public interest in genealogy. Many of us live in hope of identifying some illustrious forebear, or enjoy the frisson of having a notorious murderer somewhere in our family tree. Academic genealogies can also be traced: see this week’s That’s Maths column in The Irish Times (TM062, or search for “thatsmaths” at irishtimes.com). … Continue reading MGP: Tracing our Mathematical Ancestry
The Klein 4-Group
What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations? Answer: $latex {K_4}&fg=000000$. Symmetries of a Book --- or a Brick Take a book, place it on the table and draw a rectangle around it. How many ways can the book fit into the rectangle? Clearly, once any … Continue reading The Klein 4-Group
Perelman’s Theorem: Who Wants to be a Millionaire?
This week’s That’s Maths column in The Irish Times (TM061, or search for “thatsmaths” at irishtimes.com) is about the remarkable mathematician Grisha Perelman and his proof of a one-hundred year old conjecture. Topology During the twentieth century topology emerged as one of the pillars of mathematics, alongside algebra and analysis. Geometers consider lengths, angles and … Continue reading Perelman’s Theorem: Who Wants to be a Millionaire?
The Steiner Minimal Tree
Steiner's minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner's problem allows for additional points – now called Steiner points – to be added … Continue reading The Steiner Minimal Tree
Plateau’s Problem and Double Bubbles
Bubbles floating in the air strive to achieve a spherical form. Large bubbles may oscillate widely about this ideal whereas small bubbles quickly achieve their equilibrium shape. The sphere is optimal: it encloses maximum volume for any surface of a given area. This was stated by Archimedes, but he did not have the mathematical techniques … Continue reading Plateau’s Problem and Double Bubbles
Barcodes and QR Codes: Zebra stripes and Leopard spots
Barcodes and QR codes are described in this week’s That’s Maths column in The Irish Times (TM060, or search for “thatsmaths” at irishtimes.com). Virtually everything that you buy in your local supermarket has a curious little zebra-like pattern the size of a postage stamp printed on it. Barcodes, originally devised about forty years ago to … Continue reading Barcodes and QR Codes: Zebra stripes and Leopard spots
Seifert Surfaces for Knots and Links.
We are all familiar with knots. Knots keep our boats securely moored and enable us to sail across the oceans. They also reduce the cables and wires behind our computers to a tangled mess. Many fabrics are just complicated knots of fibre and we know how they can unravel. If the ends of a rope … Continue reading Seifert Surfaces for Knots and Links.
The MacTutor Archive
The MacTutor History of Mathematics archive is a website hosted by the University of St Andrews in Scotland. It was established, and is maintained, by Dr John O'Connor and Prof Edmund Robertson of the School of Mathematics and Statistics at St Andrews. MacTutor contains biographies of a large number of mathematicians, both historical and contemporary. … Continue reading The MacTutor Archive
2014 in review
2014 annual report of WordPress.com for this blog. Here's an excerpt: The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 37,000 times in 2014. If it were a concert at Sydney Opera House, it would take about 14 sold-out performances for that many people to see it. Click here … Continue reading 2014 in review
Fermat’s Christmas Theorem
Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions. Girard also showed how the area of a spherical triangle depends on its interior angles. If the angles of a triangle on the unit sphere … Continue reading Fermat’s Christmas Theorem
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