Science advances by overturning theories, replacing them by better ones. Sometimes, the old theories continue to serve as valuable approximations, as with Newton’s laws of motion [TM260 or search for “thatsmaths” at irishtimes.com]. Sometimes, the older theories become redundant and are forgotten. The theory of phlogiston, a fire-like element released during combustion, and the luminiferous … Continue reading Progress towards a Grand Unified Theory of Mathematics
Tag: Algorithms
Rubik’s Cube Solvable in 20 Moves
In 1974, the brilliant Hungarian professor of architecture, Ernö Rubik, invented the puzzle that has made his name familiar all over the world. When it was mass-produced, from 1980 onwards, the cube became an international craze; vast numbers were made and they brought both great fun and great frustration to millions of children of all … Continue reading Rubik’s Cube Solvable in 20 Moves
Digital Signatures using Edwards Curves
A digital signature is a mathematical means of verifying that an e-document is authentic, that it has come from the claimed sender and that it has not been tampered with or corrupted during transit. Digital signatures are a standard component of cryptographic systems. They use asymetric cryptography that is based on key pairs, consisting of … Continue reading Digital Signatures using Edwards Curves
Maths in Action at the Rugby World Cup
On September 8, I opened The Irish Times to find an A2 Poster with the programme for the Rugby World Cup. The plan showed the twenty teams who must do battle in which ultimate triumph requires survival through the preliminary rounds and victory in quarter-finals, semi-finals and the final climax. We have reached the quarter-finals … Continue reading Maths in Action at the Rugby World Cup
A Simple Formula for the Weekday
People skilled in mental arithmetic sometimes amaze friends and colleagues by calculating the day of the week on which a given date falls. Thus, given a date, say D-Day, which was on 6 June 1944, they quickly announce that it was a Tuesday. Techniques for calculating the day of the week for a given date … Continue reading A Simple Formula for the Weekday
Weather Warnings in Glorious Technicolor
Severe weather affects us all and we need to know when to take action to protect ourselves and our property. We have become familiar with the colourful spectrum of warnings issued by Met Éireann. For several years, Met Éireann has issued warnings of extreme weather. These depend on the severity of the meteorological event and … Continue reading Weather Warnings in Glorious Technicolor
The Power of the 2-gon: Extrapolation to Evaluate Pi
Richardson's extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of $latex {\pi}&fg=000000$, and show how the estimates improve dramatically with higher order extrapolation. [This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).] … Continue reading The Power of the 2-gon: Extrapolation to Evaluate Pi
Making Sound Pictures to Identify Bird Songs
A trained musician can look at a musical score and imagine the sound of an entire orchestra. The score is a visual representation of the sounds. In an analogous way, we can represent birdsong by an image, and analysis of the image can tell us the species of bird singing. This is what happens with … Continue reading Making Sound Pictures to Identify Bird Songs
Why Waffle when One Wordle Do?
Hula hoops were all the rage in 1958. Yo-yos, popular before World War II, were relaunched in the 1960s. Rubik's Cube, invented in 1974, quickly became a global craze. Sudoku, which had been around for years, was wildly popular when it started to appear in American and European newspapers in 2004. The latest fad is … Continue reading Why Waffle when One Wordle Do?
Kalman Filters: from the Moon to the Motorway
Before too long, we will be relieved of the burden of long-distance driving. Given the desired destination and access to a mapping system, electronic algorithms will select the best route and control the autonomous vehicle, constantly monitoring and adjusting its direction and speed of travel. The origins of the methods used for autonomous navigation lie … Continue reading Kalman Filters: from the Moon to the Motorway
Hanoi Graphs and Sierpinski’s Triangle
The Tower of Hanoi is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints: Only one disk can be moved at a time. No disk can be placed upon … Continue reading Hanoi Graphs and Sierpinski’s Triangle
Complexity: are easily-checked problems also easily solved?
From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a … Continue reading Complexity: are easily-checked problems also easily solved?
On what Weekday is Christmas? Use the Doomsday Rule
An old nursery rhyme begins “Monday's child is fair of face / Tuesday's child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born … Continue reading On what Weekday is Christmas? Use the Doomsday Rule
The Monte-Carlo Method
Learning calculus at school, we soon find out that while differentiation is relatively easy, at least for simple functions, integration is hard. So hard indeed that, in many cases, it is impossible to find a nice function that is the integral (or anti-derivative) of a given one. Thus, given $latex {f(x)}&fg=000000$ we can usually find … Continue reading The Monte-Carlo Method
The Mathematics of Fair Play in Video Games
Video games generate worldwide annual sales of about $150 billion. With millions of people confined at home with time to spare, the current pandemic may benefit the industry. At the core of a video game is a computer program capable of simulating a range of phenomena in the real world or in a fantasy universe, … Continue reading The Mathematics of Fair Play in Video Games
Having your Christmas Cake and Eating it
As Christmas approaches, the question of fair sharing comes into focus. Readers can rejoice that there has been a recent breakthrough in cake-cutting theory. Cake cutting may sound limited, but it is important for many practical problems. A cake is a metaphor for a parcel of land to be divided, broadcast frequencies to be allocated, … Continue reading Having your Christmas Cake and Eating it
Airport Baggage Screening with X-Ray Tomography
When you check in your baggage for a flight, it must be screened before it is allowed on the plane. Baggage screening detects threats within luggage and personal belongings by x-ray analysis as they pass along a conveyor belt. Hold-baggage and passenger screening systems are capable of detecting contraband materials, narcotics, explosives and weapons [TM175 … Continue reading Airport Baggage Screening with X-Ray Tomography
Emergence of Complex Behaviour from Simple Roots
It is exhilarating to watch a large flock of birds swarming in ever-changing patterns. Swarming is an emergent behaviour, resulting from a set of simple rules followed by each individual animal, bird or fish, without any centralized control or leadership. A murmuration of starlings is a breathtaking sight, with thousands of birds moving in harmony, … Continue reading Emergence of Complex Behaviour from Simple Roots
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
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
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
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
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
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
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
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
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
Andrey Markov’s Brilliant Ideas are still a Driving Force
Imagine examining the first 20,000 letters of a book, counting frequencies and studying patterns. This is precisely what Andrey Markov did when he analyzed the text of Alexander Pushkin's verse novel Eugene Onegin. This work comprises almost 400 stanzas of iambic tetrameter and is a classic of Russian literature. Markov studied the way vowels and … Continue reading Andrey Markov’s Brilliant Ideas are still a Driving Force
Drawing Multi-focal Ellipses: The Gardener’s Method
Common-or-Garden Ellipses In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, $latex {2a}&fg=000000$, the length of the major axis. The gardener puts down two stakes … Continue reading Drawing Multi-focal Ellipses: The Gardener’s Method
Locating the HQ with Multi-focal Ellipses
Motivation Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs … Continue reading Locating the HQ with Multi-focal Ellipses
Fractal Complexity of Finnegans Wake
Tomorrow we celebrate Bloomsday, the day of action in Ulysses. Most of us regard Joyce's singular book as a masterpiece, even if we have not read it. In contrast, Finnegans Wake is considered by some as a work of exceptional genius, by others as impenetrable bafflegab [See TM117 or search for “thatsmaths” at irishtimes.com]. Sentence Length … Continue reading Fractal Complexity of Finnegans Wake
When Roughly Right is Good Enough
How high is Liberty Hall? How fast does human hair grow? How many A4 sheets of paper would cover Ireland? How many people in the world are talking on their mobile phones right now? These questions seem impossible to answer but, using basic knowledge and simple logic, we can make a good guess at the … Continue reading When Roughly Right is Good Enough
Voronoi Diagrams: Simple but Powerful
We frequently need to find the nearest hospital, surgery or supermarket. A map divided into cells, each cell covering the region closest to a particular centre, can assist us in our quest. Such a map is called a Voronoi diagram, named for Georgy Voronoi, a mathematician born in Ukraine in 1868. He is remembered today … Continue reading Voronoi Diagrams: Simple but Powerful
The Shaky Foundations of Mathematics
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for … Continue reading The Shaky Foundations of Mathematics
A Toy Example of RSA Encryption
The RSA system has been presented many times, following the excellent expository article of Martin Gardner in the August 1977 issue of Scientific American. There is no need for yet another explanation of the system; the essentials are contained in the Wikipedia article RSA (cryptosystem), and in many other articles. The purpose of this note … Continue reading A Toy Example of RSA Encryption
Can Mathematics Keep Us Secure?
The National Security Agency is the largest employer of mathematicians in America. Mathematics is a core discipline at NSA and mathematicians work on signals intelligence and information security (US citizenship is a requirement for employment). Why is NSA so interested in mathematics? [See TM096, or search for “thatsmaths” at irishtimes.com]. Many actions are easy to … Continue reading Can Mathematics Keep Us Secure?
Lateral Thinking in Mathematics
Many problems in mathematics that appear difficult to solve turn out to be remarkably simple when looked at from a new perspective. George Pólya, a Hungarian-born mathematician, wrote a popular book, How to Solve It, in which he discussed the benefits of attacking problems from a variety of angles [see TM094, or search for “thatsmaths” … Continue reading Lateral Thinking in Mathematics
Big Data: the Information Explosion
The world is awash with data. Large data sets have been available for many decades but in recent years their volumes have grown explosively. With mobile devices and internet connections data capture is simple and with powerful computers the analysis of “big data” is feasible [see TM092, or search for “thatsmaths” at irishtimes.com]. But there are … Continue reading Big Data: the Information Explosion
Computus: Dating the Resurrection
Whatever the weather, St Patrick's Day occurs on the same date every year. In contrast, Easter springs back and forth in an apparently chaotic manner. The date on which the Resurrection is celebrated is determined by a complicated convolution of astronomy, mathematics and theology, an algorithm or recipe that fixes the date in accordance with … Continue reading Computus: Dating the Resurrection
Peano Music
The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel's incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the … Continue reading Peano Music
The Mathematics of Voting
Selection of leaders by voting has a history reaching back to the Athenian democracy. Elections are essentially arithmetical exercises, but they involve more than simple counting, and have some subtle mathematical aspects [TM085, or search for “thatsmaths” at irishtimes.com]. The scientific study of voting and elections, which began around the time of the French Revolution, is called … Continue reading The Mathematics of Voting
Prime Number Record Smashed Again
Once again the record for the largest prime number has been shattered. As with all recent records, the new number is a Mersenne prime, a number of the form Mp = 2p – 1 where p itself is a prime. Participants in a distributed computing project called GIMPS (Great Internet Mersenne Prime Search) continue without … Continue reading Prime Number Record Smashed Again
Entropy Piano Tuning
An ingenious method of tuning pianos, based on the concept of entropy, has recently been devised by Haye Hinrichsen of Würzburg University. Entropy, which first appeared in the mid-nineteenth century in thermodynamics and later in statistical mechanics, is a measure of disorder. Around 1948 Claude Shannon developed a mathematical theory of communications and used entropy … Continue reading Entropy Piano Tuning
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
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?
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
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