Which is greater, $latex {x^y}&fg=000000$ or $latex {y^x}&fg=000000$? Of course, it depends on the values of x and y. We might consider a particular case: Is $latex {e^\pi > \pi^e}&fg=000000$ or $latex {\pi^e > e^\pi}&fg=000000$? We assume that $latex {x}&fg=000000$ and $latex {y}&fg=000000$ are positive real numbers, and plot the function $latex \displaystyle z(x,y) = … Continue reading Which is larger, e^pi or pi^e?
A New Window on the World
The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. … Continue reading A New Window on the World
That’s Maths Book Published
A book of mathematical articles, That's Maths, has just been published. The collection of 100 articles includes pieces that have appeared in The Irish Times over the past few years, blog posts from this website and a number of articles that have not appeared before. The book has been published by Gill Books and copies … Continue reading That’s Maths Book Published
Thank Heaven for Turbulence
The chaotic flow of water cascading down a mountainside is known as turbulence. It is complex, irregular and unpredictable, but we should count our blessings that it exists. Without turbulence, we would gasp for breath, struggling to absorb oxygen or be asphyxiated by the noxious fumes belching from motorcars, since pollutants would not be dispersed … Continue reading Thank Heaven for Turbulence
Kepler’s Magnificent Mysterium Cosmographicum
Johannes Kepler's amazing book, Mysterium Cosmographicum, was published in 1596. Kepler's central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer … Continue reading Kepler’s Magnificent Mysterium Cosmographicum
A Ton of Wonders
Every number is interesting. Suppose there were uninteresting numbers. Then there would be a smallest one. But this is an interesting property, contradicting the supposition. By reductio ad absurdum, there are none! This is the hundredth “That's Maths” article to appear in The Irish Times [TM100, or search for “thatsmaths” at irishtimes.com]. To celebrate the … Continue reading A Ton of Wonders
Negative Number Names
The counting numbers that we learn as children are so familiar that using them is second nature. They bear the appropriate name natural numbers. From then on, names of numbers become less and less apposite. The integers include all whole numbers, both negative and positive. We are heading into difficulty: the term negative is so … Continue reading Negative Number Names
Venn Again’s Awake
We wrote about the basic properties of Venn diagrams in an earlier post. Now we take a deeper look. John Venn, a logician and philosopher, born in Hull, Yorkshire in 1834, introduced the diagrams in a paper in 1880 and in his book Symbolic Logic, published one year later. The diagrams were used long before … Continue reading Venn Again’s Awake
The next Hamilton
William Rowan Hamilton was Ireland's greatest mathematician. His name is heard thousands of times every day throughout the world when researchers use the Hamiltonian function that encapsulates the dynamics of a vast range of physical systems. He achieved fame early in life and remains one of the all-time great scientists. [TM099, or search for “thatsmaths” at … Continue reading The next Hamilton
Heron’s Theorem: a Tool for Surveyors
Heron was one of the great Greek mathematicians of Alexandria, following in the tradition of Euclid, Archimedes, Eratosthenes and Apollonius. He lived in the first century, from about AD 10 to AD 70. His interests were in practical rather than theoretical mathematics and he wrote on measurement, mechanics and engineering. He devised a steam-powered device … Continue reading Heron’s Theorem: a Tool for Surveyors
The Tunnel of Eupalinos in Samos
The tunnel of Eupalinos on the Greek island of Samos, over one kilometre in length, is one of the greatest engineering achievements of the ancient world [TM098, or search for “thatsmaths” at irishtimes.com]. Approximate course of the tunnel of Eupalinos in Samos. Modern Tunnels The Gotthard Base Tunnel opened in June and will be fully … Continue reading The Tunnel of Eupalinos in Samos
Slicing Doughnuts
It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed … Continue reading Slicing Doughnuts
Recreational Mathematics is Fun
We all love music, beautiful paintings and great literature without being trained musicians, talented artists or accomplished writers. It is the same with mathematics: we can enjoy the elegance of brilliant logical arguments and appreciate the beauty of mathematical structures and symmetries without being skilled creators of new theorems. [See TM097, or search for “thatsmaths” … Continue reading Recreational Mathematics is Fun
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?
Random Harmonic Series
We consider the convergence of the random harmonic series $latex \displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n} &fg=000000$ where $latex {\sigma_n\in\{-1,+1\}}&fg=000000$ is chosen randomly with probability $latex {1/2}&fg=000000$ of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is ``almost surely'' convergent. We are all familiar with the harmonic series … Continue reading Random Harmonic Series
Computers Speaking in Irish
Most of us use computer terminals, tablets and smart phones, absorbing information quickly and easily. How do the many thousands of Irish people who are blind or visually impaired manage to interact with computers? For them, entering data by keyboard or voice is easy, but special software is needed to convert the text on screen … Continue reading Computers Speaking in Irish
Squircles
You can put a square peg in a round hole. Shapes between circles and squares have proved invaluable to engineers and have also found their way onto our dinner tables. A plate in the shape of a `squircle' is shown in this figure . The Equation of a Squircle An ellipse with centre at the origin … Continue reading Squircles
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
Lecture sans paroles: the factors of M67
In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause. Cole, an American mathematician born in 1861, was educated at Harvard. He lectured there and later … Continue reading Lecture sans paroles: the factors of M67
Bending the Rules to Square the Circle
Squaring the circle was one of the famous Ancient Greek mathematical problems. Although studied intensively for millennia by many brilliant scholars, no solution was ever found. The problem requires the construction of a square having area equal to that of a given circle. This must be done in a finite number of steps, using only … Continue reading Bending the Rules to Square the Circle
Bloom’s attempt to Square the Circle
The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com]. The challenge … Continue reading Bloom’s attempt to Square the Circle
Prime Generating Formulae
The prime numbers have challenged and perplexed the greatest mathematicians for millennia. Shortly before he died, the brilliant Hungarian number theorist Paul Erdös said "it will be another million years, at least, before we understand the primes". The primes are scattered through the natural numbers in a manner that exhibits both order and chaos. For … Continue reading Prime Generating Formulae
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
Mathematics Everywhere (in Blackrock Station)
Mathematics is everywhere. We are often unaware of it but, when we observe our environment consciously, we can see mathematical structures all around us. Recently, while waiting for a train in Blackrock Station (Co Dublin), I photographed various objects in and around the station. There were circles and squares all about, parallel planes and lines, hexagons … Continue reading Mathematics Everywhere (in Blackrock Station)
Andrew Wiles wins 2016 Abel Prize
A recent post described the Abel Prize, effectively the Nobel Prize for Mathematics, and promised a further post when the 2016 winner was announced. This is the follow-up post [also at TM091, or search for “thatsmaths” at irishtimes.com]. Next Tuesday, HRH Crown Prince Haakon will present the Abel Medal to Sir Andrew Wiles at a ceremony … Continue reading Andrew Wiles wins 2016 Abel Prize
Ramanujan’s Astonishing Knowledge of 1729
Question: What is the connection between Ramanujan's number 1729 and Fermat's Last Theorem? For the answer, read on. The story of how Srinivasa Ramanujan responded to G. H. Hardy's comment on the number of a taxi is familiar to all mathematicians. With the recent appearance of the film The Man who Knew Infinity, this curious … Continue reading Ramanujan’s Astonishing Knowledge of 1729
Modelling Rogue Waves
There are many eyewitness accounts by mariners of gigantic waves – almost vertical walls of water towering over ocean-going ships – that appear from nowhere and do great damage, sometimes destroying large vessels completely. Oceanographers, who have had no way of explaining these 'rogue waves', have in the past been dismissive of these reports [TM090, or search for … Continue reading Modelling Rogue Waves
Sigmoid Functions: Gudermannian and Gompertz Curves
The Gudermannian is named after Christoph Gudermann (1798--1852). The Gompertz function is named after Benjamin Gompertz (1779--1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions. Sigmoid … Continue reading Sigmoid Functions: Gudermannian and Gompertz Curves
The Inexorable Rise in Life Expectancy
Life expectancy has increased in a spectacular fashion over the past 150 years. There has been a four-decade increase in average length of life over the period since 1850. World life expectancy has more than doubled, from about 25 years to around 65 years for men and 70 years for women. [TM089, or search for … Continue reading The Inexorable Rise in Life Expectancy
The Power Tower Fractal
We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here. The Operation … Continue reading The Power Tower Fractal
The Search is on for Planet Nine
Then felt I like some watcher of the skies When a new planet swims into his ken. John Keats: On First Looking into Chapman's Homer Pluto, discovered in 1930, orbits in the Kuiper Belt, a ring of asteroids and icy debris beyond Neptune. About ten years ago, it was reclassified as a "dwarf planet", so … Continue reading The Search is on for Planet Nine
The Imaginary Power Tower: Part II
This is a continuation of last week's post: LINK The complex power tower is defined by an `infinite tower' of exponents: $latex \displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,. &fg=000000$ The sequence of successive approximations to this function is $latex z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} … Continue reading The Imaginary Power Tower: Part II
The Imaginary Power Tower: Part I
The function defined by an `infinite tower' of exponents, $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ is called the Power Tower function. We consider the sequence of successive approximations to this function: $latex \displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,. &fg=000000$ As $latex {n\rightarrow\infty}&fg=000000$, the sequence $latex {\{y_n\}}&fg=000000$ converges for … Continue reading The Imaginary Power Tower: Part I
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 Abel Prize – The Nobel Prize for Mathematics
There is no Nobel Prize for mathematics, but there is a close equivalent: The prestigious Abel Medal is awarded every year for outstanding work in mathematics [TM086, or search for “thatsmaths” at irishtimes.com]. This years winner, or winners, will be announced soon. When Alfred Nobel's will appeared, the absence of any provision for a prize … Continue reading The Abel Prize – The Nobel Prize for Mathematics
Bertrand’s Chord Problem
The history of probability theory has been influenced strongly by paradoxes, results that seem to defy intuition. Many of these have been reviewed in a recent book by Prakash Gorroochurn [2012]. We will have a look at Bertrand's Paradox (1889), a simple result in geometric probability. Let's start with an equilateral triangle and add an … Continue reading Bertrand’s Chord Problem
Vanishing Zigzags of Unbounded Length
We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity. Hopping Animals Let us compare a sequence of frog hops, cricket hops and flea hops. We assume each hop is a semi-circle so that the length is easily calculated. If the … Continue reading Vanishing Zigzags of Unbounded Length
Franc-carreau or Fair-square
Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player's coin lands completely within a square, he or she wins a coin of equal value. If the … Continue reading Franc-carreau or Fair-square
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
Twin Peaks Entropy
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks. Classical Entropy Entropy was introduced in classical thermodynamics about 150 years ago and, somewhat … Continue reading Twin Peaks Entropy
Richardson’s Fantastic Forecast Factory
Modern weather forecasts are made by calculating solutions of the mathematical equations that express the fundamental physical principles governing the atmosphere [TM083, or search for “thatsmaths” at irishtimes.com] The solutions are generated by complex simulation models with millions of lines of code, implemented on powerful computer equipment. The meteorologist uses the computer predictions to produce … Continue reading Richardson’s Fantastic Forecast Factory
Squaring the Circular Functions
The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if $latex \displaystyle x = a\cos t \qquad\mbox{and}\qquad y = a\sin t … Continue reading Squaring the Circular Functions
Factorial 52: A Stirling Problem
How many ways can a deck of cards be arranged? It is very easy to calculate the answer, but very difficult to grasp its significance. There are 52 cards. Thus, the first one may be chosen in 52 ways. The next one can be any of the remaining 51 cards. For the third, there are … Continue reading Factorial 52: A Stirling Problem
How many Christmas Gifts?
We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, … Continue reading How many Christmas Gifts?
The Ping Pong Pendulum
Galileo noticed the regular swinging of a candelabra in the cathedral in Pisa and speculated that the swing period was constant. This led him to use a pendulum to measure intervals of time for his experiments in dynamics. Bu not all pendulums behave like clock pendulums. The Ping Pong Pendulum We consider a pendulum with … Continue reading The Ping Pong Pendulum
The Flight of a Golf Ball
Golf balls fly further today, thanks to new materials and mathematical design. They are a triumph of chemical engineering and aerodynamics. They are also big business, and close to a billion balls are sold every year. [TM081: search for “thatsmaths” at Irish Times ]. The golfer controls the direction and spin of the ball by … Continue reading The Flight of a Golf Ball
ThatsMaths 