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
Category: Occasional
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
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
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
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
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
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
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
Life’s a Drag Crisis
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
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
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
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?
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
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
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
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
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.
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
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
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
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
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
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
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
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.
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
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
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
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
New Curves for Old: Inversion
Special Curves A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space $latex {\mathbb{R}^2}&fg=000000$ so that it generates a plane curve as it moves. … Continue reading New Curves for Old: Inversion
Falling Bodies [2]: Philae
The ESA Rosetta Mission, launched in March 2004, rendezvoused with comet 67P/C-G in August 2014. The lander Philae touched down on the comet on 12 November and came to rest after bouncing twice (the harpoon tethers and cold gas retro-jet failed to fire). Rosetta was in orbit around the comet and, after detatchment, the lander … Continue reading Falling Bodies [2]: Philae
Falling Bodies [1]: Sky-diving
Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle's theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a … Continue reading Falling Bodies [1]: Sky-diving
Light Weight (*)
Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a … Continue reading Light Weight (*)
Waring’s Problem & Lagrange’s Four-Square Theorem
$latex \displaystyle \mathrm{num}\ = \square+\square+\square+\square &fg=000000$ Introduction We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved … Continue reading Waring’s Problem & Lagrange’s Four-Square Theorem
Triangular Numbers: EYPHKA
The maths teacher was at his wits' end. To get some respite, he set the class a task: Add up the first one hundred numbers. “That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later … Continue reading Triangular Numbers: EYPHKA
Curves with Singularities
Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it … Continue reading Curves with Singularities
Cartoon Curves
The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves
Temperamental Tuning
Every pure musical tone has a frequency, the number of oscillations per second in the sound wave. Doubling the frequency corresponds to moving up one octave. A musical note consists of a base frequency or pitch, called the fundamental together with a series of harmonics, or oscillations whose frequencies are whole-number multiples of the fundamental … Continue reading Temperamental Tuning
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