Some Fundamental Theorems of Maths

Every branch of mathematics has key results that are so important that they are dubbed fundamental theorems. The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is … Continue reading Some Fundamental Theorems of Maths →

The Wonders of Complex Analysis

If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don't be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole … Continue reading The Wonders of Complex Analysis →

Zeroing in on Zeros

Given a function $latex {f(x)}&fg=000000$ of a real variable, we often have to find the values of $latex {x}&fg=000000$ for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton's method or as the Newton-Raphson method. It usually produces highly … Continue reading Zeroing in on Zeros →

Spiralling Primes

The prime numbers have presented mathematicians with some of their most challenging problems. They continue to play a central role in number theory, and many key questions remain unsolved. Order and Chaos The primes have many intriguing properties. In his article ``The first 50 million prime numbers'', Don Zagier noted two contradictory characteristics of the … Continue reading Spiralling Primes →

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 →

The curious behaviour of the Wilberforce Spring.

The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to stretch up and down and to twist about its axis. In equilibrium, the spring hangs down with the pull of gravity … Continue reading The curious behaviour of the Wilberforce Spring. →

Billiards & Ballyards

In (mathematical) billiards, the ball travels in a straight line between impacts with the boundary, when it changes suddenly and discontinuously We can approximate the hard-edged, flat-bedded billiard by a smooth sloping surface, that we call a ``ballyard''. Then the continuous dynamics of the ballyard approach the motions on a billiard. Elliptical Billiards We idealize … Continue reading Billiards & Ballyards →

Boxes and Loops

We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions. Libration and Rotation of a Pendulum The simple pendulum, with one degree of freedom, provides a … Continue reading Boxes and Loops →

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 →

Bernard Bolzano, a Voice Crying in the Wilderness

Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but … Continue reading Bernard Bolzano, a Voice Crying in the Wilderness →

Symplectic Geometry

For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum ``physics is geometry'' is a guiding principle of modern theoretical physics. Einstein's General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of … Continue reading Symplectic Geometry →

Chase and Escape: Pursuit Problems

From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved … Continue reading Chase and Escape: Pursuit Problems →

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 →

K3 implies the Inverse Square Law.

Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler's Astronomia Nova. The first law (K1) describes the orbit of a planet as an ellipse with the Sun at one focus. The … Continue reading K3 implies the Inverse Square Law. →

Massive Collaboration in Maths: the Polymath Project

Sometimes proofs of long-outstanding problems emerge without prior warning. In the 1990s, Andrew Wiles proved Fermat's Last Theorem. More recently, Yitang Zhang announced a key result on bounded gaps in the prime numbers. Both Wiles and Zhang had worked for years in isolation, keeping abreast of developments but carrying out intensive research programs unaided by … Continue reading Massive Collaboration in Maths: the Polymath Project →

A Chirping Elliptic Rocker

Sitting at the breakfast table, I noticed that a small cereal bowl placed within another larger one was rocking, and that the period became shorter as the amplitude died down. What was going on? The handles of the smaller bowl appeared to be elliptical in cross-section, so I considered how a rigid body shaped … Continue reading A Chirping Elliptic Rocker →

The Kill-zone: How to Dodge a Sniper’s Bullet

Under mild simplifying assumptions, a projectile follows a parabolic trajectory. This results from Newton's law of motion. Thus, for a fixed energy, there is an accessible region around the firing point comprising all the points that can be reached. We will derive a mathematical description for this kill-zone (the term kill-zone, used for dramatic effect, … Continue reading The Kill-zone: How to Dodge a Sniper’s Bullet →

Don’t be Phased by Waveform Distortions

For many years there has been an ongoing debate about the importance of phase changes in music. Some people claim that we cannot hear the effects of phase errors, others claim that we can. Who is right? The figure below shows a waveform of a perfect fifth, with components in the ratio $latex {3 : … Continue reading Don’t be Phased by Waveform Distortions →

Folding Maps: A Simple but Unsolved Problem

Paper-folding is a recurring theme in mathematics. The art of origami is much-loved by many who also enjoy recreational maths. One particular folding problem is remarkably easy to state, but the solution remains elusive: Given a map with M × N panels, how many different ways can it be folded? Each panel is considered to … Continue reading Folding Maps: A Simple but Unsolved Problem →

Our Dearest Problems

A Colloquium on Recreational Mathematics took place in Lisbon this week. The meeting, RMC-VI (G4GEurope), a great success, was organised by the Ludus Association, with support from several other agencies: MUHNAC, ULisboa, CMAF-IO, CIUHCT, CEMAPRE, and FCT. It was the third meeting integrated in the Gathering for Gardner movement, which celebrates the great populariser of … Continue reading Our Dearest Problems →

From a Wide Wake to the Width of the World

The finite angular width of a ship's turbulent wake at the horizon enables the Earth's radius to be estimated. By ignoring evidence, Flat-Earthers remain secure in their delusions. The rest of us benefit greatly from accurate geodesy. Satellite communications, GPS navigation, large-scale surveying and cartography all require precise knowledge of the shape and form of the … Continue reading From a Wide Wake to the Width of the World →

Really, 0.999999… is equal to 1. Surreally, this is not so!

The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory. In school we learn that … Continue reading Really, 0.999999… is equal to 1. Surreally, this is not so! →

Gaussian Curvature: the Theorema Egregium

One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his ``Disquisitiones generales circa superficies curvas'', or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by … Continue reading Gaussian Curvature: the Theorema Egregium →

The 3 : 2 Resonance between Neptune and Pluto

For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto. The Harmony of the Spheres Pythagoras based his musical analysis on two ratios: the octave 2 : 1 and the perfect fifth 3 : 2. He … Continue reading The 3 : 2 Resonance between Neptune and Pluto →

The Two Envelopes Fallacy

During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed … Continue reading The Two Envelopes Fallacy →

Gravitational Waves & Ringing Teacups

Newton's law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide. Energy and angular momentum are carried … Continue reading Gravitational Waves & Ringing Teacups →

Listing the Rational Numbers III: The Calkin-Wilf Tree

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce … Continue reading Listing the Rational Numbers III: The Calkin-Wilf Tree →

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 … Continue reading Saving Daylight with Hip-hop Time: a Modest Proposal →

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. The Stern-Brocot Tree We … Continue reading Listing the Rational Numbers II: The Stern-Brocot Tree →

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. How can we make a list that includes all rationals? For the present, let us consider rationals in the interval $latex {[0,1]}&fg=000000$. It would be nice if … Continue reading Listing the Rational Numbers: I. Farey Sequences →

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. An isoceles trapezoidal prism A prism … Continue reading A Trapezoidal Prism on the Serpentine →

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. Life-cycle … Continue reading A Zero-Order Front →

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? There … Continue reading The Flight of the Bumble Bee →

Euler’s “Degree of Agreeableness” for Musical Chords

The 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 – … Continue reading Euler’s “Degree of Agreeableness” for Musical Chords →

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 $latex \displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots &fg=000000$ This is a divergent series: the sequence of partial sums is $latex {\{ 1, … Continue reading Grandi’s Series: A Second Look →

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 … Continue reading Grandi’s Series: Divergent but Summable →

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. Counting the Nines It is a simple … Continue reading Numbers with Nines →

“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] In Ithaca, the narrator takes every … Continue reading “Dividends and Divisors Ever Diminishing” →

A Glowing Geometric Proof that Root-2 is Irrational

It 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 … Continue reading A Glowing Geometric Proof that Root-2 is Irrational →

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. For centuries, mathematicians have struggled to find roots of polynomials like p(x) ≡ xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + … a1 x … Continue reading Marden’s Marvel →

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. Interference of two waves A single sinusoidal wave is infinite in extent and periodic in space and time. When waves interact, the dynamics are more interesting. The … Continue reading Waves Packed in Envelopes →

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? The General Equation for Geodesics Open a typical text … Continue reading Geodesics on the Spheroidal Earth – I →

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 … Continue reading Fourier’s Wonderful Idea – I →

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 $latex \displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1 &fg=000000$ This is an essential step in deriving an expression for the derivative of … Continue reading Subtract 0 and divide by 1 →

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. Suppose the curve $latex {\gamma}&fg=000000$ is specified in parametric form $latex {(x(t), y(t))}&fg=000000$ for $latex {t \in [0,1]}&fg=000000$. The centre of curvature $latex {\Gamma = (X, Y)}&fg=000000$ at a particular point … Continue reading The Evolute: Envelope of Normals →

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 … Continue reading Hardy’s Apology →

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 is remembered today for a range of curious mathematical patterns that he discovered. The best known … Continue reading Kaprekar’s Number 6174 →

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. The Möbius band may be characterised in … Continue reading Moebiquity: Ubiquity and Versitility of the Möbius Band →