## Archive for the 'Occasional' Category

### The Intermediate Axis Theorem

In 1985, cosmonaut Vladimir Dzhanibekov commanded a mission to repair the space station Salyut-7. During the operation, he flicked a wing-nut to remove it. As it left the end of the bolt, the nut continued to spin in space, but every few seconds, it turned over through ${180^\circ}$. Although the angular momentum did not change, the rotation axis moved in the body frame. The nut continued to flip back and forth, although there were no forces or torques acting on it.

Flipping nut [image from Veritasium].

Continue reading ‘The Intermediate Axis Theorem’

### Archimedes and the Volume of a Sphere

One of the most remarkable and important mathematical results obtained by Archimedes was the determination of the volume of a sphere. Archimedes used a technique of sub-dividing the volume into slices of known cross-sectional area and adding up, or integrating, the volumes of the slices. This was essentially an application of a technique that was — close to two thousand years later — formulated as integral calculus.

Cone, sphere and cylinder on the same base. The volumes are in the ratios  1 : 2 : 3 [image from mathigon.org].

Continue reading ‘Archimedes and the Volume of a Sphere’

### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Introduction

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ using the integral

$\displaystyle u = \int_0^{\phi} \frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin^2\phi}} \,.$

He called ${\phi}$ the amplitude and wrote ${\phi = \mathop\mathrm{am} u}$. It can be difficult to understand what motivated his definitions. We will define the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ in a more intuitive way, as simple ratios associated with an ellipse.

### An Attractive Spinning Toy: the Phi-TOP

It is fascinating to watch a top spinning. It seems to defy gravity: while it would topple over if not spinning, it remains in a vertical position as long as it is spinning rapidly.

There are many variations on the simple top. The gyroscope has played a vital role in navigation and in guidance and control systems. Many similar rotating toys have been devised. These include rattlebacks, tippe-tops and the Euler disk. The figure below shows four examples.

(a) Simple top, (b) Rising egg, (c) Tippe-top, (d) Euler disk. [Image from website of Rod Cross.]

### 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 then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

### The Wonders of Complex Analysis

Augustin-Louis Cauchy (1789–1857)

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 world of wonder is within your grasp.

In the early nineteenth century, Augustin-Louis Cauchy (1789–1857) constructed the foundations of what became a major new branch of mathematics, the theory of functions of a complex variable.

### Zeroing in on Zeros

Given a function ${f(x)}$ of a real variable, we often have to find the values of ${x}$ 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 accurate approximations to the roots of the equation ${f(x) = 0}$.

A rational function with five real zeros and a pole at x = 1.

### Spiralling Primes

The Sacks Spiral.

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 distribution of prime numbers. The first is the erratic and seemingly chaotic way in which the primes “grow like weeds among the natural numbers”. The second is that, when they are viewed in the large, they exhibit “stunning regularity”.

### 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 from a shuffled deck and reveal it. Then place it on the bottom and draw another card. The odds have changed: if the first card was an ace, the chances that the second is also an ace have diminished.

### 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.

Wilberforce spring [image from Wikipedia Commons].}

In equilibrium, the spring hangs down with the pull of gravity balanced by the elastic restoring force. When the weight is pulled down and released, it immediately oscillates up and down.

However, due to a mechanical coupling between the stretching and torsion, there is a link between stretching and twisting motions, and the energy is gradually converted from vertical oscillations to axial motion about the vertical. This motion is, in turn, converted back to vertical oscillations, and the cycle continues indefinitely, in the absence of damping.

The conversion is dependent upon a resonance condition being satisfied: the frequencies of the stretching and twisting modes must be very close in value. This is usually achieved by having small adjustable weights mounted on the pendulum.

There are several videos of a Wilberforce springs in action on YouTube. For example, see here.

### 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 tray in the form of a Ballyard.

### 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.

Phase portrait for a simple pendulum. Each line represents a different orbit.

### 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 uses a counting board or abacus (tabula) and – presumably – a less convenient number system. Arithmetica is looking with favour towards Boethius. He smiles smugly while Pythagoras is looking decidedly glum.

The figure aims to show the superiority of the Hindu-Arabic number system over the older Greek and Roman number systems. Of course, it is completely anachronistic: Pythagoras flourished around 500 BC and Boethius around AD 500, while the Hindu-Arabic numbers did not arrive in Europe until after AD 1200.

### Bernard Bolzano, a Voice Crying in the Wilderness

Bernard Bolzano (1781-1848)

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 nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.

### 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 this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:

$\displaystyle \mbox{Matter tells space how to curve. Space tells matter how to move.}$

### 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 chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.

### Bouncing Billiard Balls Produce Pi

There are many ways of evaluating ${\pi}$, 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.

### K3 implies the Inverse Square Law.

Johannes Kepler. Stamp issued by the German Democratic Republic in 1971, the 400th anniversary of Kepler’s birth.

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 second law (K2) states that the radial line from Sun to planet sweeps out equal areas in equal times; we now describe this in terms of conservation of angular momentum.

The third law (K3), which appeared in 1619 in Kepler’s Harmonices Mundi, is of a different character. It does not relate to a single planet, but connects the motions of different planets. It states that the squares of the orbital periods vary in proportion to the cubes of the semi-major axes. For circular orbits, the period squared is proportional to the radius cubed.

### 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 others. This ensured that they did not have to share the glory of discovery, but it may not be an optimal way of making progress in mathematics.

Polymath

Timothy Gowers in 2012 [image Wikimedia Commons].

Is massively collaborative mathematics possible? This was the question posed in a 2009 blog post by Timothy Gowers, a Cambridge mathematician and Fields Medal winner. Gowers suggested completely new ways in which mathematicians might work together to accelerate progress in solving some really difficult problems in maths. He envisaged a forum for the online discussion of problems. Anybody interested could contribute to the discussion. Contributions would be short, and could include false routes and failures; these are normally hidden from view so that different workers repeat the same mistakes.

### 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?

A small bowl with its handles resting on the rim of a larger bowl. The handles are approximately elliptical in cross-section.

### 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, is the region embracing all the points that can be reached by a sniper’s bullet, given a fixed muzzle velocity).

Family of trajectories with fixed initial speed and varying launch angles. Two particular trajectories are shown in black. 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 ${3 : 2}$ for various values of the phase-shift. Despite the different appearances, all sound pretty much the same.

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 be distinct, so two foldings are equivalent only when they have the same vertical sequence of the L = M × N layers.

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 maths, Martin Gardner. For more information about the meeting, see http://ludicum.org/ev/rm/19 .

### 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 Earth and a precise value of its radius.

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.

[Image Wikimedia Commons]

### Gaussian Curvature: the Theorema Egregium

Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

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 his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his Theorema Egregium. The Gaussian curvature ${K}$ characterizes the intrinsic geometry of a surface.

### 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.

Unstable (left) and stable (right) orbital configurations.

### 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 on the web, with much misunderstanding and confusion. I will try to avoid adding to that.

### 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.

Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].

### 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 another tree structure, 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 already bright before most of us wake up but, in Winter, the mornings would be too dark unless we reverted to Standard Time.

### 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.

### 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.

### 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.

Changing perspective on approach to the Mastaba

### 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.

### 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?

### 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 – Harmony of the World, and René Descartes’ first work was a compendium of music.

### 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

$\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots$

This is a divergent series: the sequence of partial sums is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$, which obviously does not converge, but alternates between ${0}$ and ${1}$.

### 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 “off” by ${1}$ and ${0}$, the sequence of states over the first minute is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### 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.

### “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]

Joyce in Zurich: did he meet Zermelo?

### Motifs: Molecules of Music

Motif: A short musical unit, usually just few notes, used again and again.

A recurrent short phrase that is developed in the course of a composition.

A motif in music is a small group of notes encapsulating an idea or theme. It often contains the essence of the composition. For example, the opening four notes of Beethoven’s Fifth Symphony express a musical idea that is repeated throughout the symphony.

### 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 meant that there was no number to describe the diagonal of a unit square.

### 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.

Cubic with roots at x=1, x=2 and x=3.

### 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.

### 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?

A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

### 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 Lagrange, entirely incredible.”

Introduction

Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

### Sophus Lie

It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002).

Sophus Lie (1842-1899)

Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

### 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

$\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$

This is an essential step in deriving an expression for the derivative of ${\sin x}$.