Archive for the 'Occasional' Category

TeX: A Boon for Mathematicians

Donald E Knuth, designer of the \TeX mathematical typesetting system.

Mathematicians owe a great debt of gratitude to Donald Knuth. A renowned American computer scientist and mathematician, Knuth is an emeritus professor at Stanford University. He is author of many books, including the multi-volume work, The Art of Computer Programming.

Knuth is the author of the powerful and versatile mathematical typesetting system called TeX. The original version, designed and written by Knuth, was released in 1978.

TeX is a powerful system for typesetting mathematical formulae. It is ideal both for simple mathematical notes with few formulas and for more complex documents and books involving subtle and sophisticated mathematical typography. TeX is used by almost all research mathematicians. It is also popular in computer science, engineering, physics, statistics, and and many other sciences.

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Jung’s Theorem: Enclosing a Set of Points

Let us imagine that we have a finite set {P} of points in the plane {\mathbb{R}^2} (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle?  The answer is given by Jung’s Theorem.

Left: a set P of points in the real plane. Right: The span s is the maximum distance between two points of P.

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Think of a Number: What are the Odds that it is Even?

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?


Probability distribution {P(n)=1/(\zeta(s)n^s)} for s=1.1 (red), s=1.01 (blue) and s=1.001 (black).

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Berry’s Paradox and Gödel’s Incompleteness Theorem



Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form


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Does Numerical Integration Reflect the Truth?

Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:

Does the numerical solution resemble the true solution of the equation?

The answer is: “Not necessarily”.

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.


Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).

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Buridan’s Ass


Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

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The Online Encyclopedia of Integer Sequences

Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.


The Home page of OEIS:

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Dimension Reduction by PCA

We live in the age of “big data”. Voluminous data collections are mined for information using mathematical techniques. Problems in high dimensions are hard to solve — this is called “the curse of dimensionality”. Dimension reduction is essential in big data science. Many sophisticated techniques have been developed to reduce dimensions and reveal the information buried in mountains of data.


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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 {f(x)} we can usually find {d f /d x}, whereas we may not be able to find {\int f(x)\,d x}.


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A New Perspective on Perspective

The development of perspective in the early Italian Renaissance opened the doors of perception just a little wider. Perspective techniques enabled artists to create strikingly realistic images. Among the most notable were Piero della Francesca and Leon Battista Alberti, who invented the method of perspective drawing.


School of Athens, a fresco painted by Raphael in 1509-11 illustrates the power of perspective.

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Order in the midst of Chaos

We open with a simple mathematical puzzle that is easily solved using only elementary reasoning. Imagine a party where some guests are friends while others are unacquainted. Then the following is always true:

No matter how many guests there are at the party, there are
always two guests with the same number of friends present.

If you wish, try proving this before reading on. The proof is outlined at the end of this post.


Complete graphs with 6 to 10 vertices.

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John Horton Conway: a Charismatic Genius


John H Conway in 2009
[image Denise Applewhite, Princeton University].

John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which he was appointed Lecturer in Pure Mathematics.


In 1986, Conway moved to Princeton University, where he was Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics. He was awarded numerous honours during his career. Conway enjoyed emeritus status from 2013 until his death just two weeks ago on 11 April.

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The Ross-Littlewood Paradox


Ross-Littlewood Paradox [Image from Steemit website:  here. ]

A most perplexing paradox appeared in Littlewood’s book A Mathematician’s Miscellany. It was later analysed in detail by Sheldon Ross in his 1988 book A First Course in Probability.

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Bang! Bang! Bang! Explosively Large Numbers


Typical Comic-book `bang’ mark [Image from vectorstock ].

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is

\displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

which is approximately {8\times 10^{53}}. The number of atoms in the universe is estimated to be about {10^{80}}. When we consider permutations of large sets, even more breadth-taking numbers emerge.

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Samuel Haughton and the Twelve Faithless Hangmaids

In his study of humane methods of hanging, Samuel Haughton (1866) considered the earliest recorded account of execution by hanging (see Haughton’s Drop on this site). In the twenty-second book of the Odyssey, Homer described how the twelve faithless handmaids of Penelope “lay by night enfolded in the arms of the suitors” who were vying for Penelope’s hand in marriage. Her son Telemachus, with the help of his comrades, hanged all twelve handmaids on a single rope.


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Zhukovsky’s Airfoil

A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

\displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)}

and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section Joukowsky Airfoil.


Visualization of airflow around a Joukowsky airfoil. Image generated using code on this website.

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A Ring of Water Shows the Earth’s Spin

Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus.


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The Rambling Roots of Wilkinson’s Polynomial

Finding the roots of polynomials has occupied mathematicians for many centuries. For equations up to fourth order, there are algebraic expressions for the roots. For higher order equations, many excellent numerical methods are available, but the results are not always reliable.


A 10th-order polynomial (blue) and a slightly perturbed version, with the coefficient of {x^9} changed by one part in a million.

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Adjoints of Vector Operators

We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results:

\displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all vector functions\ }\boldsymbol{\psi} \end{array}

Question: Is there a connection between these identities?


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Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis

Vector analysis can be daunting for students. The theory can appear abstract, and operators like Grad, Div and Curl seem to be introduced without any obvious motivation. Concrete examples can make things easier to understand. Weather maps, easily obtained on the web, provide real-life applications of vector operators.


Fig. 1. An idealized scalar field representing the mean sea-level atmospheric pressure over the North Atlantic area.

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Divergent Series Yield Valuable Results

Mathematicians have traditionally dealt with convergent series and shunned divergent ones. But, long ago, astronomers found that divergent expansions yield valuable results. If these so-called asymptotic expansions are truncated, the error is bounded by the first term omitted. Thus, by stopping just before the smallest term, excellent approximations may be obtained.

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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].

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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].

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


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.

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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.]

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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.

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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.

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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.

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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”.

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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.

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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.

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 tray in the form of a Ballyard.

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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.

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Cumbersome Calculations in Ancient Rome

Arithmetica-WoodcutTypus 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.

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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.

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Symplectic Geometry

Albert-EinsteinFor 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.}

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Chase and Escape: Pursuit Problems

Jolly-RogerFrom 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.

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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.


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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.

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 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.



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.

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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.

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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).

Sniper-Killzone-1 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’

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.

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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 .

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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.

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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]

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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.

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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.

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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.


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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].

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