Category: Occasional

A Model for Elliptic Geometry

For many centuries, mathematicians struggled to derive Euclid's fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the … Continue reading A Model for Elliptic Geometry →

Circles, polygons and the Kepler-Bouwkamp constant

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is $latex {\cos \pi/3 = 0.5}&fg=000000$. More generally, for an N-gon the ratio is easily shown to be $latex {\cos \pi/N}&fg=000000$. Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular … Continue reading Circles, polygons and the Kepler-Bouwkamp constant →

Was Space Weather the cause of the Titanic Disaster?

Space weather, first studied in the 1950's, has grown in importance with recent technological advances. It concerns the influence on the Earth's magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of … Continue reading Was Space Weather the cause of the Titanic Disaster? →

Differential Forms and Stokes’ Theorem

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds … Continue reading Differential Forms and Stokes’ Theorem →

The Basel Problem: Euler’s Bravura Performance

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to $latex {\ln 2}&fg=000000$. The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers, $latex \displaystyle … Continue reading The Basel Problem: Euler’s Bravura Performance →

The p-Adic Numbers (Part I)

The motto of the Pythagoreans was ``All is Number''. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers $latex {\mathbb{N}}&fg=000000$, and ratios of these, the positive rational numbers $latex {\mathbb{Q}^{+}}&fg=000000$. It came as a great shock that the diagonal of a … Continue reading The p-Adic Numbers (Part I) →

From Impossible Shapes to the Nobel Prize

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology. Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for … Continue reading From Impossible Shapes to the Nobel Prize →

TeX: A Boon for Mathematicians

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 … Continue reading TeX: A Boon for Mathematicians →

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? Since the set $latex {\mathbb{N}}&fg=000000$ of natural numbers is infinite, there are difficulties in assigning probabilities to subsets of $latex {\mathbb{N}}&fg=000000$. We require the probability … Continue reading Think of a Number: What are the Odds that it is Even? →

Resolution of Paradox: a Gateway to Mathematical Progress

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the … Continue reading Resolution of Paradox: a Gateway to Mathematical Progress →

Berry’s Paradox and Gödel’s Incompleteness Theorem

  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 THE SMALLEST NATURAL NUMBER THAT CANNOT BE DEFINED IN FEWER THAN TWENTY WORDS. This appears to indicate a specific positive integer, which we denote $latex {\mathcal{B}}&fg=000000$. But there is … Continue reading Berry’s Paradox and Gödel’s Incompleteness Theorem →

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 … Continue reading Does Numerical Integration Reflect the Truth? →

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 … Continue reading Dimension Reduction by PCA →

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. For centuries, artists have painted scenes on a sheet … Continue reading A New Perspective on Perspective →

John Horton Conway: a Charismatic Genius

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 … Continue reading John Horton Conway: a Charismatic Genius →

Bang! Bang! Bang! Explosively Large Numbers

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is $latex \displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 &fg=000000$ which is approximately $latex {8\times 10^{53}}&fg=000000$. The number of atoms in the universe is estimated to be about $latex {10^{80}}&fg=000000$. When we consider permutations of large sets, even … Continue reading Bang! Bang! Bang! Explosively Large Numbers →

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 … Continue reading Samuel Haughton and the Twelve Faithless Hangmaids →

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 $latex \displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)} &fg=000000$ and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with … Continue reading Zhukovsky’s Airfoil →

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. Compton (1892--1962) won the Nobel Prize in Physics in 1927 for his work on scattering of EM radiation. This phenomenon, now called the Compton effect, confirmed the particle nature of … Continue reading A Ring of Water Shows the Earth’s Spin →

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. James Wilkinson (1963) examined the behaviour of a high-order polynomial $latex \displaystyle p(x,\epsilon) = … Continue reading The Rambling Roots of Wilkinson’s Polynomial →

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: $latex \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 … Continue reading Adjoints of Vector Operators →

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. Weather charts provide great examples of scalar … Continue reading Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis →

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. Astronomical Series Many of … Continue reading Divergent Series Yield Valuable Results →

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 … 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 … Continue reading Elliptic Trigonometry: Fun with “sun”, “cun” and “dun” →