Category: Occasional

A Symbol for Global Circulation

The recycling symbol consisting of three bent arrows is found on bottles, cartons and packaging of all kinds. It originated in 1970 when the Chicago-based Container Corporation of America (CCA) held a competition for the design of a symbol suitable for printing on cartons, to encourage recycling and re-use of packaging materials. The competition for … Continue reading A Symbol for Global Circulation →

Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868--1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term ``phugoid'' to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the … Continue reading Inertial Oscillations and Phugoid Flight →

A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all $latex {n}&fg=000000$ such that $latex {1<n<777{,}451{,}915{,}729{,}368}&fg=000000$,  but equality is violated for this enormous value and, intermittently, for larger values of $latex {n}&fg=000000$. Hypercube Tic-Tac-Toe The simple game of tic-tac-toe, or noughts and crosses, has been generalized in several ways. The number of cells in … Continue reading A Remarkable Pair of Sequences →

A Geometric Sieve for the Prime Numbers

In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying. A Nomogram for Multiplication The graph of a parabola $latex {y=x^2}&fg=000000$ can be used … Continue reading A Geometric Sieve for the Prime Numbers →

Torricelli’s Trumpet & the Painter’s Paradox

    Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli's Trumpet. It is the surface generated when the curve $latex {y=1/x}&fg=000000$ for $latex {x\ge1}&fg=000000$ is rotated in 3-space … Continue reading Torricelli’s Trumpet & the Painter’s Paradox →

Numerical Coincidences

A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising. Simple Examples A simple example is the near-equality between 2 cubed … Continue reading Numerical Coincidences →

Brun’s Constant and the Pentium Bug

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler's conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges: $latex \displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty &fg=000000$ Obviously, this could not happen if there were … Continue reading Brun’s Constant and the Pentium Bug →

Raphael Bombelli’s Psychedelic Leap

The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book Ars Magna. The method, often called Cardano's method, gives the solution of a depressed cubic equation t3 … Continue reading Raphael Bombelli’s Psychedelic Leap →

Taylor Expansions from India

  The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor's formula. In 1772, Lagrange described it as ``the main foundation of differential calculus'' (Wikipedia: Brook Taylor). Taylor also wrote a treatise on … Continue reading Taylor Expansions from India →

Which is larger, e^pi or pi^e?

Which is greater, $latex {x^y}&fg=000000$ or $latex {y^x}&fg=000000$? Of course, it depends on the values of x and y. We might consider a particular case: Is $latex {e^\pi > \pi^e}&fg=000000$ or $latex {\pi^e > e^\pi}&fg=000000$? We assume that $latex {x}&fg=000000$ and $latex {y}&fg=000000$ are positive real numbers, and plot the function $latex \displaystyle z(x,y) = … Continue reading Which is larger, e^pi or pi^e? →

Kepler’s Magnificent Mysterium Cosmographicum

  Johannes Kepler's amazing book, Mysterium Cosmographicum, was published in 1596. Kepler's central idea was that the distance relationships between the six planets (only six were known at that time) could be represented by six spheres separated by the five Platonic solids. For each of these regular polyhedra, there is an inner and an outer … Continue reading Kepler’s Magnificent Mysterium Cosmographicum →

Heron’s Theorem: a Tool for Surveyors

Heron was one of the great Greek mathematicians of Alexandria, following in the tradition of Euclid, Archimedes, Eratosthenes and Apollonius. He lived in the first century, from about AD 10 to AD 70. His interests were in practical rather than theoretical mathematics and he wrote on measurement, mechanics and engineering. He devised a steam-powered device … Continue reading Heron’s Theorem: a Tool for Surveyors →

Random Harmonic Series

We consider the convergence of the random harmonic series $latex \displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n} &fg=000000$ where $latex {\sigma_n\in\{-1,+1\}}&fg=000000$ is chosen randomly with probability $latex {1/2}&fg=000000$ of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is ``almost surely'' convergent. We are all familiar with the harmonic series … Continue reading Random Harmonic Series →

Squircles

You can put a square peg in a round hole. Shapes between circles and squares have proved invaluable to engineers and have also found their way onto our dinner tables. A plate in the shape of a `squircle' is shown in this figure . The Equation of a Squircle An ellipse with centre at the origin … Continue reading Squircles →

Lecture sans paroles: the factors of M67

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause. Cole, an American mathematician born in 1861, was educated at Harvard. He lectured there and later … Continue reading Lecture sans paroles: the factors of M67 →

Mathematics Everywhere (in Blackrock Station)

Mathematics is everywhere. We are often unaware of it but, when we observe our environment consciously, we can see mathematical structures all around us. Recently, while waiting for a train in Blackrock Station (Co Dublin), I photographed various objects in and around the station. There were circles and squares all about, parallel planes and lines, hexagons … Continue reading Mathematics Everywhere (in Blackrock Station) →

Sigmoid Functions: Gudermannian and Gompertz Curves

The Gudermannian is named after Christoph Gudermann (1798--1852). The Gompertz function is named after Benjamin Gompertz (1779--1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions. Sigmoid … Continue reading Sigmoid Functions: Gudermannian and Gompertz Curves →