A Simple Growth Function

Three Styles of Growth Early models of population growth represented the number of people as an exponential function of time, $latex \displaystyle N(t) = N_0 \exp(t/\tau) &fg=000000$ where $latex {\tau}&fg=000000$ is the e-folding time. For every period of length $latex {\tau}&fg=000000$, the population increases by a factor $latex {e\approx 2.7}&fg=000000$. Exponential growth was assumed by … Continue reading A Simple Growth Function →

Hyperbolic Triangles and the Gauss-Bonnet Theorem

Poincaré's half-plane model for hyperbolic geometry comprises the upper half plane $latex {\mathbf{H} = \{(x,y): y>0\}}&fg=000000$ together with a metric $latex \displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,. &fg=000000$ It is remarkable that the entire structure of the space $latex {(\mathbf{H},ds)}&fg=000000$ follows from the metric. The … Continue reading Hyperbolic Triangles and the Gauss-Bonnet Theorem →

Poincaré’s Half-plane Model

For two millennia, Euclid's geometry held sway. However, his fifth axiom, the parallel postulate, somehow wrankled: it was not natural, obvious nor comfortable like the other four. In the first half of the nineteenth century, three mathematicians, Bolyai, Lobachevesky and Gauss, independently of each other, developed a form of geometry in which the parallel postulate … Continue reading Poincaré’s Half-plane Model →

New Estimate of the Speed of Light

A team of German scientists have recently discovered a new method of measuring the speed of light using Einstein's famous equation E = m c2 Scientists from SFZ, the Spätenheim Forschungszentrum in Bavaria, assembled a group of twenty volunteer climbers at a local mountain, Schmerzenberg. Using high-precision Mettler balance equipment, each climber was weighed at … Continue reading New Estimate of the Speed of Light →

Paddling Uphill

Recently, I kayaked with two friends on the River Shannon, which flows southward through the centre of Ireland. Starting at Dowra, Co. Cavan, we found it easy paddling until we reached Lough Allen, when the going became very tough. It was an uphill struggle. Could we really be going uphill while heading downstream?  That seems … Continue reading Paddling Uphill →

The Ups and Downs of Hailstone Numbers

Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers. The Collatz Conjecture There are many simply-stated … Continue reading The Ups and Downs of Hailstone Numbers →

Computer Maths

Will computers ever be able to do mathematical research? Automatic computers have amazing power to analyze huge data bases and carry out extensive searches far beyond human capabilities. They can assist mathematicians in checking cases and evaluating functions at lightning speed, and they have been essential in producing proofs that depend on exhaustive searches.  The … Continue reading Computer Maths →