Tag: Geometry

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 →

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 →

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 →

Bloom’s attempt to Square the Circle

The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com]. The challenge … Continue reading Bloom’s attempt to Square the Circle →

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) →

Mode-S: Aircraft Data improves Weather Forecasts

A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com]. It has often happened that an instrument designed for one purpose has proved invaluable for another. Galileo observed the regular swinging of a pendulum. Christiaan Huygens derived a mathematical … Continue reading Mode-S: Aircraft Data improves Weather Forecasts →

Earth’s Shape and Spin Won’t Make You Thin

Using a simple pendulum we can determine the shape of the Earth. That amazing story is told in this week’s That’s Maths column in The Irish Times (TM057 or search for “thatsmaths” at http://www.irishtimes.com ). Many of us struggle to lose weight, or at least to keep our weight within a manageable range. There is no … Continue reading Earth’s Shape and Spin Won’t Make You Thin →

Cartoon Curves

The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves →

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 →

Shackleton’s spectacular boat-trip

A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives. Endurance For eight months, Ernest Shackleton's expedition ship Endurance was carried along, ice-bound, until it was finally … Continue reading Shackleton’s spectacular boat-trip →