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 →

Poincare’s Half-plane Model (bis)

In a previous post, we considered Poincaré's half-plane model for hyperbolic geometry in two dimensions. The half-plane model comprises the upper half plane $latex {H = \{(x,y): y>0\}}&fg=000000$ 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 … Continue reading Poincare’s Half-plane Model (bis) →

Geometry in and out of this World

Hyperbolic geometry is the topic of the That’s Maths column in the Irish Times this week (TM031 or click Irish Times and search for “thatsmaths”). Living on a Sphere The shortest distance between two points is a straight line. This is one of the basic principles of Euclidean geometry. But we live on a spherical … Continue reading Geometry in and out of this World →

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 →

The Simpler the Better

This week’s That’s Maths in The Irish Times ( TM030 ) is about Linear Programming (LP) and about how it saves millions of Euros every day through optimising efficiency. A Berkeley graduate student, George Dantzig, was late for class. He scribbled down two problems written on the blackboard and handed in solutions a few days … Continue reading The Simpler the Better →