## Archive for October, 2013

### Hyperbolic Triangles and the Gauss-Bonnet Theorem

Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane ${\mathbf{H} = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space ${(\mathbf{H},ds)}$ follows from the metric.
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 ${H = \{(x,y): y>0\}}$ together with a metric

$\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.$

It is remarkable that the entire structure of the space follows from the metric.

In the earlier post, we derived the total curvature by evaluating the Riemann tensor. Here, we compute the curvature directly, using Gauss’s “Remarkable Theorem”.
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 Earth, and we cannot travel the straight line path from Dublin to New York: we have to stick to the surface of the globe, and the geometry we need is more complicated than the plane geometry of Euclid. Spherical geometry is central for the study of geophysics and astronomy, and vital for navigation.

### 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 no longer applied. This later bacame known as hyperbolic geometry.