Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane 
It is remarkable that the entire structure of the space 
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’
Archive for October, 2013
Hyperbolic Triangles and the Gauss-Bonnet Theorem
Published October 31, 2013 Occasional ClosedTags: Gauss, Geometry, Topology
Poincare’s Half-plane Model (bis)
Published October 24, 2013 Occasional ClosedTags: Gauss, Geometry
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 
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)’
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.
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The Simpler the Better
Published October 3, 2013 Irish Times ClosedTags: Algorithms, Applied Maths, Computer Science
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 later. But the problems on the board were not homework assignments; they were two famous unsolved problems in statistics. The solutions earned Dantzig his Ph.D.
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ThatsMaths 