Tags » Hyperbolic Geometry

SnapPy 2.4 released

A new version of SnapPy, a program for studying the topology and geometry of 3-manifolds, is available.  Added features include a census of Platonic manifolds, rigorous computation of cusp translations, and substantial improvements to its link diagram component.

3-manifolds

The geodesic flow

Given any Riemannian manifold M, we may define a geodesic flow  on the unit tangent bundle which sends a point  1,058 more words

Hyperbolic Geometry

Hyperbolic Geometry Note #2: On Möbius transformations as isometries of the upper half plane

Möbius transformations are conformal (i.e., angle-preserving), orientation-preserving, area-preserving isometries in the upper half-plane model of hyperbolic space. It piqued my interest to discover that the form of the metric in this model (which I discussed in… 732 more words

Hyperbolic Geometry

More Maths Art

Wednesday is here, the middle of the week, just a couple of days until the weekend. In the past weeks I have done more artistic and relaxing Wednesdays and this one will be the same. 273 more words

Mathematics

Hyperbolic Geometry Note #1: Strange behaviour of length calculations in the Poincaré half-plane model of hyperbolic space

In the Poincaré half-plane model of hyperbolic space, the upper half-plane is the set of complex numbers with positive imaginary part:

The boundary of , , is the real axis together with the point . 762 more words

Hyperbolic Geometry

The many faces of the hyperbolic plane

is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

1. It is diffeomorphic to as a topological space (or, indeed, isometric as a Riemannian manifold, if we give a left-invariant metric): to show this, we note that has isometry group , and the subgroup of isometries which stabilize any given is isomorphic to .
2. 211 more words
Snippets

Fuchsian vs. Kleinian

Or, all the fun you could have in one additional dimension.

The former, of course, may be viewed as a special case of the latter; the more general Kleinian case shares many, but not all, of the characteristics of the simpler Fuchsian case. 776 more words

Hyperbolic Geometry