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.


Ergodicity of the Geodesic Flow

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


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

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