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Approximating Heisenberg Geodesics

Tuesday, 23 June 2009 17:38 | Author: Anton

A Riemannian approximation of the Heisenberg group is given by declaring the following frame field orthonormal in $\mathbb{R}^3$ :

$X_s = \frac{\partial}{\partial x} - y \frac{s}{2} \frac{\partial}{\partial t}, \;\; Y_s = \frac{\partial}{\partial x} - y \frac{s}{2} \frac{\partial}{\partial t} \;\; T_s = \sqrt{1-s} \frac{\partial}{\partial t}$

(note that the $T_s$ has changed from the earlier versions).

There are many ways to approximate horisontal curves in $\mathbb{H}$ by geodesics in the approximations $\mathbb{H}_s$ . Here's one approximation, going from a sphere in Euclidean space to the apple sphere in Heisenberg space, by way of apple sets in the intermediate spaces:

Last Updated ( Tuesday, 23 June 2009 18:10 )

ComplexHyperbolic0.3.nb

Math - ComplexHyperbolic09

Tuesday, 23 June 2009 04:57 | Author: Anton

The notebook has been updated with a proper routine for the Clifford Torus animation (see CH0.2.nb), a quick description of extors, and an animation showing a spinal sphere turning into a fan and then into a Clifford torus.

ComplexHyperbolic0.3.nb

(right click to play)

Last Updated ( Wednesday, 24 June 2009 19:32 )

Kissinger on Iran - June 17

Et Cetera Res - etc

Thursday, 18 June 2009 15:29 | Author: Anton

Thursday, 11 June 2009 18:48 | Author: Anton

ComplexHyperbolic0.2.nb

Math - ComplexHyperbolic09

Tuesday, 16 June 2009 04:20 | Author: Anton

The notebook now has the routines for drawing fans and Clifford tori, objects defined similarly to the spinal spheres:

Last Updated ( Friday, 19 June 2009 17:47 )

Approximating Heisenberg Geodesics

ComplexHyperbolic0.3.nb

Kissinger on Iran - June 17

ComplexHyperbolic0.2.nb

Spinach Dip (cold)