Geodesic Balls in the Nil-Space
W. Heisenberg's real matrix group provides a noncommutative translation group of an affine 3-space. The Nil-geometry, which is one of the eight Thurston 3-geometries, can be derived from this group. It was proved by E. Molnár that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere ). In this Demonstration we visualize the geodesic balls of the Nil-space with the origin as the center, radius in , and translated by .
We get the geodesic ball by rotating the following curve about the axis (lying in the plane ): , if ; if : , .
If , then the curve is basically half of the intersection of the geodesic sphere with the [, ] plane and looks like this:
The coordinates of a point rotated by around the axis are (, , ).
Finally, we can translate the geodesic sphere with a vector (, , ) to get , , ). This translation is defined by left multiplication with Heisenberg's matrix:
It is a good idea to zoom in for a better view as well as to rotate the image.
If the radius is less than , then the ball is convex in the affine-Euclidean sense of our model, but if the radius is in , then it is not convex. Also the geodesic sphere exists in Nil if and only if . For example if , then the curve used to rotate about the axis would be:
J. Szirmai, "The Densest Geodesic Ball Packing by a Type of Nil Lattices," Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry), 48(2), 2007 pp. 383-397.