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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 .


	Contributed by: Benedek Schultz, János Pallagi Suggested by: Jenő Szirmai
	Show Initialization Code

We get the geodesic ball by rotating the following curve about the

axis (lying in the plane

; 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:

Reference:

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.

Geodesic (Wolfram MathWorld)

"Geodesic Balls in the Nil-Space" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/GeodesicBallsInTheNilSpace/

Contributed by: Benedek Schultz, János Pallagi

Suggested by: Jenő Szirmai

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