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Torus in Nil-Space


	W. Heisenberg's real matrix group provides a noncommutative translation group of an affine three-space. The Nil-geometry, which is one of the eight Thurston three-geometries, can be derived from this group. E. Molnár proved that the homogeneous three-spaces have a unified interpretation in the projective three-sphere ). Here, the tori of the Nil-space are visualized.


	Contributed by: Benedek Schultz and János Pallagi Suggested by: Jenő Szirmai
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By creating an intersection of an origin-centered geodesic ball with the

plane in the Nil-space we get a geodesic sphere lying in

plane.

;  if

Now on this intersection we can use a translation defined by right multiplication by Heisenberg's matrix: 

In the case

, this is the ordinary Euclidean translation in the

plane.

Finally, we rotate the sphere around the

axis with the following as

As we can see the

-translation changes the shape of the torus.

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 (The Wolfram Demonstrations Project)

"Torus in Nil-Space" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TorusInNilSpace/

Contributed by: Benedek Schultz and János Pallagi

Suggested by: Jenő Szirmai

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