Torus in Nil-Space

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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 (June 2009)
Suggested by: Jenő Szirmai
Open content licensed under CC BY-NC-SA



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 ,

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.


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.

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