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.



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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 ,
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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