Details

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.

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.