Here's a quick description of the objects (I'll start a more formal description of complex hyperbolic geometry soon):

 

Fans

The standard spinal sphere is the XY plane, and is defined by the points zero and infinity: the boundary points of its real spine. A fan is the limit of spinal spheres as their vertices converge along a chain in the spinal sphere. In the standard case, one can just move the origin out to infinity along a ray in the XY plane (just left-translate in the Heisenberg group). The spinal sphere remains a plane, but shears (see the Heisenberg distribution picture). In the limit it becomes the v-Z plane, where v is the vector it was translated along. For example, we might get the XZ plane. Any rotation around the Z axis by angle is also a fan.

 

For a more interesting example, we can apply a conformal map sending to the origin and the origin to {1,0,1} to the XZ plane. This gives the more exotic fan in the picture above. The red circle is the image of the Z axis and does not move. Precomposing with a rotation around the Z axis produces the following animation (right click to play):

 

 

A Clifford torus is in a sense the opposite of a spinal sphere. The basic example is shown above, defined the two green chains. Fixing the vertical chain and constricting the finite circular one produces the following animation:

 

Here's the calculation of the parametrization. (Note: the correct Clifford torus command will appear in CH0.3.nb).

 

Here is the new notebook: