Summary

Four copies of this part together make
an object, whose surface is called
Roman or Steiner surface (see figure).
For more information see:

http://en.wikipedia.org/wiki/Roman_surface

(Klein's bottle is mentioned under
"remix" as it might be of interest as well).

Instructions

The Roman surface also known as Steiner surface
can be described by different equations, see:
http://en.wikipedia.org/wiki/Roman_surface

Maple 18 offers the new feature to export plots
outputs as STL files - it was used for creating
this object (using the command plot3d and
the equations for x,y,z in polar coordinates).

with(plottools)
p2 := plot3d([cos(u)sin(u)cos(v)^2, sin(u)cos(v)sin(v), cos(u)cos(v)sin(v)], u = (1/2)Pi .. Pi, v = 0Pi .. .5*Pi, grid = [144, 144]) # output size limited to 1 MB, at least in academic version
stlp := "............stl"
exportplot(stlp, p2)

Important hint: If you want to create and print
your own objects, the spherical angles should
be chosen such that no point is generated
twice (e.g. if phi=0..Pi is enough, then it
should not be extended to phi=-Pi..Pi).
Otherwise no useable .gcode could be created
by the 3d printer software.

Roman surface / Steiner surface