Michael Spivak, the author of the famous textbook in differential geometry
explains a beautiful shape and shows using self built glasswork how this object
is equivalent to a three handled cup. We are interested in surfaces also in
this course. But in topology, one consideres surfaces equivalent if one can deform
one into the other. Like a doughnut is equivalent to a cup with one handle.
Here is an attempt to model the original surface in Mathematica:
And a GIF animation