The Two-Piece Property
These are two images created for an article by Thomas Banchoff
and Wolfgang Kühnel that surveys the current state of the
study of tight surfaces. A geometric characterization of tightness is the
following: a closed surface is tight if, and only if, every plane cuts the
surface into at most two pieces. The first image shows an example of a
tight surface: a torus of revolution. Here, a sample slicing plane cuts
the surface into two parts, and it turns out that this is typical of any
plane. The second image gives the canonical counter-example: there is a
plane that cuts this surface into three pieces, so it is not tight.
The article also included other computer-generated and hand-drawn
images that I produced based on the descriptions given by Banchoff.