Kuiper’s original analysis of
tight surfaces showed that every surface has a tight immersion in three-space except
for the Klein bottle and the projective plane, which have none, and the projective
plane with one handle, for which he was unable to determine whether a tight
immersion was possible. The latter obtained a unique position among surfaces when
it was shown that no smooth tight immersion of it can be formed, while a polyhedral
one does exist. Continuing in its role as an unusual example, this surface has another
unexpected property, demonstrated here: Any tight immersion is necessarily
asymmetric, while every other surface can be immersed tightly and symmetrically in
space.
