##
Non-Rigidity Theorems for Tight Polyhedra

by Thomas F. Banchoff

Given a surface M in E^3, M is said to be *rigid in a given class
of surfaces* if the intrinsic properties of M completely determine
(within that class) its extrinsic properties. Cauchy [5] proved the
first rigity theorems, which applied to convex polyhedra,
Cohn-Vossen [7] has proven rigity theorems for convex real analytics
surfaces, and Herglotz [8] proved rigity theorems for convex surfaces
which are several times differentiable. A. D. Alexandroff [1]
generalized the idea of convexity to apply to a particular set
of real analytic surfaces which he named *T-surfaces*
and proved to be rigid. Alexandroff's condition, although stated
in terms of minimal total absolute curvature, is equivalen to the
notion of a *tight* surface. (A surface is M^2 in E^3 is
tight if the intersection of M^2 and H^3 is
connected for every half space H^3 in E^3.)
Nirenberg, in 1963, proved a rigidity theorem for several times
differentiable tight surfaces under additional assumptions.

The paper primarily seeks to prove that the corresponding statement regarding
rigidity of tight polyhedra is false, even for "square tori." Examples
of isometric but non-congruent tight tori are given to demonstrate
this point.

Stoker [11] has recently provided rigidity theorems for some classes
of non-convex polyhedra with vertices of prescribed types. A
discussion of Stoker's restrictions on square tori is given, and
it is proven that any square torus satisfying these conditions is
necessarily rigid in the class of square tori.