Steffen's Flexible Polyhedron
Steffen's polyhedron is able to flex without self-intersection and without distorting any of its faces. It has 14 triangular faces, 21 edges, and 9 vertices. Euler had conjectured no polyhedron could flex. Cauchy proved that convex polyhedra cannot flex.[more]
The bellows conjecture (proven in 1997) asserts that a flexible polyhedron has constant volume as it is flexed.[less]
R. Connelly, I. Sabitov, and A. Walz, "The Bellows Conjecture," Contrib. Algebra Geom. 38, pp. 1–10, 1997.
 P. R. Cromwell, Polyhedra, New York: Cambridge University Press, pp. 245 and 247, 1997.