*Geombinatorics*, 2004, 14(2):55-61.

In the present paper we would like to draw attention to some unusual geometric and topological properties that polyhedral approximations of even so simple a surface as a 2-dimensional sphere may have. We prove the existence of infinitely many polyhedral surfaces (with vertices belonging to the sphere) such that the polytopes bounded by them are mutually disjoint and, for each epsilon > 0, one can find, among these surfaces, any number of congruent surfaces of any genus, each of which is an epsilon-approximation of the sphere and has area less than epsilon. Such approximations may be interesting not only from a geometric point of view but also as an illustration of possible geometric and topological difficulties for surface polyhedral approximation algorithms.