  It is well-known that every isosceles tetrahedron (disphenoid) admits
infinitely many simple closed geodesics on its surface. They can be naturally
enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases
$(1,0)$ and $(1,1)$. The (n,m)-geodesic is a broken line with $4(n+m)$
vertices, its length tends to infinity as $m\to \infty$. Are there other
polyhedra possessing this property? The answer depends on convexity. We give an
elementary proof that among convex polyhedra only disphenoids admit arbitrarily
long closed simple geodesics. For non-convex polyhedra, this is not true. We
present a counterexample with the corresponding polyhedron being a union of
seven equal cubes. Several open problems are formulated
