On polyhedra inscribed in $S^2$, with approximately equal edges

Abstract

We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and the shortest edges. If the number of faces tends to infinity, and the polyhedron with this number of faces varies, then the limit inferior of this quotient is $2 \sin 36^{\circ } = 1.1756 \ldots $.