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

TL;DR

This paper investigates convex polyhedra inscribed in a sphere with triangular faces, analyzing how their edge length ratios behave as the number of faces increases, revealing a specific limit inferior value.

Contribution

It provides a new limit analysis of edge length ratios in inscribed convex polyhedra with many faces, connecting geometric properties to a specific limit value.

Findings

Limit inferior of edge length ratio approaches 1.1756 as faces increase

Edge length ratios are bounded and tend towards a specific value

Insights into the geometry of inscribed convex polyhedra with many faces

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 $.