Surface areas of equifacetal polytopes inscribed in the unit sphere $\mathbb{S}^2$

TL;DR

This paper investigates optimal arrangements of seven or eight points on the unit sphere to maximize the convex hull's surface area, focusing on configurations with congruent triangular facets.

Contribution

It provides solutions for maximizing surface area of convex hulls with specific congruent triangular facets for seven or eight points on the sphere.

Findings

Optimal point configurations identified for maximum surface area.

Convex hulls with congruent isosceles or equilateral triangles analyzed.

Explicit solutions for seven and eight point arrangements provided.

Abstract

This article is concerned with the problem of placing seven or eight points on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ so that the surface area of the convex hull of the points is maximized. In each case, the solution is given for convex hulls with congruent isosceles or congruent equilateral triangular facets.