The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}

Area of Science:

• Geometry
• Computational Geometry
• Crystallography

Background:

• The problem of maximizing the surface area of a convex hull for a fixed number of points on a sphere is a classic geometric challenge.
• Previous work, including numerical searches, suggested potential optimal configurations but lacked analytical proof.

Purpose of the Study:

• To analytically determine the optimal placement of five points on the unit sphere to maximize the surface area of their convex hull.
• To confirm a conjecture regarding the optimal configuration.
• To apply these findings to measure distortion in coordination polyhedra in crystallography.

Main Methods:

• Analytical determination of optimal point placement on the unit sphere.
• Geometric analysis of convex hull properties.
• Formulation of a surface area discrepancy measure.

Main Results:

• The optimal configuration for five points on the unit sphere is a trigonal bipyramid.
• This structure features two vertices at the poles and three vertices forming an equilateral triangle on the equator.
• A formula for the surface area discrepancy of five-vertex coordination polyhedra was derived.

Conclusions:

• The analytical solution confirms the trigonal bipyramidal structure as the maximizer for convex hull surface area with five points.
• The derived formula provides a quantitative measure for assessing distortions in coordination polyhedra, relevant to crystallographic analysis.