The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$
| Autor: | Donahue, Jessica, Hoehner, Steven, Li, Ben |
|---|---|
| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1107/S2053273320015089 |
| Popis: | This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices. Comment: 15 pages, 3 figures, 1 table. To appear in Acta Crystallographica A77 (2021) |
| Databáze: | arXiv |
| Externí odkaz: |
|