Landscapes of the Octahedron

TL;DR

This paper extends existing methods for cubes and tetrahedra to develop formulas for shortest paths on octahedron surfaces, enhancing geometric understanding of polyhedral landscapes.

Contribution

It introduces new formulas for shortest paths on octahedron surfaces, expanding the landscape analysis to a new class of polyhedra.

Findings

Formulas for shortest path lengths on octahedron surfaces.

Extension of landscape analysis from cubes and tetrahedra to octahedra.

Enhanced geometric understanding of polyhedral shortest paths.

Abstract

The landscapes of a polyhedron are subsets of its nets one must consider to identify all shortest paths. Landscapes of cubes and tetrahedra have been used to identify coordinate based formulas for the lengths of the shortest paths between points on these surfaces. We extend these results to develop formulas for the lengths of the shortest paths between points on the surface of octahedra.