Landscapes of the Tetrahedron and Cube: An Exploration of Shortest Paths on Polyhedra

TL;DR

This paper develops coordinate-based formulas to compute shortest surface paths between points on tetrahedra and cubes by leveraging their symmetries and unfolding techniques.

Contribution

It introduces a novel coordinate system and formulae for shortest path calculations on specific polyhedra surfaces, enhancing geometric analysis methods.

Findings

Derived explicit formulas for shortest paths on tetrahedra and cubes

Utilized symmetries to simplify path computation

Provided a framework for coordinate-based distance calculations

Abstract

We consider the problem of determining the length of the shortest paths between points on the surfaces of tetrahedra and cubes. Our approach parallels the concept of Alexandrov's star unfolding but focuses on specific polyhedra and uses their symmetries to develop coordinate based formulae. We do so by defining a coordinate system on the surfaces of these polyhedra. Subsequently, we identify relevant regions within each polyhedron's nets and develop formulae which take as inputs the coordinates of the points and produce as an output the distance between the two points on the polyhedron being discussed.