Geodesic trajectories on regular polyhedra

TL;DR

This paper investigates the structure and quantity of shortest paths (geodesics) on regular polyhedra, revealing recursive patterns and differences in geodesic counts between vertex pairs on tetrahedra and cubes.

Contribution

It characterizes all geodesics from vertices to points on a tetrahedron and uses the Stern--Brocot tree to analyze geodesic counts on a cube, highlighting new recursive structures.

Findings

All geodesics from a vertex to a point on a tetrahedron are described.

There are twice as many geodesics between certain vertex pairs on a cube.

No geodesics start and end at the same vertex on either polyhedron.

Abstract

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.