Vertex to vertex geodesics on platonic solids

TL;DR

This paper proves, using symmetry arguments, that geodesics cannot connect a vertex to itself on the Platonic solids: cube, tetrahedron, octahedron, and icosahedron.

Contribution

It provides a straightforward symmetry-based proof for the non-existence of vertex-to-vertex geodesics on these Platonic solids.

Findings

No vertex-to-vertex geodesics in the cube, tetrahedron, octahedron, and icosahedron.

Symmetry considerations simplify the proof.

Results clarify geodesic properties on Platonic solids.

Abstract

We give a simple proof based on symmetries that there are no geodesics from a vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.