Geodesics on the regular tetrahedron and the cube
type: Item
instance of: scholarly article
title: Geodesics on the regular tetrahedron and the cube
zbMATH Open document ID: 1360.52010
author: Diana Davis
author: Victor Dods
author: Jed Yang
author: Cynthia Traub
published in: Discrete Mathematics
publication date: 2016-10-21T00:00:00Z
reviewed by: Zdeněk Dušek
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 52C05
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 6642332
zbMATH Keywords: geodesics on polyhedra
zbMATH Keywords: regular tetrahedron
zbMATH Keywords: cube
MaRDI profile type: Publication
full work available at URL: j.disc.2016.07.004
OpenAlex ID: W2513684333
cites work: Convex Polyhedra
cites work: Q5427478
cites work: Geodesics on regular polyhedra with endpoints at the vertices
cites work: Q5421078
cites work: Star unfolding convex polyhedra via quasigeodesic loops
cites work: The Discrete Geodesic Problem
cites work: The intrinsic diameter of the surface of a parallelepiped
cites work: Sets of tetrahedra, defined by maxima of distance functions
cites work: On Shortest Paths in Polyhedral Spaces
DOI: 10.1016/J.DISC.2016.07.004
review text: Given a convex polyhedron \(S\) in \({\mathbb{R}}^3\), a \textit{geodesic} on \(S\) is a curve \(\gamma\) on \(S\) such that for any sufficiently close points \(p\) and \(q\) on \(\gamma\), the shortest path on \(S\) from \(p\) to \(q\) coincides with \(\gamma\). A natural way how to study geodesics on polyhedra is by unfolding the polyhedron into the plane and, if possible, tiling the plane with faces of the polyhedron. In this way, geodesics on \(S\) correspond to straight lines in the plane.NEWLINENEWLINEIn the present paper, the regular tetrahedron and the cube are studied, because their faces give tiling of the plane. For the tetrahedron, geodesics from an arbitrary point to an arbitrary vertex are characterized in terms of corresponding directions. In particular, vertex-to-vertex geodesics are characterized this way. For the cube, vertex-to-vertex geodesics are studied. Depending on the pair of vertices chosen, the number of geodesic differs. Surprisingly, for both the tetrahedron and the cube, there are no geodesics which start and end in the same vertex.
description: scientific article; zbMATH DE number 6642332