Regular simple geodesic loops on a tetrahedron
type: Item
instance of: scholarly article
title: Regular simple geodesic loops on a tetrahedron
zbMATH Open document ID: 0754.51008
DOI: 10.1007/BF00147545
author: Yang Lu
author: John B. Strantzen
published in: Geometriae Dedicata
publication date: 1992-09-27T00:00:00Z
review text: A simple geodesic loop on a tetrahedron which does not pass through any vertex of the tetrahedron is called regular. Such a loop must meet each face on the tetrahedron. It is called minimal if each face is met exactly once. Considering developments of the faces of the tetrahedron in the plane the authors obtain necessary and sufficient conditions for the existence of numerical loops on a tetrahedron. It is shown that for the existence of three distinct minimal loops through any point on the face of a tetrahedron it is necessary and sufficient that the tetrahedron be isosceles. Furthermore, on such a tetrahedron all other regular simple geodesic loops are classified using an associated tiling of the plane. Finally conditions are discussed under which similar loops exist on a general tetrahedron.
Mathematics Subject Classification ID: 51M04
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 57579
zbMATH Keywords: elementary Euclidean geometry
zbMATH Keywords: geodesic loops
zbMATH Keywords: existence of numerical loops
zbMATH Keywords: tetrahedron
reviewed by: Bernd Wegner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147545
OpenAlex ID: W1973189607
description: scientific article; zbMATH DE number 57579