Regular simple geodesic loops on a tetrahedron

Abstract

A regular simple geodesic loop on a tetrahedron is a simple geodesic loop which does not pass through any vertex of the tetrahedron. It is evident that such loops meet each face of the tetrahedron. Among these loops, the minimal loops are those which meet each face exactly once. Necessary and sufficient conditions for the existence of minimal loops are obtained. These conditions fall naturally into two categories, conditions in the first category being called coherence conditions and conditions in the second category being called separation conditions. 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, which, in turn, amounts to the tetrahedr0n satisfying three coherence conditions. All other regular simple geodesic loops on an isosceles tetrahedron are then classified. Finally, coherence conditions for the existence of similar loops on an arbitrary tetrahedron are found. CONVENTIONSThe arguments to be presented are elementary and easy to follow with the aid of supporting diagrams. In order that the arguments may be seen to be essentially diagram independent we shall establish some conventions in notation.We denote points by upper-case letters, A, B, C, etc., and lines by lower-case letters a, b, c, etc. The line containing points A and B is also denoted by AB. AB will also be used to denote the distance between points A and B with the context making it clear to which concept we refer. The line segment with endpoints A and B is denoted (AB) if it is an open line segment, and AB flit is a closed line segment. The ray from A which passes through B is denoted AB.In any given argument, all angles in a plane are measured in the same sense. If the chosen sense is clockwise, and A, B and C are non-collinear points, then ----+ /_ ABC is the angular measure from BA to BC measured in a clockwise sense. By this convention. 0 < / ABC < 2re, and /_ CBA = 2re--/_ ABC.The triangle with vertices A, B and C is denoted by AABC. Note that if /_ ABC is to denote the measure of the angle at vertex B of AABC then we will require 0 < /ABC < n. This will fix the sense in which we measure the angles in AABC, and our convention will dictate that/_ BCA and /_ CAB be used to denote the measure of the remaining angles in AABC, and that /_CBA, /_BAC and /__ACB be not used to denote the angles in AABC.The tetrahedron with vertices A, B, C and D is denoted AABCD. ZA is used to denote the sum of the (measures of the) angles at vertex A of AABCD. Note Geometriae Dedicata 42: 139-153, 1992.

Simple Closed Quasigeodesics on Tetrahedra

“…Alongside the above results, we also obtain a characterization of isosceles tetrahedra, defined in Section 2. It complements results in [9,10]. A geodesic loop is a simple closed curve that is a geodesic everywhere except at one point, the loop point.…”

“…This lemma complements the remark in [9] and was reproved in [15], that a regular tetrahedron has no geodesic loop. See also [10] for a characterization of isosceles tetrahedra as the only tetrahedra having three distinct minimal loops through any point on the face. With a different argument, briefly presented next, one can prove a stronger result.…”

Simple Closed Quasigeodesics on Tetrahedra

“…This lemma complements the remark in [DDTY17], that a regular tetrahedron has no geodesic loop. See also [SL92] for a characterization of isosceles tetrahedra as the only tetrahedra having three distinct minimal loops through any point on the face. 3 Q 1 : 1-Vertex Quasigeodesics…”

83.46 Equifacial tetrahedra and a famous location problem

“…• each face has the same area, • the four faces are congruent triangles, • every edge has the same length as the opposite (skew) one, • the circumscribed parallelepiped, each of whose faces contains exactly one of the tetrahedral edges, is a box (Figure 1). Note that the existence of such a circumscribed parallelepiped C is assured by the well-known fact that two skew lines (respectively skew edges of a tetrahedron) determine a unique pair of parallel planes (respectively faces of C), These and more characterisations of equifacial tetrahedra can be found in [7,8,9,10,11,12], and not so obvious characterisations (even within the family of all convex polyhedra) are given by [13] and [14].…”