Q806036
type: Item
instance of: scholarly article
title: Über eine Äquivalenzrelation auf der Menge der (orientierten) Dreiecke einer affinen Ebene und Kennzeichnungen des papposschen Falles. (About an equivalence relation on the set of (oriented) triangles of an affine plane and characterizations of the Pappian case)
zbMATH Open document ID: 0729.51003
DOI: 10.1007/BF03323287
author: Wilhelm Junkers
published in: Results in Mathematics
publication date: 1991-01-01T00:00:00Z
review text: This article gives a characterization of pappian affine planes in terms of a property called smooth (schlicht) for equivalence relations on triangles with a common side. Let \({\mathcal A}\) be an affine plane. Consider two triangles \(\Delta_ 1=(A,B,C_ 1)\) and \(\Delta_ 2=(A,B,C_ 2)\) with a common side AB. The triangles \(\Delta_ 1\) and \(\Delta_ 2\) are primary-equivalent if the points \(C_ 1\) and \(C_ 2\) lie on a line parallel to the line AB. An arbitrary equivalence relation \(\sim\) on the set of triangles in \({\mathcal A}\) is smooth, and the plane \({\mathcal A}\) is smooth, if \((A,B,C_ 1)\sim (A,B,C_ 2)\) implies the two triangles are primary-equivalent. The author proves that an affine plane \({\mathcal A}\) is pappian if and only if it is smooth with respect to some equivalence relation on the set of triangles in \({\mathcal A}\). More, however, is accomplished. The author shows that the Pappus configuration theorem is equivalent to five other conditions; these include an interesting configuration theorem and two statements about an area mapping on the set of triangles.
reviewed by: Michael J. Kallaher
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 4205260
zbMATH Keywords: pappian affine planes
zbMATH Keywords: equivalence relations on triangles
zbMATH Keywords: Pappus configuration theorem
zbMATH Keywords: area mapping
MaRDI profile type: Publication
cites work: Die Orientierungsfunktionen eines affinen Raumes
cites work: Beziehungen zwischen mehrwertigen Ordnungsfunktionen und Inhaltsfunktionen auf Inzidenzstrukturen
cites work: Théorie algébrique de l'orientation et de la mesure des simplexes
cites work: Sur la mesure des triangles en géométrie affine plane
cites work: Mesure des triangles et mesure des vecteurs à supports parallèles dans une géométrie plane affine
full work available at URL: https://doi.org/10.1007/bf03323287
OpenAlex ID: W2087321931
label: Über eine Äquivalenzrelation auf der Menge der (orientierten) Dreiecke einer affinen Ebene und Kennzeichnungen des papposschen Falles. (About an equivalence relation on the set of (oriented) triangles of an affine plane and characterizations of the Pappian case)
description: scientific article; zbMATH DE number 4205260