Pappus's theorem and the modular group
type: Item
instance of: scholarly article
title: Pappus's theorem and the modular group
zbMATH Open document ID: 0806.57004
DOI: 10.1007/BF02712918
published in: Publications Mathématiques
publication date: 1995-02-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104091
review text: The author treats Pappus's theorem in the real projective plane \({\mathcal P} (\mathbb{R})\) as a dynamical system defined on so-called marked boxes. The latter are essentially quadrilaterals with distinguished top and bottom edges to be taken as initial data for the iterative application of the Pappian theorem. The resulting orbit may be indexed by the modular group \(\mathbb{Z}/2* \mathbb{Z}/3\). This group also occurs as the group of projective symmetries of an orbit: Varying the initial box one has two (commuting) modular group actions on the space of all marked boxes. If the marked boxes are convex then certain distinguished points of the boxes of an orbit form a set which is dense in some subspace of \({\mathcal P} (\mathbb{R})\) homeomorphic to the unit circle \(S^ 1\). Dual statements for distinguished lines hold true as well. These circles give rise to a subset \({\mathcal N}\) of the projective tangent bundle; furthermore the modular group has a representation \(\overline{M}\) in the Lie group of symmetries of the tangent bundle. From this the author finally gets a quotient space of \({\mathcal N}/ \overline{M}\) which is a three-dimensional analytic manifold with the homotopy type of the trefoil knot complement in \(S^ 3\).
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 51A30
zbMATH DE Number: 569542
zbMATH Keywords: Pappus theorem
zbMATH Keywords: real projective plane
zbMATH Keywords: dynamical system
zbMATH Keywords: modular group
zbMATH Keywords: marked boxes
zbMATH Keywords: trefoil knot complement
author: Richard Evan Schwartz
reviewed by: Horst Szambien
MaRDI profile type: Publication
cites work: Q3310101
cites work: Elastic waves in rods and clad rods
Wikidata QID: Q125588308
OpenAlex ID: W2000650052
description: scientific article; zbMATH DE number 569542