The geometry of Markoff numbers
type: Item
instance of: scholarly article
title: The geometry of Markoff numbers
zbMATH Open document ID: 0566.10024
DOI: 10.1007/BF03025802
author: Caroline Series
published in: The Mathematical Intelligencer
publication date: 1985-01-01T00:00:00Z
review text: This article offers a very fine exposition of the result of Andrew Haas to the effect that the Markoff quadratic irrationalities and simple closed geodesics on the punctured torus H/\(\Gamma\) ' are identified. In addition, the article contains a discussion of the analogue of the necessary dynamics for Haas' theorem in the case of a torus, uniformized by the plane. Some interesting historical data are provided for this case, some of it concerning ancient astronomy. The article also discusses the author's own work on the description of the exact connection between continued fractions and the tracing of geodesics on the modular surface, H/\(\Gamma\) (1). The paper ends with the interesting speculation that perhaps there is also a nice structure for geodesics on a punctured torus that self-intersect precisely once. One typo: The citations for references 6 (Haas' paper) and 7 are reversed.
Mathematics Subject Classification ID: 11J04
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 3902752
zbMATH Keywords: Markoff spectrum
zbMATH Keywords: self-intersections
zbMATH Keywords: Markoff quadratic irrationalities
zbMATH Keywords: simple closed geodesics
zbMATH Keywords: punctured torus
zbMATH Keywords: continued fractions
zbMATH Keywords: modular surface
reviewed by: Mark Sheingorn
MaRDI profile type: Publication
cites work: Approach to Markoff's minimal forms through modular functions
cites work: Representation of Markoff's binary quadratic forms by geodesics on a perforated torus
cites work: The Modular Surface and Continued Fractions
cites work: Minimum of quadratic forms with respect to Fuchsian groups. I.
full work available at URL: https://doi.org/10.1007/bf03025802
OpenAlex ID: W2073116343
description: scientific article; zbMATH DE number 3902752