Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-31T09:47:37.170Z Has data issue: false hasContentIssue false
  • English
  • Français

Barycentric subdivision of triangles and semigroups of Möbius maps

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  26 February 2010

I. Bárány ,
A. F. Beardon  and
T. K. Carne
I. Bárány
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364, Hungary.
A. F. Beardon
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
T. K. Carne
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
Get access

Extract

The following question of V. Stakhovskii was passed to us by N. Dolbilin [4]. Take the barycentric subdivision of a triangle to obtain six triangles, then take the barycentric subdivision of each of these six triangles and so on; is it true that the resulting collection of triangles is dense (up to similarities) in the space of all triangles? We shall show that it is, but that, nevertheless, the process leads almost surely to a flat triangle (that is, a triangle whose vertices are collinear).

MSC classification

Secondary: 60D05: Geometric probability and stochastic geometry

Information

Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 165 - 171
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Beardon, A. F.. The geometry of discrete groups, Graduate Texts, 91 (Springer-Verlag, 1983).CrossRefGoogle Scholar
2.Bougerol, P. and Lacroix, J.. Products of random matrices with applications to Schrödinger operators. Progress in Probability and Statistics, 8 (Birkhäuser, 1985).Google Scholar
3.Conway, J. H. and Jones, A. J.. Trigonometric Diophantine equations (on vanishing sums of roots of unity). Acta Arithmetica, 30 (1976), 229–240.CrossRefGoogle Scholar
4.Dolbilin, N.. Private correspondence.Google Scholar
5.Furstenberg, H.. Noncommuting random products. Trans. Amer. Math. Soc, 108 (1963), 377–428.CrossRefGoogle Scholar
6.Kendall, D. G.. Shape manifolds, Procrustean metrics, and complex projective space. Bull. London Math. Soc, 16 (1984), 81–121.CrossRefGoogle Scholar
7.Ledrappier, F.. Quelques propriétés des exposants caractéristiques, (from École d'été de Saint-Flour, XII, edited by Hennequin, P. L.) Lecture Notes in Mathematics 1097 (Springer-Verlag, 1984).CrossRefGoogle Scholar
8.Mannion, D.. Convergence to collinearity of a sequence of random triangle shapes. Adv. Appl. Prob., 22 (1990), 831–844.CrossRefGoogle Scholar
9.Mannion, D.. The invariant distribution of a sequence of random collinear triangle shapes. Adv. Appl. Prob., 22 (1990), 845–865.CrossRefGoogle Scholar