Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-01T06:51:25.855Z Has data issue: false hasContentIssue false
  • English
  • Français

Buffon’s problem determines Gaussian curvature in three geometries

Part of: Geometric probability and stochastic geometry Global differential geometry

Published online by Cambridge University Press:  08 April 2024

Aizelle Abelgas ,
Bryan Carrillo ,
John Palacios ,
David Weisbart Open the ORCID record for David Weisbart [Opens in a new window]  and
Adam M. Yassine
Aizelle Abelgas*
Affiliation:
University of California, Riverside
Bryan Carrillo*
Affiliation:
Saddleback College
John Palacios*
Affiliation:
University of California, Irvine
David Weisbart*
Affiliation:
University of California, Riverside
Adam M. Yassine*
Affiliation:
Pomona College
*
*Postal address: Department of Mathematics, 900 University Ave., Riverside, CA 92521, Skye Hall.
***Postal address: Saddleback College, Department of Mathematics, 28000 Marguerite Parkway, Mission Viejo, California 92692. Email: bcarrillo@saddleback.edu
****Center for Complex Biological Systems, University of California, Irvine, 2620 Biological Sciences III, Irvine, CA 92697-2280. Email address:jjpalac2@uci.edu
*Postal address: Department of Mathematics, 900 University Ave., Riverside, CA 92521, Skye Hall.
******Postal address: Pomona College, Department of Mathematics and Statistics, 610 N. College Avenue, Claremont, CA 91711. Email: adam.yassine@pomona.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.

Keywords

Buffon’s problemGaussian curvaturearea deficitcircumference deficitBertrand–Diguet–Puiseux theoremgeometric probability

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 53C30: Homogeneous manifolds
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 12
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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.)

References

Barbier, E. (1860). Note sur le problème de l’aiguille et le jeu du joint couvert. J. Math. Pures et Appliquées, 2e série 5, 273–286.Google Scholar
Bertrand, J., Diguet, C. F. and Puiseux, V. (1848). Démonstration d’un théorème de Gauss. J. Math. 13, 80–90.Google Scholar
Buffon, G. (1733). Editor’s note concerning a lecture given in 1733 by Mr. Le Clerc de Buffon to the Royal Academy of Sciences in Paris. Histoire de l’Acad. Roy. des Sci., 43–45.Google Scholar
Buffon, G. (1777). Essai d’arithmatique morale. Histoire naturelle, générale et particulière, Vol. 4.Google Scholar
Calegari, D. (2020). On the kinematic formula in the lives of the saints. Notices Amer. Math. Soc. 67, 1042–1044.Google Scholar
Diaconis, P. (1976). Buffon’s problem with a long needle. J. Appl. Prob. 13, 614–618.CrossRefGoogle Scholar
Falconer, K. J. (1986). The Geometry of Fractal Sets (Cambridge Tracts in Mathematics 85). Cambridge University Press.Google Scholar
Isokawa, Y. (2000). Buffon’s short needle on the sphere. Bull. Faculty Ed. Kagoshima Univ. Nat. Sci. 51, 17–36.Google Scholar
Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge University Press.Google Scholar
Peres, Y. and Solomyak, B. (2002). How likely is Buffon’s needle to fall near a planar Cantor set? Pacific J. Math. 204, 473–496.CrossRefGoogle Scholar
Peter, E. and Tanasi, C. (1984). L’aiguille de Buffon sur la sphere. Elem. Math. 39, 10–16.Google Scholar
Solomon, H. (1978). Geometric Probability. Society for Industrial and Applied Mathematics, Philadelphia, PA. CrossRefGoogle Scholar