No CrossRef data available.
Article contents
Buffon’s problem determines Gaussian curvature in three geometries
Published online by Cambridge University Press: 08 April 2024
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
MSC classification
Secondary:
53C30: Homogeneous manifolds
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust
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