Methods for counting the intersections of slopes in the flat torus

Document Type : Research Paper


Department of Mathematical Sciences, Rhode Island College, 600 Mt. Pleasant Ave. Providence, RI 02908


We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in $\mathbb{R}^2$. The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point $(0,0)$. In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus.


Main Subjects

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[2] S. I. R. Costa, J. E. Strapasson, M. M. S. Alves and T. B. Carlos, Circulant graphs and tessellations on flat tori, Linear Algebra Appl., 432 (2010) 369–382.
[3] J. E. Nelson and R. F. Picken, Theory of intersecting loops on a torus, Adv. Theor. Math. Phys., 18 (2014) 709–740.
Volume 13, Issue 4 - Serial Number 4
December 2024
Pages 305-317
  • Receive Date: 08 November 2022
  • Revise Date: 25 August 2023
  • Accept Date: 29 August 2023
  • Published Online: 01 December 2024