New York Journal of Mathematics
Volume 21 (2015) 637-656

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Jayadev Athreya, Sneha Chaubey, Amita Malik, and Alexandru Zaharescu

Geometry of Farey-Ford polygons

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Published: July 29, 2015
Keywords: Ford circles, Farey fractions, distribution
Subject: 37A17, 11B57, 37D40
Abstract
The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in the geometry of numbers and hyperbolic geometry. We define two sequences of polygons associated to these objects, the Euclidean and hyperbolic Farey-Ford polygons. We study the asymptotic behavior of these polygons by exploring various geometric properties such as (but not limited to) areas, length and slopes of sides, and angles between sides.
Acknowledgements

J.S.A partially supported by NSF CAREER grant 1351853; NSF grant DMS 1069153; and NSF grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric structures And Representation varieties" (the GEAR Network)."
Author information

Jayadev Athreya:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA.
jathreya@illinois.edu

Sneha Chaubey:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA.
chaubey2@illinois.edu

Amita Malik:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA.
amalik10@illinois.edu

Alexandru Zaharescu:
Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania.
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA.
zaharesc@illinois.edu