Continued fractions and lines across the Stern–Brocot diagram

Abramson, Heather; Chesebro, Eric; Cummins, Vivian; Emlen, Cory; Grady, Ryan; Ke, Kenton

doi:10.2140/involve.2025.18.373

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Continued fractions and lines across the Stern–Brocot diagram

Heather Abramson, Eric Chesebro, Vivian Cummins, Cory Emlen, Ryan Grady and Kenton Ke

Vol. 18 (2025), No. 2, 373–385

DOI: 10.2140/involve.2025.18.373

Abstract

This paper concerns the relationships between continued fractions and the geometry of the Stern–Brocot diagram. Each rational number can be expressed as a continued fraction $[a_{0}; a_{1}, \dots, a_{n}]$ whose terms $a_{i}$ are integers and are positive if $i \geq 1$ . Select an index $i \in {1, \dots, n}$ and replace $a_{i}$ with an integer $m$ to obtain a continued fraction expansion for an extended rational $α_{m} \in ℚ \cup {\infty}$ . This paper shows that the vertices of the Stern–Brocot diagram corresponding to the numbers ${α_{m}}_{m \in ℤ}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L = ([a_{0}; a_{1}, \dots, a_{i - 1}], 0) \in ℝ^{2}$ . Moreover, as $| m | \to \infty$ , the associated vertices move down these lines and converge to $L$ . This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston’s work on hyperbolic Dehn surgery.

Keywords

Stern–Brocot diagram, 2-bridge links, continued fractions

Mathematical Subject Classification

Primary: 11B57, 57M50

Milestones

Received: 12 September 2023

Accepted: 11 October 2023

Published: 26 February 2025

Communicated by Gaven Martin

Authors

Heather Abramson
Department of Mathematical Sciences Montana State University Bozeman, MT United States

Eric Chesebro
Department of Mathematical Sciences University of Montana Missoula, MT United States

Vivian Cummins
Department of Mathematical Sciences University of Montana Missoula, MT United States

Cory Emlen
Department of Mathematical Sciences University of Montana Missoula, MT United States

Ryan Grady
Department of Mathematical Sciences Montana State University Bozeman, MT United States

Kenton Ke
Department of Mathematics University of California Davis Davis, CA United States

Vol. 18, No. 2, 2025