Rational Spherical Triangles

2023/12/04 by Haiyang Wang, Wang, Haiyang
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.01633

openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of π. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this work, we are interested in determining all the rational spherical triangles. We introduce a conjecture on the solutions to a trigonometric Diophantine equation. An implication of the conjecture is that the only rational spherical triangles are the ones given by Little and Coxeter. We prove some partial results towards the conjecture.

Related

• Spherical Heron triangles and elliptic curves
• A simple method for generating rational triangles
• Spherical tetrahedra with rational volume, and spherical Pythagorean triples
• An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally
• Some diophantine problems concerning a pair of rational triangles with a\n common circumradius