Rational Angled Hyperbolic Polygons

TL;DR

This paper proves that hyperbolic polygons with rational angles cannot have algebraic side lengths, showing all such polygons have transcendental sides, and conjectures this extends to polygons with more sides.

Contribution

It establishes the transcendental nature of side lengths in rational angled hyperbolic triangles and quadrilaterals, and conjectures this property for polygons with more sides.

Findings

Rational angled hyperbolic triangles have transcendental side lengths.

Rational angled hyperbolic quadrilaterals have at least one transcendental side.

No rational angled hyperbolic triangle or quadrilateral with algebraic side lengths exists.

Abstract

We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle or quadrilateral with algebraic side lengths. We conjecture that there does not exist a rational angled hyperbolic polygon with algebraic side lengths.