# Rational Elliptic Surfaces and the Trigonometry of Tetrahedra

## TL;DR

This paper explores the connection between non-Euclidean tetrahedra and rational elliptic surfaces, revealing new geometric and algebraic insights into their properties and symmetries.

## Contribution

It establishes a novel bijection between tetrahedra and elliptic surfaces, interpreting geometric data as period maps and linking Regge symmetries to Weyl group actions.

## Key findings

- Edge lengths and angles correspond to period map values.
- Cross-ratio of solid angles equals that of face perimeters.
- Regge symmetries relate to Weyl group actions on the surface's Picard lattice.

## Abstract

We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron as values of period maps for the corresponding surface. As a corollary we show that the cross-ratio of the exponents of the solid angles of a tetrahedron is equal to the cross-ratio of the exponents of the perimeters of its faces. The Regge symmetries of a tetrahedron are related to the action of the Weyl group $W(D_6)$ on the Picard lattice of the corresponding surface.