# The Regge symmetry, confocal conics, and the Schl\"afli formula

## TL;DR

This paper provides a simple geometric proof of the Regge symmetry, revealing its connections across Euclidean, spherical, and hyperbolic geometries, and relates it to confocal conics and the Schl"afli formula.

## Contribution

It introduces a straightforward geometric proof of the Regge symmetry applicable to multiple geometries, expanding understanding of its geometric foundations.

## Key findings

- Regge symmetry holds in Euclidean, spherical, and hyperbolic geometries.
- The proof connects Regge symmetry with confocal conics and the Schl"afli formula.
- The approach simplifies understanding of the symmetry's geometric basis.

## Abstract

The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.