An analytical approach to the Rational Simplex Problem

TL;DR

This paper introduces an analytical method involving a specific integral function to address the open Rational Simplex Problem in spherical geometry, aiming to determine whether rational dihedral angles imply rational volume multiples.

Contribution

It proposes a novel analytical approach using a function derived from elementary integrals to investigate the Rational Simplex Problem.

Findings

If a rational parameter t near zero yields an irrational f(t), the problem's answer is negative.

Provides a new perspective on the Rational Simplex Problem through integral analysis.

Lays groundwork for future analytical or computational verification of the problem.

Abstract

In 1973, J. Cheeger and J. Simons raised the following question that still remains open and is known as the Rational Simplex Problem: Given a geodesic simplex in the spherical 3-space so that all of its interior dihedral angles are rational multiples of $\pi$, is it true that its volume is a rational multiple of the volume of the 3-sphere? We propose an analytical approach to the Rational Simplex Problem by deriving a function $f(t)$, defined as an integral of an elementary function, such that if there is a rational $t$, close enough to zero, such that the value $f(t)$ is an irrational number then the answer to the Rational Simplex Problem is negative.