Asymptotics of the number of waves on rational polyhedra

TL;DR

This paper investigates the asymptotic behavior of wave counts on rational polyhedra, establishing subexponential growth bounds and solving the problem explicitly for Platonic solids using advanced number theory techniques.

Contribution

It provides the first asymptotic analysis of wave counts on rational polyhedra and introduces subexponential bounds applicable to all such polyhedra.

Findings

Wave counts on Platonic solids grow subexponentially.

A subexponential upper bound is established for all rational polyhedra.

The analysis leverages number theory to connect geometric and wave phenomena.

Abstract

The problem of counting the number of waves arriving at the vertex of a polyhedron is motivated by physics. In the article it was solved for the case of Platonic solid using three nontrivial results from number theory. This growth turns out to be subexponential. Also we prove a subexponential upper bound for all polyhedra with rational total angles at vertices.