Lattice Platonic Solids and their Ehrhart polynomial

TL;DR

This paper explores the Ehrhart polynomial of lattice cubes and derives relationships between the polynomials of regular lattice tetrahedrons and octahedrons, simplifying their computation.

Contribution

It introduces formulas for Ehrhart polynomials of lattice cubes and establishes relations linking tetrahedrons and octahedrons, reducing computational complexity.

Findings

Derived Ehrhart polynomial for lattice cubes.

Established relations between tetrahedron and octahedron Ehrhart polynomials.

Simplified calculation of these polynomials by reducing to one coefficient.

Abstract

First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.