Ehrhart's polynomial for equilateral triangles in $\mathbb Z^3$

TL;DR

This paper derives a simple formula for Ehrhart's polynomial of equilateral triangles in three-dimensional integer lattice, based on vertex coordinates and a key divisor-related parameter.

Contribution

It provides a novel explicit calculation of Ehrhart's polynomial for 2D equilateral triangles in Z^3, linking it to geometric and number-theoretic properties.

Findings

Ehrhart's polynomial depends on the divisor structure of a specific parameter d

The polynomial's form is explicitly expressed in terms of vertex coordinates

The approach simplifies counting lattice points in equilateral triangles in Z^3

Abstract

In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the polytope and it depends heavily on the value $d$ and its divisors, where $d=\sqrt{\frac{a^2+b^2+c^2}{3}}$ and $(a,b,c)$ ($\gcd(a,b,c)=1$) is a vector with integer coordinates normal to the plane containing the triangle.