Ehrhart's polynomial for equilateral triangles in \mathbb Z³

2011/07/04 by Eugen J. Ionaşcu, Eugen J. Ionascu, Ionascu, Eugen J.
Mathematics · #05A15 #52C07 #68R05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.CO #math.NT #msc:05A15 #msc:52C07 #msc:68R05

paper · pdf · doi:10.48550/arxiv.1107.0695

13 pages and three figures

openalex publication_date 2011/07/04 · arxiv created 2011/07/11 · arxiv updated 2011/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in \mathbb Z³. 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=√((a²+b²+c²)/(3)) and (a,b,c) (gcd(a,b,c)=1) is a vector with integer coordinates normal to the plane containing the triangle.

Related

• Ehrhart polynomials of 3-dimensional simple integral convex polytopes
• A characterization of all equilateral triangles in \Bbb Z³
• Ehrhart polynomial for lattice squares, cubes and hypercubes
• The distribution of roots of Ehrhart polynomials for the dual of root polytopes of type C
• Ehrhart polynomials of cyclic polytopes