Hilbert basis in the face-centered cubic grid -- mathematical proofs

TL;DR

This paper characterizes the Hilbert basis of the face-centered cubic grid, revealing it consists of 11 types of basic cycles, using geometric, combinatorial, algebraic, and operations research methods.

Contribution

It provides a detailed classification of the Hilbert basis for the FCC grid, including the number of elements in each type, with rigorous mathematical proofs.

Findings

Basic cycles of FCC grid belong to 11 types

Quantitative analysis of elements in each cycle type

Mathematical proofs using multiple methods

Abstract

The Hilbert basis is fundamental in describing the structure of the integer points of a polyhedral cone. The face-centered cubic grid is one of the densest packing of the 3-dimensional space. The cycles of a grid satisfy the constraint set of a pointed, polyhedral cone which contains only non-negative integer vectors. The Hilbert basis of a grid gives the structure of the basic cycles in the grid. It is shown in this paper that the basic cycles of the FCC grid belong to 11 types. It is also discussed that how many elements are contained in the individual types. The proofs of the paper use geometric, combinatorial, algebraic, and operations research methods.