Overlapping Unit Cells in 3d Quasicrystal Structure

TL;DR

This paper constructs a 3D quasiperiodic lattice with overlapping unit cells using grid and projection methods, analyzing the frequencies of overlaps through Kronecker's theorem, advancing understanding of quasicrystal structures.

Contribution

It introduces a novel 3D quasiperiodic lattice with overlapping unit cells and applies Kronecker's theorem to analyze overlap frequencies, extending previous 2D models.

Findings

Constructed a 3D quasiperiodic lattice with overlapping units.

Identified the frequency distribution of overlaps using Kronecker's theorem.

Described the structure of unit cells with shared interior points.

Abstract

A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of a convex polytope P, and 4 are interior points also shared with other neighboring unit cells. Using Kronecker's theorem the frequencies of all possible types of overlapping are found.