Multilattice Tilings and Coverings

• Main Author: Linnell, Joshua Randall
• Format: Others
• Published: BYU ScholarsArchive 2021
• Subjects: Multilattice; Covering; Density; Word-length; Simplex; Physical Sciences and Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/8911; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=9920&context=etd

Let L be a discrete subgroup of \mathbb{R}^n under addition. Let D be a finite set of points including the origin. These two sets will define a multilattice of \mathbb{R}^n. We explore how to generate a periodic covering of the space \mathbb{R}^n based on L and $D$. Additionally, we explore the problem of covering when we restrict ourselves to covering \mathbb{R}^n using only dilations of the right regular simplex in our covering. We show that using a set D= {0,d} to define our multilattice the minimum covering density is 5-\sqrt{13}. Furthermore, we show that when we allow for an arbitrary number of displacements, we may get arbitrarily close to a covering density of 1.