Multiple arrangements

TL;DR

This paper reviews the development of multiple packings and coverings, focusing on recent advances in decomposing multiple coverings and characterizing multiple tilings, highlighting progress since the 1960s.

Contribution

It provides a comprehensive survey emphasizing recent research on decomposing multiple coverings and characterizing multiple tilings, expanding understanding of multiple arrangements.

Findings

Recent progress in decomposing multiple coverings into simple coverings

Characterization of multiple tilings on the plane

Historical overview of multiple arrangements since the 1960s

Abstract

This paper surveys the theory of multiple packings and coverings. The study of multiple arrangements started in the 60s of the last century, and it was restricted mostly to lattice arrangements on the plane or of general arrangements of balls. We emphasize two new topics which were intensively investigated recently: decomposition of multiple coverings into simple coverings and characterization of multiple tilings.