Introduction and Problem History

Abstract

The general mathematical problem of finding the densest packing of equal objects in a bounded shape arises in many fields of the natural sciences, in engineering design, and also in everyday life. Some applications that involve the packing of identical circles are the following:

• coverage: place radio towers in a geographical region such that the coverage of the towers is maximal, with as little interference as possible;

• storage: place as many identical objects as possible (e.g. barrels) into a storage container;

• packaging: determine the smallest box into which one can pack a given number of bottles;

• tree exploitation: plant trees in a given region such that the forest is as dense as possible, but the trees allow each other to grow up to their maximal desired size;

• cutting industry: cut out as many identical disks as possible from a given (in the general case, irregular) piece of material.