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# Packing Geometric Objects with Optimal Worst-Case Density (Multimedia Exposition)

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LIPIcs.SoCG.2019.63.pdf
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## Cite As

Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Sebastian Morr, and Christian Scheffer. Packing Geometric Objects with Optimal Worst-Case Density (Multimedia Exposition). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 63:1-63:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.63

## Abstract

We motivate and visualize problems and methods for packing a set of objects into a given container, in particular a set of {different-size} circles or squares into a square or circular container. Questions of this type have attracted a considerable amount of attention and are known to be notoriously hard. We focus on a particularly simple criterion for deciding whether a set can be packed: comparing the total area A of all objects to the area C of the container. The critical packing density delta^* is the largest value A/C for which any set of area A can be packed into a container of area C. We describe algorithms that establish the critical density of squares in a square (delta^*=0.5), of circles in a square (delta^*=0.5390 ...), regular octagons in a square (delta^*=0.5685 ...), and circles in a circle (delta^*=0.5).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
• Theory of computation → Computational geometry
##### Keywords
• Packing
• complexity
• bounds
• packing density

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## References

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