LQ placeholderPacking things into a crowded place

Packing things into a crowded place

Predicting the geometry of densely packed objects in the natural and man-made worlds, from rocks to foams to polydisperse spheres.

Background Densely packed objects of a similar shape, from sedimentary rocks to foams to spheres, have intrigued mathematicians since Apollonius of Perga. The mechanical properties of these materials are determined by how big the gaps are between them—in particular, how close they are to their maximum random-packed density. Predicting this maximum, and how the material properties vary around this point, is a geometric and physical challenge.

Project We investigate the packing of spheres of the same size and mixtures of spheres with a range of sizes. While simulating packing is easy to program, the computational time scales poorly. To get around this obstacle, we develop new mathematical techniques for determining the properties of dense packings using a simplified 1d version. This powerful approach captures essential features of higher-dimensional systems, including an estimator of the maximum density.

Consequences Understanding how objects fit closely together under pressure or by design is at the foundation of materials science and branches of fluid dynamics. Predicting the behaviour of densely packed systems also sets the stage of reverse-engineering new composite materials to have desired properties.

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