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The Kelvin problem (3-D packing)

In 2 dimensions, the most efficient packing mechanism is an array of hexagons - a honeycomb. In 1887, William Thomson (Lord Kelvin) asked the question 'What is the most efficient 3-D packing system?"

See (the original paper) : On the Division of Space with Minimla Partitional Area Open Access

He suggested that it was the 'bi-truncated cubic honeycomb' which, until 1993, was widely recognised as the most efficient possible.

It was succeeded, however, by the Weaireā€“Phelan Structure (see Wikipedia) - found by computerised simulations of foam generation - which is currently the most efficient form yet found.

The question of whether the Weaireā€“Phelan Structure is the most efficient 3-D packing structure - having the smallest surface area per cell - is however still open. Subsequent mathematical simulations suggest that it is optimal - but this remains unproven.

Also see : Ulam's packing conjectureplugin-autotooltip__plain plugin-autotooltip_bigUlam's packing conjecture

When packing convex identical 3-dimensional objects into a defined space, is a sphere the most 'efficient' shape when considering the amount of free space in the gaps?

According to the conjecture, the sphere is the convex solid which forces the largest fraction of space to remain empty.

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