It is well known that spheres do not naturally order themselves in a predicable lattice. If thrown in a box, the spheres are disordered, and occupy roughly 60% of the space in the box.  If you shake the box,   the spheres settle, becoming more densely packed in the box, and eventually the spheres occupy about 64% of the volume in the box.  If you continue to shake the spheres, they do not settle any further.  No amount of shaking, or type of shaking, or special forces, or anything tried so far has gotten spheres to pack denser than this 64%.
        Oddly it is possible for the sphere to be placed denser than this.  If you make a hexagonal lattice in a plane

and place another hexagonal lattice plane on top of this, offset a little, you are packing spheres in the hexagonal close packed manner.  In this way, you pack the spheres so that they occupy 74% of the space in the box.
        The logical question is why don't the spheres proceed to the densest state possible?  They would have a lower gravitational energy in the denser state, and systems love to have low energy, so the behavior of the spheres is puzzling.  If the spheres are placed in the densest state, when shaken they actually become less dense, to approach the 64% final state.
        Figuring out why spheres don't self order is important for many industries.  The most obvious is shipping.  If we could understand how to get spheres to order, it would be a small logical step figuring out how to get other shapes to order.  If we could get gravel to attain an ordered state, it would occupy less volume and be cheaper to ship.  But we shouldn't confine our selves to gravel.  Coffee beans, rice, sand, verily all granular materials would be easier and cheaper to ship.
        Another useful product of understanding how granular materials settle would be in the concrete mixing industry.  For good concrete, it is desirable to have the large grains mixed homogeneously with the small grains.  During transport, however, the small grains always settle to the bottom.  If we understood the mechanics behind granular material settling, mixing concrete homogeneously might be easier.

        Actually doing settling experiments in three dimensions with spheres is difficult.  If you have the spheres settle in some way, seeing what is happening on the inside, away from the visible surface, is very hard.  If you can't accurately measure where all the balls are, it is impossible to get useful data on how the balls settle in relation to each other.
        For this reason we did our experiments in a plane.  It is plausible that the data that we get from two dimensional experiments will help explain the three dimensional case.  When confined to a plane, it is easy to measure the exact position of every ball, and so data analysis is easy as well.

        Two dimensions is a little different from three dimensions in that while some shapes still never order on any long term scale, some shapes DO naturally self order.  For example, trapezoids cannot be coaxed into ordering by the methods we used in our experiments, but circles (two dimensional spheres) easily form a lattice, as seen above.
        We did our experiment with the hope of finding correlations between the characteristics of a shape and how well the shape orders.

This material is based upon work supported by the National Science Foundation under Grant No. 9733898. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.