It is clear hexagons ordered the best, and trapezoids didn't order well at all.  We tried to explain why by looking at the multipole moment expansions but this had very little correspondence.  We thought perhaps how circular an object was determined how well it ordered, but this too had very little correspondence.  We then realized how symmetrical a hexagon was and how asymmetrical a trapezoid was, and when we examined the symmetries of the other shapes, we found very good correspondence with the data.

To count symmetries, we found how many angles a shape could be rotated by to make the shape return to its original configuration.  For example, if you rotate a hexagon by Pi/3 radians, it is the same as if you didn't rotate the hexagon at all.  Thus, this is one symmetry of the hexagon. It is true that the more symmetries a shape had, the better it ordered.  There is a logical reason this might be so.  If your have a disordered region that would be more ordered if one particular particle would orient itself correctly, with every hit the shape has a chance of rotating and attaining the desired orientation.  If the shape has many symmetries, then the amount of rotating the shape must do to attain the desired orientation is minimal.  If the shape has very few symmetries, then the shape must rotate much more to attain the desired orientation.  We were using hits that didn't really disrupt the particles too much.  If the particles are hit too hard, any momentarily created order is destroyed on the next hit.  So with a hit that doesn't disrupt the particles too much, a small rotation is more probable than a large rotation, so shapes with many symmetries attain the desired orientation more often than shapes with few symmetries, so shapes with many symmetries change disordered regions into ordered regions more often.   Thus, they order better.

As a result of doing this experiment, we have created a method of measuring the order of a system, and the calculated order of a system agrees with our intuitive image or order.  We have found that hexagons ordered better than any other shape, and trapezoids order the worst.  In addition, we have a good method of predicting the amount of order a shape will exhibit, based on how many symmetries a shape contains.

Next, we are going to examine the time dependence of the order, and we will see the effects of mixing shapes.  It is our hope that the date from these experiments will help explain why disorder prevails in three dimensions.