We made an assortment of shapes and studied the settling characteristics of each shape. The  shapes were made by taking carbon steel balls, putting them in contact with each other, and running a  large current through the balls.  The balls melted together, in effect welding them to each other.  Using this  welding technique, we made many shapes.  We were concerned that the welding would distort the circularity of the balls, but apon inspection through a microscope, they appeared fine.  As further eveidence, the balls, after being welded, still formed a lattice.  If they were significantly distorted, they would not form a lattice anymore. We took our shapes, and placed them in a 2 dimensional plane, similar to a double paned  windowpane.  It consisted of two pieces of parallel plexiglass separated by .14 inch.  While the balls were  inside it, they were free to move in two dimensions but not the third.  The plane was attached to an  kicking machine that would, every three seconds, give the plane a kick, shaking the balls. There was, in effect, a hammer attached to the plane by a spring.  The machine would pull back  the hammer a fixed distance from the plane, and release it. The spring would accelerate the hammer and the hammer would strike the plane.

The machine was set up so that the angle of the plane with respect to horizontal could be varied.   Also, we had three standard distances the hammer could be pulled back to.  With a longer  distance, the spring was stretched more than for a short distance, and so the plane got a bigger hit. We placed about 2500 balls in the plane, attached the plane to the machine, and let the machine kick the balls for about an hour.  At one kick every three seconds, this amounted to about 1000 kicks.   We noticed the balls settled very much in the first few kicks, and barely moved at the end of an hour.  It  appeared the balls were asymptotically approaching a final amount of order.  The exact dependence of  ordering as a function of time would be a very interesting follow up experiment.

At the end of an hour, we took a picture of the balls in their final state with a digital camera.  We then had a  computer program scan the photograph for the center of each ball.  It did this by finding the local brightest spots, and storing these pixel numbers in memory.  Since we had limited resolution with the digital camera, the brightest spot on a ball might  not actually be the center of a ball.  If a ball consisted of four pixels, the brightest pixel would be the edge  of the ball, never the center.  In our experiment each ball was about fifteen pixels in diameter, so the effect  is smaller, but still present.  To minimize this effect, we had another computer program examine the pixels  surrounding the brightest pixel, and do a weighted average of the brightness and distance.  This was in  effect the same calculation as finding the center of mass of a bunch of particles.  By finding the center of brightness, we were able to more accurately locate the center of the balls.

The computer then calculated the distance from every ball to every other ball, and put these distances in bins.  We then made a histogram of bin number, corresponding to distance, and how many  occurrences were in that bin.  The computer never found any distances less than one diameter; if you did it would mean balls were overlapping, which can't happen.  In a perfect hexagonally close packed lattice, the  distances between the balls are 1, sqrt{3}, 2, sqrt{7},3 diameters.  These distances are a naturally occuring result from this kind of  lattice.  Thus, with a perfect lattice, balls were never 1.5 diameters apart.  So the histogram for a perfect lattice has very high and sharp peaks at all of the characteristic distances, and very few balls found  between characteristic distances.  The more disordered the balls became, the less predictable the distances  became.  If you had a ball not in the lattice, then it COULD be 1.5 diameters away from another ball.  So  fewer balls were the characteristic distances apart from each other, and more balls were non-characteristic  distances apart from each other.  As a result, the histogram for a disordered state had relatively low peaks,  and many balls were found between characteristic distances, so noisy valleys occured.

The way we would turn a photograph into a number representing the amount of order present  was to take the sum of the first five peaks and divide by the sum of the first few valleys.  The more ordered a state was, the higher the peaks and the lower the valleys were, so the higher the order number  was.  The disordered states had low peaks and high valleys (relatively) so the order number was low. With each shape, power setting, and angle we then got a number representing the amount of  order that the balls showed.  For a given shape, we considered the top five ordering configurations,  ignored the top one, and averaged the other four.  This was a number that represented the best a shape  could order, and we then compared order versus shape.