Limited ball packs lead to sausage disaster

For a few balls, the sausage is on top in the 3D case

But what about lead? It would not be surprising that the three-dimensional case raises more questions than the optimal circular packing. There is at least one piece of evidence: Kepler’s conjecture states that an infinite number of similar spheres fill three-dimensional space better if they are stacked like cannonballs. In the first level, you arrange them along a triangular grid like coins in the 2D case, and in the second level, you place a ball in each gap. Then the third level again matches the first and so on.

However, if you only consider a limited number of fields, the situation is completely different, so we go back to the above example, where the oranges are wrapped in wrapping paper. If you have only one or two oranges, it is immediately clear how to arrange them optimally. But with three pieces the task is more complicated. You can arrange them in a row (sausage box) or form a triangle (pizza box) with them as before. So you’re in a similar situation to Three Coins, except now you’re dealing with balls. In order to find out which package is the most space-saving in this case, the sizes of the two arrangements can be compared.

It helps to divide the shell of the balls into individual geometric shapes and add their sizes. In the case of a sausage bundle, this is very simple: the shape can be divided into a cylinder and a ball with a total size¹⁶⁄₃piright³ & almost ; 16.76right³ to have. A pizza box is a bit more complicated: you get three half-cylinders, a triangular prism, and a ball that’s all the same size ¹³⁄₃piright³ +2√3right³ & almost ; 17.08right³ results. In this case, sausage packaging is more space-saving. And as it turns out, a sausage order really can be perfectly packaged.

Sausage, pizza or clusters: what’s the best way to arrange the balls?

If you add another ball, that is n = 4, one distinguishes between three different arrangements: again, the balls can be lined up one after the other (sausage) or distributed in plane (pizza), but you can also use all three spatial dimensions and stack them (“cluster fill”). Even for four balls, a sausage tin can be demonstrated to be perfect; It requires a smaller size.

However, with more balls, it gets more complicated: Experts assume soThat sausage pack up n = 55 balls is optimal – but for 56 balls, cluster packaging is clearly more space saving. Jurge Wells and Pierre Mario Gandini showed it in 1992. What exactly this grouping looks like is not clear: a better arrangement of balls than sausages can be found, but this cannot be proven to be optimal. Another arrangement may take up less volume.

The sudden transition from an ordered one-dimensional chain to a three-dimensional stack is known in specialized circles as the “sausage disaster”. Wills and Gandini were able to demonstrate that arrangements with 59, 60, 61, and 62 plus all groups containing at least 65 balls optimally form a cluster. This is for all numbers of other balls n n = 57, 58, 63 and 64 A sausage tin is assumed to be optimal. This means: with up to 55 balls the sausage package should be perfect, with 56 balls suddenly a cluster bundle and with 57 and 58 balls again the sausage would be the most space-saving – to be replaced by clustering again with 59, 60 and 61 ball. This doesn’t seem particularly intuitive. No one was able to prove this beyond a reasonable doubt.