# 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 size163piright3 & 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 133piright3 +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.