Spheres are (almost) always packed most efficiently as sausages

Physical experiments and simulations offer more insight into mathematical phenomenon ’sausage catastrophe’

Researchers from Utrecht University and the University of Twente have investigated the mathematical sphere packing problem through physical experiments and computer simulations. The study illustrates how a finite quantity of spheres can be packed most efficiently. Their findings complement mathematical proof of the so-called ’sausage catastrophe’. The researchers published the study in the scientific journal Nature Communications on November 30.

How do you transport 50 oliebollen (traditional Dutch pastries) to your friends’ New Year’s party in the most efficient way? Decades ago, mathematical calculations showed a long sausage-shaped bag to be the answer. While one might assume that tightly clustering a large number of spheres would be the most efficient way to pack them, science reveals this isn’t always the case. At least not for every quantity of spheres.

There’s something peculiar at play here: the sausage-shaped bag being the best way to pack spheres only applies to a quantity of up to 55. For 56 bulbs, a cluster becomes more practical. It’s a sudden shift that mathematicians refer to as the ’sausage catastrophe’.

An infinite number of spheres

The sausage catastrophe is part of the sphere packing problem - a mathematical puzzle that has occupied scientists for centuries. The problem revolves around arranging non-overlapping spheres in such a way that the space between them is minimized, occupying as little space as possible, including both the spheres and the air in between.

What makes this matter so complex? Here’s the gist of it: when scientist Johannes Kepler pondered over the sphere packing problem in 1611, his calculations were based on an infinite quantity of spheres, whereas in reality, you always deal with a finite quantity.

Experiments in the lab

So, what happens when you need to pack 40 spheres? Or 94? That’s what the team led by Marjolein Dijkstra set out to explore. They didn’t rely on mathematical calculations but instead conducted physical experiments and computer simulations.

The researchers placed spherical nanoscale particles (colloids) into a microscopic, deformable container called a vesicle, both in the lab and in simulations, which they could observe in real-time under a microscope. By varying both the quantity of colloids inside the vesicles and the external pressure applied to the ensemble, the researchers mapped out how the arrangement of the particles changed under different conditions.

Simulations

Testing such a mathematical theory in a lab can be challenging, as Marjolein’s team discovered. "The vesicles kept rupturing with a quantity of nine particles. This prevented us from testing how the stacking of particles would change if we added more than nine," explained Dijkstra. In the computer simulations, the properties of the vesicles could be adjusted to prevent rupturing with a larger quantity of particles. The researchers thus studied the shapes the vesicles took up to a quantity of 150 particles.

Sausage-shaped packaging

The computer simulations revealed that packing spheres in a sausage-shaped bag was optimal up to a quantity of 55 spheres. Marjolein herself was surprised: "I even made sketches of 55 spheres in a row; you wouldn’t think that’s the most compact cluster."

When attempting to pack 56 spheres in a vesicle, however, the most efficient packing method changed to a compact, three-dimensional cluster. And for 57 spheres, it’s back to a sausage shape. While mathematics asserts that for 58 and 64 bulbs, a sausage is also the most efficient packing, the current study shows that compact clusters actually work better. The researchers’ findings demonstrate that the sausage catastrophe can be observed not only through mathematics but also in (simulated) experimental settings.