Spheres are (almost) always packed most efficiently as sausages

Experiments and simulations provide deeper insights into the mathematical phenomenon known as the 'sausage catastrophe'

Researchers from Utrecht University and the University of Twente have investigated the mathematical sphere packing problem via experiments and computer simulations. Their study demonstrates the most efficient way to pack a finite number of spheres, complementing the mathematical proof of the so-called ‘sausage catastrophe’. The researchers published their study in the scientific journal Nature Communications on November 30.

What’s the most efficient way to transport 50 oliebollen (traditional Dutch pastries) to your friends' New Year's party? Mathematicians calculated already decades ago that a linear, sausage-like arrangement or bag provides the best packing. While one might assume that tightly clustering a large number of spheres would be the most efficient way to pack them, mathematicians have shown this is not always the case. At least not for all numbers of spheres.

There's something rather peculiar going on: the sausage-shaped bag only the most efficient packing for quantities up to 55 spheres. Beyond 55, a cluster becomes the best packing. It's an abrupt change that mathematicians refer to as the 'sausage catastrophe'.

Een hoeveelheid sinaasappels, (inefficiënt) gestapeld.
A number of (inefficiently) arranged spheres.

An infinite number of spheres

The sausage catastrophe is part of the sphere packing problem — a mathematical problem that has occupied scientists for centuries. It explores the most efficient way to pack equally-sized spheres in such a way that the packing occupies as little space as possible.

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

Experiments in the lab

So, what happens when you are faced with packing 40 spheres? Or 94? That's precisely what the team led by Marjolein Dijkstra set out to explore. Rather than relying solely on mathematical calculations, they conducted experiments and computer simulations.

The researchers placed spherical nanoscale particles (colloids) into a microscopic, flexible container known as a vesicle. They investigated these vesicles in real-time under a microscope as well as in simulations. By varying both the quantity of colloids inside the vesicles and the external pressure applied to the vesicle, the researchers studied the arrangement of the particles under different conditions.

Afbeelding van bollen, gerangschikt langs een lijn. De afbeelding bestaat uit twee delen: het bovenste laat een beeld zien dat gemaakt is met een microscoop, het onderste een beeld van een computersimulatie.
Nine colloids packed in a vesicle, in a lab experiment (upper image) and a computer simulation (lower image).

Simulations

Testing a mathematical theory in the lab is challenging, discovered Marjolein's team. "The vesicles kept rupturing with more than nine particles. This prevented us from testing how the stacking of particles would change if we added more than nine," explains Dijkstra. In 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 assumed with up to 150 particles in simulations.

Sausage-shaped packaging

The computer simulations revealed that packing spheres in a sausage is most efficient up to a quantity of 55 spheres. Marjolein was surprised herself: "I even drew sketches of 55 spheres in a row. At first glance, you wouldn't expect this to be the most compact cluster."

However, when attempting to pack 56 spheres in a vesicle, the most efficient packing method altered to a compact, three-dimensional cluster. Strikingly, for 57 spheres, the packing reverted to a sausage shape. While mathematicians found that a sausage is the most eficient packing for for 58 and 64 spheres, a sausage is also the most efficient packing, the present study contradicts this, showing that compact clusters perform better. These findings demonstrate that the sausage catastrophe can be observed not only through mathematical proofs but also in (simulated) experimental settings.