A 120-Year-Old Puzzle, Photographed in a Droplet of Helium

Here is a puzzle that sounds simple and turns out to be very hard. Take a sphere. Put a number of identical electric charges on its surface. The charges repel each other, so they push apart until they settle into the arrangement with the lowest possible energy. What does that arrangement look like?

Joseph J. Thomson posed the question in 1904, while working on his "plum-pudding" model of the atom. For a handful of charges, the answers are known: four charges sit at the corners of a tetrahedron, six at the corners of an octahedron. For larger numbers, the number of possible arrangements grows so fast that finding the best one becomes extremely difficult. In 1998, mathematician Stephen Smale listed the Thomson problem as Problem No. 7 among his 18 major mathematical challenges for the 21st century.

Now, for the first time, physicists have photographed such arrangements directly. The work, led by the University of Innsbruck and published in Physical Review Letters on 26 October 2026, turns an abstract optimisation problem into something that can be seen under an electron microscope.

Gold dots mark where the charges sat

The experiment, by Paul Scheier's team at Innsbruck's Department of Ion Physics and Applied Physics, with collaborators from the University of Leicester, Durham University and the University of Cambridge, uses droplets of superfluid helium — helium cooled to a few tenths of a degree above absolute zero, where it flows without internal friction.

The researchers produce tiny helium nanodroplets and charge them several times with an electron beam. The charges move to the droplet surface and arrange themselves there, almost undisturbed. The droplets then pass through gold vapour. Gold atoms attach to the charge centres and grow into singly charged nanoparticles of about 700 atoms each, marking exactly where each charge sat.

When a droplet lands on a transmission electron microscopy grid, the helium evaporates. The gold particles remain, as an imprint of the original charge arrangement. "By tagging the charge centres with gold nanoparticles, we can preserve and image these arrangements after the helium has evaporated," the team explains.

The Innsbruck team analysed 25 imprints containing between 5 and 36 charges per droplet. The observed patterns closely matched the lowest-energy configurations predicted by theoretical calculations. Larger droplets, with radii up to about one micrometre, showed a different picture: the gold particles formed a hexagonal lattice interrupted by isolated fivefold defects — the kind of flaw that geometry forces on a curved surface, where a perfect hexagonal pattern cannot close neatly around a sphere.

What it does — and does not — prove

This is an experimental way to see charge ordering, not a general mathematical solution to the Thomson problem. The researchers compared a limited set of small configurations with existing predictions, and larger droplets presented reconstruction problems that limit detailed comparison at higher charge numbers. What the method opens up is a new experimental route: a way to watch how charge order develops from small, well-defined configurations to much larger systems on curved surfaces, and to check mathematical models against real patterns.

For students, the experiment is a beautiful meeting point of subjects. The question is mathematical — how to spread points evenly on a sphere. The laboratory is physics — ultra-cold helium, electric charge and electron microscopy. And the answer matters beyond the puzzle: similar patterns appear in the arrangement of proteins on virus shells, in the structure of fullerenes, and wherever nature must pack identical units onto a curved surface.

If your child likes questions that sound easy until you try to answer them, a first lesson in physics or maths can start with four charges on a ball — and end with gold dots in a droplet colder than deep space. Book your first lesson

Sources:

Pohl, T. et al., "Real-Space Imaging of Thomson Charge Configurations in Superfluid Helium Nanodroplets", Physical Review Letters 137, 183001 (2026), published 26 October 2026. DOI: 10.1103/qby4-bm6q: https://journals.aps.org/prl/abstract/10.1103/qby4-bm6q
Phys.org, "Mathematical puzzle from 1904 takes physical form in gold-studded helium droplets", 27 October 2026: https://phys.org/news/2026-10-mathematical-puzzle-physical-gold-studded.html

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