Ever since the quasicrystals were discovered in the 1980s, I have been fascinated by their peculiar structure. They have something that has long been considered mathematically impossible, namely 5- or 10-fold symmetry. In 2011, Dan Shechtman was awarded the Nobel Prize in Chemistry for his discovery of these crystals.

In my years as a researcher at Stockholm University, I have published more than 20 articles, most of them together with my wife and colleague Xiaodong Zou, on quasicrystals and their closely related approximants. Approximants are traditional crystals with the “permitted” 2-, 3-, 4-, and 6-fold symmetries, but parts of their structures also have atomic configurations similar to those of quasicrystals. Quasicrystals and approximants are found primarily in complex metal alloys. One of the crystals’ many fascinating properties is that the dimensions of their building blocks, the unit cells, are related to each other by the number τ (tau), approximately 1.618.

One of Markus Döblinger’s beautiful images of a diffraction pattern.

Illuminating a crystal with X-rays or electrons will produce a diffraction pattern. Quasicrystals and approximants have exceptionally beautiful and complex diffraction patterns in which the “forbidden” 10-fold symmetry becomes visible.

At a conference I attended ten years ago, a doctoral student named Markus Döblinger presented a poster with an incredibly beautiful diffraction pattern. The pattern was from the approximant PD1. I asked him on the spot if he would be able to come to Stockholm University as a postdoc to solve the atomic structure of PD1, that is, to investigate the relationship between the hundreds of atoms in the crystal. I did not know if I had the money to hire him, but with such beautiful images of diffraction patterns, money was a trivial problem that just had to be resolved.

A few months later, when Markus had defended his thesis, he came to Stockholm. We started working on PD1 and several other approximants in the PD series. He brought an amazing collection of electron microscope images and diffraction patterns with him, but in spite of hard work involving several other people, we did not get anywhere. After six months, Markus gave up and took a job in Germany. I was very angry and disappointed in myself.

The following year, a few colleagues and I found a rather large PD4 crystal and managed to solve its atomic structure using traditional methods of X-ray crystallography. Now we knew what one member of the PD series looked like. Since they are all related, we should have been able to solve the PD1 structure as well, but no.

Over the years, I occasionally encountered Markus’s amazing images of diffraction patterns. I used them in my teaching to promote the beauty of crystallography. Every now and then, I took on the diffraction patterns and worked hard for a few weeks to no avail. My frustration grew.

In 2011, I was asked to write an article in a special issue of Philosophical Transactions of the Royal Society dedicated to Alan Mackay. Alan Mackay is one of the world’s smartest crystallographers and the first person who came up with the idea that 5-fold symmetry is actually possible. I accepted, of course, and started to think about the content of the article; it should be about quasicrystals and approximants. Once again, I took out Markus Döblinger’s beautiful images and tried to solve their structure, but made no progress, as usual.

This time was different, however. One Saturday, as I sat hunched over the annoyingly beautiful images, I had an insane idea. Why not ask my youngest son, Linus, aged 10, if he would like to help? The night before, Linus had turned down a bedtime story in favour of a few sudoku puzzles. I had quickly noticed that he was smarter than me. He saw the number patterns much more clearly and corrected me:

“But dad, don’t you see, if we put an eight here, this row will be like this, and this column will be like that. Then we’ll end up with two eights on the same row, and that won’t work!”

I asked Linus if he would like to sit next to me and try to figure out the approximant structures. Linus, who is a good boy, said yes, and I started to explain what I was doing. Naturally, he knew nothing about metals, alloys, crystallography, diffraction patterns, or approximants. On the other hand, I had taught him the Fibonacci sequence (1 1 2 3 5 8 13 21 34 55 and so on, where each number is the sum of the previous two; for example, 5+8=13). He also knew that the division of two adjacent numbers in the sequence results in a value of approximately τ; 8/13 = 1.625, 34/21 = 1.619.

The structure of PD1. From Sven Hovmöller, Linus Hovmöller Zou, Xiaodong Zou, and Benjamin Grushko in Transactions of the Royal Society (2011).

Linus quickly got into it. As my brain became saturated with all the different (yet similar) images, Linus’s unspoiled brain was able to sort out the impressions. We sat at the kitchen table, which was full of images, and worked intensively all Saturday. After about eight to ten hours, we had succeeded – we had solved both PD1 and PD2! We continued to work just as intensively on Sunday and managed to solve PD3 and PD5 as well. In one weekend, Linus had solved what four qualified researchers had failed to do for eight years.

I was both happy and proud when I wrote the article about our work. But what was I to do about Linus? Had he been a doctoral student or researcher, the answer would have been obvious, but he had just finished fourth grade. At any rate, I decided to include Linus as a co-author. Just in case, I wrote to the editor of the journal: NB! One of the authors is only ten years old. I thought he might reply that no, this is not possible; this is a serious research journal (the world’s oldest, actually) and nothing for children. Or maybe he would reply cheerfully that he thought it was great and touching. But it was neither. I did not hear a word from the editor for ten months until the article was published. But then!

I received an e-mail from the editor that the article was finally in print. And yes, he had written on their website that one of the co-authors was ten years old. The same day, The Scientist and BBC Radio called and asked for interviews with Linus. Linus was sceptical, but the professional journalists took it easy and allowed him to give good answers to all the questions about what his contribution was. All in English. One of the questions was: “Was it harder than your school homework?” “Yes, a lot”, was the answer.

A Google search for Sven Hovmöller returns 3,000 hits, Xiadong Zou returns 10,000 hits, but Linus Hovmöller Zou returns 27,000 hits.

Now, that was a real story! as H.C. Andersen said.