Imagine trying to stack oranges as tightly as possible so that no space is wasted. Pretty simple at the grocery store. Now imagine doing this in a world of eight dimensions, a place that exists not in physical reality but in the abstract architecture of mathematics, and proving without a doubt that you have found the most perfect arrangement possible. For more than 400 years, the greatest mathematical minds on Earth couldn’t do it. In 2016, a quietly determined Ukrainian mathematician named Maryna Viazovska did just that. She was 31 years old and working alone in Berlin. What he created wasn’t just a solution, it was a masterpiece.

Marina solved a 400 year old problem! viazovska

The problem of packing spheres seems deceptively simple: What is the most efficient way to arrange identical spheres so that they fill space with as few wasted gaps as possible? In three dimensions, the answer is intuitive, a pyramid shape, the way fruit is stacked in a market. Johannes Kepler proposed this arrangement as optimal in 1611, but it took until 1998 for a formal mathematical proof to be completed, and even that proof required long, controversial computer calculations running hundreds of pages.However, move beyond three dimensions, and the problem becomes an entirely different animal. In dimensions four, five, six, seven, mathematicians had almost nothing. As MIT’s Henry Cohn described it after Viazowska’s success: “It is this catastrophic gap in our knowledge that is almost shameful to humanity.”Of all the higher dimensions, eight were special. Mathematicians long suspected that the answer lay in a structure called the E8 lattice, an arrangement of extraordinary symmetries that exists only in eight-dimensional space. More than a decade before Viazovska’s proof, Kohn and mathematician Noam Elkies had calculated that the E8 lattice was accurate to within one billionth of one percent of the theoretical optimum. They could almost touch the north. But they could not prove it. No one could do it.

Evidence that will shock the world in just 23 pages

Viazovska had been struggling with this problem for years. The key insight came from an unexpected direction: his doctoral work on modular forms, a type of highly symmetric mathematical function that normally resides in the world of number theory, which appears to be very far from geometry. He studied under the renowned mathematician Don Zagier at the Max Planck Institute for Mathematics in Bonn, where he completed his PhD in 2013.His success was in using the tools of Fourier analysis and modular forms to construct a special “magic function” that could serve as an exact upper bound, a mathematical ceiling for which dense regions could possibly be packed. When he compared that roof to the E8 mesh, they matched perfectly. This type of exact match is exceptionally rare. In mathematics, this is the equivalent of cutting a key in the dark and finding it to open a lock.

Visualization of the E8 lattice, a highly symmetric structure that solves spherical packing in eight dimensions.

The proof was uploaded to the academic preprint server in March 2016. It was 23 pages long. Earlier attempts at related problems spanned hundreds of pages. The mathematical community was stunned not only by the solution, but also by its beauty. Experts described it as “surprisingly simple” and praised its clarity and originality. Within a week of its publication, Viazowska had solved that version of the problem using a structure called a Leech lattice, joining with four colleagues to extend the same approach to 24-dimensional space. Two big problems, both solved within a few days.

Why are any of these matters beyond pure mathematics?

It would be easy to dismiss this as beautiful but impractical mathematics conducted in a rarefied universe that normal life never touches. That would be a mistake.Sphere packing in higher dimensions is deeply linked to error correction codes, the technique that allows information to be accurately transmitted across noisy channels. Every time you stream a video, make a phone call, or receive a file without any corruption, error correction codes are working quietly in the background. The mathematical structures that govern optimal field packing are the same structures that underlie how to efficiently encode and decode information. Viazowska’s work not only satisfied centuries of curiosity, but it expanded the theoretical foundations on which applied mathematicians and engineers work.Their results have also opened new doors in theoretical physics and cryptography, fields where the geometry of high-dimensional space has direct and practical consequences.

the woman behind the math

Viazovska was born in Kyiv in 1984, the eldest of three sisters, and showed an early passion for mathematics through school competitions and Olympiads. He received a bachelor’s degree from Taras Shevchenko National University of Kyiv, a master’s degree from the University of Kaiserslautern in Germany, and a doctorate from the University of Bonn. She is now full professor and chair of number theory at the École Polytechnique Fédérale de Lausanne in Switzerland.

Her teenage son Michael once recalled being the last child picked up from a Berlin kindergarten while his mother was busy working on an E8 proof. When he later learned of the Fields Medal, he reportedly said: “Now I understand why he worked so hard.”In July 2022, Viazovska was awarded the Fields Medal, widely considered the Nobel Prize of mathematics and restricted to mathematicians under the age of 40. She became only the second woman to receive it in the 86-year history of the prize, after Iranian mathematician Maryam Mirzakhani in 2014. The award was announced just weeks after Russia began its full-scale invasion of Ukraine. In interviews, Viazowska spoke quietly but strongly about her country: “Tyrants can’t stop us from doing mathematics. At least there’s something they can’t take away from us.”More than four centuries after Kepler first asked the question, the answer in eight dimensions came not through brute computational force, but through a woman’s creative intuition and a set of tools borrowed from an unrelated corner of mathematics. This is what makes Maryna Viazowska’s story so remarkable and worth remembering.