September 26, 2022

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Math Prize: How do oranges stack in eight dimensions?

Math Prize: How do oranges stack in eight dimensions?

“Three weeks ago, my life changed forever in a dramatic way I could never have imagined.” This is how Ukrainian mathematician Marina Vyazovska began on March 16 her presentation At the Eidgenössische Technische Hochschule in Zurich. Her story, About “Fourier Interpolation,” is dedicated to 21-year-old compatriot Yulia Zhdanovska, a rising star in mathematics and computer science, whose life ended abruptly during the bombing of Kharkov, Ukraine’s second city.

The 37-year-old, who has been a professor at the Federal Polytechnic School in Lausanne, Switzerland since 2018, was awarded the prestigious Field Medal in Helsinki on July 5, among other things for her account of how to optimally package high-dimensional spheres. Her name has already circulated on Internet forums, sometimes accompanied by comments along the lines of “freedom for Ukraine” – as if she deserved the award because of her nationality. But Fields Medals are all about one thing: groundbreaking mathematics. It is not the Eurovision Song Contest.

For Viazovska, the university in Kyiv was “the best place to study mathematics”. The level was very high. “After obtaining a bachelor’s degree in Ukraine, most of them choose to look for a job,” she once said in an “But” interview. Viazovska did not like it, she felt that she had not yet graduated.

Marina Vyazovska. Photo by Fred Merz / Lundy 13

So in 2005 I traveled abroad. She did her master’s degree in Kaiserslautern, Germany, and moved to Bonn for her Ph.D. In 2016 – now a postdoctoral student in Berlin – Vyazowska announced a solution to a notoriously difficult problem. I have shown how 8-D domains can be stacked as efficiently as possible. breakthrough. Its solution attracted the world’s attention.

How do you search for such fields? It is impossible to imagine them. But it can be described by formulas. In our familiar three-dimensional world, points are named with three coordinates. Similarly, the eight dimensional points are marked with eight coordinates.

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In two dimensions, the balls are flat circles. To be able to fit as many circles as possible on a surface, divide the surface into a grid of regular hexagons, the well-known honeycomb. Then the circles occupy approximately 91 percent (pi divided by the square root of 12) of the entire plane. This was proven to be the most efficient way of laying out circles in the 1890s.

right in the well

3D Version – How do you stack oranges as efficiently as possible? – He also has a clear answer. Lay one layer of oranges following the circles pattern, then place another layer on top of them with the same pattern, so that each orange falls into the well of the first layer completely, and keep repeating this layer after layer. The area occupied by an orange is about 74 percent (pi divided by the square root of 18). It was only proven in 1998 that there is no more efficient stacking. At the time, the evidence was met with some skepticism because it relied heavily on computer calculations, but now no one doubts it.

The next logical question for mathematicians is how to optimally stack balls of higher dimensions. The difficulty is that there is no solving method that can be used in all dimensions. Each dimension has its own quirks. The only general thing you can say about it is that the distance between the balls increases as the distance increases.

Dimensions 8 and 24 are special. The spacing will be large enough to fit a new field in between. The balls fit a mesh perfectly, just like a honeycomb in two dimensions. The professional talks about the E8 network and the Leech network. In these grids, the balls fill over 25 percent (pix 4 divided by 384) and 0.19 percent (pix 12 divided by factor 12), respectively. In this last 24-dimensional case, where each sphere touches at least 196,560 contiguous spheres, the vast majority of space remains empty.

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It’s been known for some time that these nets exist, just like you can easily stack balls in them. But the lack of stacking that saves more space has not been proven yet. In 2003 there were developments that could lead to proof. These attempts failed, but in 2016 Vyazovska succeeded in the 8th dimension. Shortly thereafter, she and some colleagues were able to extend her proof to the 24th dimension. Unlike “normal” 3D spheres, no computer calculations of proof were required.

Corrupted QR Code

Searching for higher-dimensional field stacks has applications in error-correcting code theory. Can a QR code printed with torn corners or coffee spots be read? Can you restore the undamaged copy of a digitally sent message that has become corrupted? In principle, yes, as long as the damage is not too great.

The message to be sent is converted into tokens matching the domain centers. If errors occur, the bits in the password change, but as long as there aren’t many, the code word remains within the domain. The damaged code can then be corrected by attaching it to the center of the ball.

For this application, Viazovska’s guide to optimization is not necessary. Previously, E8 and Leech networks were also useful for debugging applications. But for Viazovska, these types of applications are just a byproduct. Thanks to her work, new horizons in pure mathematics appeared.