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A maths prize for packing balls

By , 21 July 2022

Box bursting open with green balls spilling out.

Packing spheres in 8D and beyond won Maryna Viazovska a Fields Medal
Image: ©iStock.com/Gearstd

Have you ever struggled to pack balls back into your toy box? The Fields Medal, one of the most famous prizes in mathematics, was recently announced. One of this year’s winners, Ukrainian mathematician Maryna Viazovska, is an expert in packing balls in boxes. But there’s a catch! Her findings work in 8 dimensions (D) and beyond.

Pipes stacked together, seen from the end looks like a honeycomb shape.

This is the best way to pack circles together
Image: Flickr.com/brewbooks CC BY-SA 2.0

Before we look at this achievement, let’s look at some examples that are easier to come to terms with. In 2D, balls are just circles. We can ask ‘what’s the best way to arrange lots of equal circles, so there’s not much wasted space’? You can try working it out on paper. Put the circles in a hexagon pattern, similar to honeycomb.

From 3D, balls are called spheres, and the maths gets more complex. There are actually 2 solutions that give the tightest packing. You can arrange spheres in a flat honeycomb pattern, and then keep adding honeycomb layers on top. Or you can start with a square arrangement of spheres, and then put spheres on top in the biggest gaps.

Beyond 3D, it’s extremely difficult to imagine what a ball could even be, let alone work out how well they pack together. That hasn’t stopped mathematicians from coming up with good arrangements of spheres in 4D, 5D and beyond!

Maryna found a way of packing spheres in 8D that looked particularly good. She was able to calculate how much of the packing was spheres and how much was gaps, and then proved that it was completely impossible to pack them any better. As a follow-up, she worked with several other mathematicians to do the same thing in 24D too!

The Fields Medal

The Fields Medal is one of the most famous prizes in mathematics. Up to 4 prizes are given out once every 4 years, and only to mathematicians under the age of 40. Unlike the million-dollar Nobel Prize, the Fields Medal comes with about $15,000 in prize money.

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  1. What about the obvious gaps between the groups of packed pipes in your example photo? Surely the most efficient way to pack them would be in triangular bundles that alternate base to apex to fill a flat/rectangular area, or can be formed into the hexagonal shape to fill a rounder space? Thinking of pool and snooker frames…

    1. Yup! The best way is really just to keep the pattern inside the hexagon going, kinda like a honeycomb. I just really liked that pipe picture!


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