We’ve gone from one to two to 8 to 24 dimensions, and the packing density dropped below 0.2%

It’s a constant battle. After every meal, we take the dishes off the dining table and hope to fit them into a fridge that already looks full. We usually manage it somehow, though it’s hard. But every time, I wish the dishes were square rather than circular. That way, we’d fit more of them more easily into the fridge.

Though there are reasons to make dishes circular rather than square. They are structurally more sound, for one thing. But more important: for a given capacity, a square container needs over 13% more material than a round one does.

Which means my problem of packing those round boxes in is likely to persist. No complaints, though. After all, plenty of mathematicians have wrestled with the problem of packing, to the extent that it is a well-known and widely-investigated area of mathematical inquiry. And just last week, a Ukrainian mathematician called Maryna Viazovska won a Fields Medal for her work with packing.

It’s not often that I can see a connection between what goes in my fridge and a Fields winner. Though let it be said, the connection is tenuous. Viazovska’s work is not just far more intricate, it involves other dimensions altogether.

Let me try to explain what I mean by that. Start with one-dimensional geometry—a line 10cm long. If I gave you several 1cm line segments, how many can you “pack" onto the longer line without overlap? Easy: 10 of them, end to end. You’ll have covered the entire line, or 100%.

Nothing really interesting there. Now think of a two-dimensional space. Consider a square table that’s 1m on a side. I have a number of circular coasters, each 10cm in diameter. I want to lay them out on the table, as closely placed as possible with no overlap. How many coasters can I place?

Easy again. Here’s one arrangement: One row of 10 coasters will go from one edge of the table to the opposite edge, and I can place 10 such rows. That is, I can “pack" this table-top with 100 round coasters. What fraction of the total area of the table-top have my 100 coasters covered? Well, the table-top is 1 square metre, or 10,000 sqcm. Each coaster has a radius of 5cm, which gives it an area of 25π, or about 78.54 sqcm. 100 of those cover an area of 7,854 sqcm, 78.54% of the table-top. (In contrast, 100 perfectly square coasters—remember my fridge yearnings?—each 10cm by 10cm, would have covered 100%.)

But you can improve on my 78.54%. After placing the first row, don’t line up the second so each coaster is directly below another, as I did above. Instead, shift it slightly so each coaster nestles in the space between two coasters immediately above. Only nine will fit in that row. Place the third row similarly, and it will hold 10 again. Continue this way—a “honeycomb" arrangement, each coaster touching six others—and you will fit 11 rows on the table. Some basic geometry will actually prove that. These alternating rows of 10 and nine coasters better my count of 100 coasters—this arrangement takes in 105, for about 8,247 sqcm, or 82.47% of the table. (With a more precise mathematical definition, this “packing density" in 2-D space is actually just over 90%.)

Move to three dimensions now. No doubt you’ve seen fruit vendors stack oranges or watermelons, invariably in the shape of a pyramid. If you take a hollow pyramid and pack it with spherical watermelons of the same size, you’ll find that when you can pack in no more, you’ve used a little less than 75% of the pyramid’s volume. The celebrated 17th Century German astronomer Johannes Kepler was intrigued by the problem of packing spheres. He suggested that this pyramid structure is the most efficient method, meaning that 75% can’t be bettered. It wasn’t until 1998 that his conjecture was proved.

So we’ve gone from one to two to three dimensions, and our packing density has sunk to under 75%. This is where Viazovska comes in. She works in spaces of dimensions higher than three— conceptually and imaginatively, of course, because in reality we are confined to our three-dimensional world. What’s the most efficient way to pack spheres—or really, sphere-like conceptual objects—in those higher dimensions? This may seem esoteric, maybe even useless. Far from it. How will I send this column to my editor, for example, and how will it reach the device on which you may be reading it? Such transmission of data can generate errors. Exploring sphere-packing in higher dimensions actually has implications for how we detect and correct those errors.

But sphere-packing is a hard problem. We don’t know the arrangement that will pack spheres most densely in four dimensions. Nor in five, six or seven. But Viazokska found an arrangement in eight dimensions that she called the E8-lattice. In a 2017 paper, she proved this theorem: “No packing of unit balls in [eight dimensions] has density greater than that of the E8-lattice packing." That density, she also proved, is about 25.37%.

But wait! Her paper caused something of a sensation in mathematical circles. From the Massachusetts Institute of Technology (MIT), the mathematician Henry Cohn wrote to ask if she would collaborate on investigating the 24-dimensional space, and a packing arrangement called the “Leech lattice". Within a week—yes, one week!—they, and two more colleagues proved that this lattice “achieves the optimal sphere packing density in (24 dimensions)." That density? Just under 0.2%.

So, we’ve gone from one to two to three to eight to 24 dimensions, and the packing density has dropped to below 0.2%.

I invite you to think in four dimensions. I’ve tried. It’s hard. So eight and 24? Give Maryna Viazovska that Fields Medal, I say.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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