| "Packings" are a perfect gateway into pure mathematics. They're the subject of a serious area of research filled with open problems — the kind mathematicians get major awards for solving — and yet you encounter them all the time in regular life.
When you arrange cylindrical cans in a cabinet, you're solving a packing problem. Those pyramids of oranges you see in the produce sections of grocery stores settle an important 17th-century conjecture by the astronomer Johannes Kepler. At the start of the Covid-19 pandemic, everyone became extremely conscious of the challenges of social distancing when many people are packed into a room.
Almost every packing problem is similarly simple to state and straightforward to visualize, especially compared to the abstract and abstruse concepts that populate other areas of math.
But they're incredibly hard to solve — so much so that experts still haven't figured out how to answer some of the most obvious-seeming questions, like how best to arrange balls (albeit four-dimensional balls) in a box. This makes packing a nice entrée into Quanta's math section: The questions mathematicians ask aren't that different from the ones you might.
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Let's start with the prototypical packing problem: How do you arrange a bunch of identically sized spheres to leave as little space between them as possible? The grocery-store arrangement, a pyramid, may seem obvious, but it wasn't until 1998 that the mathematician Thomas Hales finally proved that it's the best solution. And his proof, which relied heavily on the use of computers, only worked for three-dimensional space — higher dimensions continued to evade solution.
Then in 2016, the Ukrainian mathematician Maryna Viazovska figured out the best way to pack eight-dimensional balls into eight-dimensional boxes. Along with colleagues, she then used a similar argument, involving "magic" functions called modular forms, to prove the 24-dimensional case, and to attack a vast suite of other problems. In 2022 this work earned her a Fields Medal, math's highest honor.
When you keep at least a certain distance from all your neighbors, you're really surrounding yourself with an imaginary sphere of impenetrability. So any situation that involves objects in a space repelling one another is somewhat related to sphere packing. Viazovska found that this connection becomes particularly powerful in either eight- or 24-dimensional space. The arrangements of spheres she'd found in these dimensions, called an E8 lattice and a Leech lattice, respectively, turned out to be "universally optimal" — meaning that they're also the best solution to all kinds of other problems, like what the lowest-energy configuration of a bundle of electrons looks like.
What about other dimensions? Mathematicians have been trying to estimate how efficiently you can pack spheres in any dimension since the turn of the 20th century. But progress more or less stalled after 1947 — until late last year, when a team gave the best recipe yet for packing spheres in an arbitrary number of dimensions.
Understanding packing isn't just about finding the most stuffable shapes. Since the 1920s, mathematicians have studied shapes that are downright awful at living together. For example, they've long believed that an octagon with rounded edges is the least packable 2D shape there is, but proving it has been deceptively tricky. After almost a century with little progress on the problem, two mathematicians proved a conjecture in May that takes them partway toward this goal.
Apollonian circles are my personal favorite packings. You can't pack circles without creating gaps, so this problem is all about shoving successively smaller circles into those spaces. The resulting pattern — circles all the way down — is one of the earliest documented fractals. Last year, two students showed that circles with certain properties, which had been expected to always appear in these packings, sometimes don't. The work casts fresh doubt on other important expectations in number theory.
Because they merge conceptual simplicity with technical complexity, packing problems — and related problems about tiling — might offer the clearest peek into how modern math works. Maybe these more accessible problems can whet your appetite for other important mathematical breakthroughs that you don't encounter day to day. | 
