The Art of Stacking Spheres, Cans, and Dimensions Beyond Imagination

 The Science Behind Packing

Ever found yourself standing in the grocery store, marveling at those perfectly stacked pyramids of oranges, feeling a bit smug because, hey, you just solved a packing problem? No? Well, congratulations anyway! You’ve just dipped your toe into the vast, perplexing, and often award-worthy world of “Packings” in mathematics. You see, packing problems are deceptively simple, yet maddeningly difficult — like trying to solve a Rubik’s Cube while blindfolded or deciding which leftovers to fit in your already crammed fridge.

What’s the Deal with Packing?

Packing is all about fitting objects into a confined space with as little wasted room as possible. Sounds easy enough, right? But here’s the kicker: while your neatly arranged cans of soup seem straightforward, mathematicians are out there scratching their heads trying to figure out how to pack spheres in 24-dimensional space. Yes, 24. The kind of space that makes our measly 3D world look like child’s play.

Let’s start simple — the packing problem you already know: stacking a bunch of identical spheres (or oranges) to fill as little space as possible. This was famously cracked by Johannes Kepler in the 1600s, who proposed that a pyramid arrangement is the most efficient way to pack spheres. Great. Done. Except… not quite. Despite this intuitive answer, it wasn’t until 1998 that a mathematician named Thomas Hales officially proved it. And it required computers. Lots of computers. That’s three-dimensional packing. But what about higher dimensions? That’s where Maryna Viazovska, the rock star of packing problems, comes into play. In 2016, she solved the eight-dimensional packing problem (yes, that’s a thing) and even tackled the mind-boggling 24-dimensional case with some magical mathematical functions called “modular forms.” For this, she earned the Fields Medal in 2022, which is essentially the Nobel Prize of mathematics but with fewer televised award shows and red carpets.

But Wait, There’s More!

Packing isn’t just about spheres. It’s about everything! M&Ms, for instance, have their own special place in the packing pantheon. Physicist Paul Chaikin showed that oblate spheroids (fancy talk for “M&Ms”) pack more efficiently than spheres when randomly thrown together. Now that’s a sweet discovery — literally. Rumor has it Chaikin received a lifetime supply of M&Ms for his troubles. Who said math couldn’t be delicious?

And it doesn’t stop there. Mathematicians have also been hard at work figuring out what happens in different shapes and dimensions. For example, some shapes, like octagons with rounded edges, are just *terrible* at fitting together. It’s like trying to stuff square pegs into round holes. Mathematicians have long believed these octagons are the least packable two-dimensional shapes, but actually proving it has been a century-long headache — kind of like trying to find the end of a roll of tape. However, in May 2023, two mathematicians got one step closer to solving this frustrating puzzle.

The Ultimate Fractal Party

Now, let’s get a bit fancy. Apollonian circles are my personal favorite packing puzzle — and yes, you’re allowed to have a favorite math problem, thank you very much. The idea here is that you can’t pack circles without leaving gaps, so the goal is to cram successively smaller circles into those gaps, creating a stunning fractal pattern. Think of it as trying to fit an infinite number of coins in a jar, with each new coin getting smaller and smaller. It’s fractals all the way down, folks. Last year, two brave students found that certain expected patterns in Apollonian circle packings don’t always show up, challenging long-held beliefs in number theory. It’s like showing up to a formal dinner party and realizing everyone’s wearing jeans. Chaos!

What’s Next in Packing?

Now that you’ve dipped your toes into this wild world of packings, it’s time to sit back and appreciate just how much these seemingly simple problems have to offer. Whether it’s stacking oranges, spheres, or entire dimensions, packing problems give us a tantalizing peek into the complex beauty of modern mathematics.

If you’re itching for more, dive into Erich Friedman’s website for an endless list of packing problems that will make your head spin. Or take a trip down the mathematical rabbit hole with Maryna Viazovska’s Fields Medal-winning work on higher-dimensional sphere packings. Just be sure to pack some snacks. It’s going to be a wild ride.