THREAD: My latest @QuantaMagazine is the blow-by-blow tale of how a small team unraveled a conspiracy hidden deep in the laws of nature.

But it all started with simple math. https://www.quantamagazine.org/geometry-reveals-how-the-world-is-assembled-from-cubes-20201119/

If you take a polygon and slice it up with random lines a few times, the resulting smaller shapes will average... four vertices. In an average sense, they'll be rectangles no matter what shape you started with.

You can even try this at home with printer paper like I did.

Okay, then make it harder. Go to 3d. Slice open a block with random planes, not lines, like you're taking a knife to a head of cabbage.

The resulting smaller shapes will average...eight vertices, six faces. On average, you've got yourself some cubes.

A few years ago, Hungarian mathematician Gabor Domokos started wondering about these geometry problems.

To be sure his 3d answer -- cubes -- he needed to check a book called Stochastic Geometry. Either the same answer would be in the book in some form, or it was wrong.

Correction: the book is Stochastic and Integral Geometry.

Regardless, it's a "bible." It's brilliant but inscrutable, a forbidden tome with few figures. "Only five people in the world can read this book," Domokos told me, "and I am not one of those people."

Real Name of the Rose/Borges vibes here, and that won't be the last time in this story.

Luckily, one of those chosen people helped. They found that there was a formula in the book that showed a 3d block sliced by random planes would produce, on average, cubes. Confirmed.

Okay, so geometry likes cubes. Big deal; abstract math says all sorts of things. But then the Hungarian team started looking at reality. They found eroded shards of dolomite on a mountaintop. Painstakingly, a lab tech counted them.

On average? Six faces, eight vertices. Cubes.

This wasn't Minecraft or a LEGO set. These were real rocks. And not just these but A LOT of broken rocks, once you counted, averaged out to a cube. Just how deep did this conspiracy go?

This is where I add that Plato, you know that Plato, of Athens, had associated the cube with the element Earth. Not sure if this means anything and neither are the researchers, but damn it's cool.

But then again this journey was far from finished. They had geometry offering a fantasy -- dare I say "ideal" or "Platonic" -- view of polygons and polytropes being sliced by lines and planes.

Then they had some real rocks that had apparently gotten the message. But how??????

You see, at least in my own mangled philosophical view, math exists in it's little perfect realm, and physics translates it into the world and into things like rocks. Like a game of telephone.

But this case worked almost **too** well.

"It was geometry with an exact prediction that was borne out in the natural world, with essentially no physics involved,” said Douglas Jerolmack, a geophysicist recruited to help out. “How in the hell does nature let this happen?”

Basically, they had to figure out a very big missing piece: the physics that would inscribe this subtle geometry into real stone.

And for that, they needed a way to describe how things fall apart.

Imagine a mosaic, made of cells that tile together with no gaps or overlaps. Anything that fragments apart needs to meet this definition.

But those rectangles and cubic averages represented only one way to make a mosaic, albeit one nature seemed to prefer.

Here's the new descriptive language for mosaics, for how things fall apart. You count up the average number of vertices per cell. Then you count up how many cells, on average, vertices share; are a vertex for.

So a mosaic of rectangles: 4, 4
Of hexagons: 6, 3
Of cubes: 8, 8

The important thing here is that now you can look at any mosaics in nature (or made by humanity) and express them this way.

So you can talk about when natural mosaics follow the rectangle/cube average laws from geometry, and when the disobey.

And here the team discovered something new, which geologists told me they are actually more excited about then the whole "world is made of cubes" top line. The geometry of real broken rocks tells a story about their geology. Specifically, it speaks to stress.

Consider a flat sheet of isotropic rock, a pretty 2d system. Subject it to compressive forces; squeeze it in. When it cracks, it will follow the geometry. It will break into, on average, rectangles.

Similarly, consider a brick. Compress it, or shear it like pushing your finger into the side of a deck of cards. On average it will break into cubes per the geometric prediction.

Here's the key thing: compression and shear are super prevalent, geologically. Aha, you say. The world really is* made of cubes! Nature listens to what geometry tells it to do!

*caveats applying

But the team also realized that there's another geometrical story out there, which points to a separate geological origin.

So mudflats that crack open and then heal? Their cells look six-sided, like hexagons. Not listening to the "average" geometry predictions at all.

And this isn't just mudflats. In a 3-day sprint in Budapest, the team also starting thinking about the grandest possible scales they could apply their theories to.

They came up with the mosaic formed by Earth's tectonic plates.

"The pattern looked familiar, and Jerolmack called the others over. 'We were like, oh shit*,' he said."

*"wow" in the published piece but you get the raw truth here

The tectonic plates are roughly hexagonal!

Notice I said "roughly." Okay, they averaged 5.77 vertices. Not 6 like perfect hexagons. And that was kind of a bummer.

But they kept thinking. Then lightbulb moment. Perfect hexagons tile together in Euclidean space, on a plane. But the Earth isn't a plane, y'all.

Suppose instead you wanted to tile across the skin of a sphere, hewing mostly to hexagon-ish shapes, and you had the Earth's 52 tectonic plates to do it... in this case, geometry says you would have **exactly** 5.77 vertices. Just like the real plates.

This seems like a detour. It's not. In trying to prove the rule of cubes, the team had also stumbled on this rarer case: hexagons in 2d, and then hexagonal volcanic columns in 3d.

In these cases, the rock broke when forces pulled it apart instead of compressing it together.

So at the end of the day, it's not just that the world, on average, is made of cubes. It's that you have a descriptive language for how things fall apart, and that language has **meaning**.

You see rectangular and cube-ish pieces, like most of the world, in most geological cases? -> that rock was smashed, compressed

You see hexagon-ish pieces, like mudflats or the tectonic plates? -> that rock was pulled apart

Geologists I spoke to were hopeful and excited that this research can ultimately be used that way, diagnostically, to take a fractured landscape and discern the kind of forces that fractured it.

Which leaves the researchers, at the end or perhaps the beginning of a quest through pure math and dense volumes and real rocks and computer simulations, still wondering what it all means.

This Plato guy, after all, talked about ideal geometric forms that were only visible to us as shadows.

“This is literally the most direct example we can think of. The statistical average of all these observations is the cube,” Jerolmack said.

“But the cube never exists.”

END