She sells hexagonal rocks by the seashore

Hexagon-shaped volcanic rocks at St Mary’s Island, near Udupi in Karnataka.Premium
Hexagon-shaped volcanic rocks at St Mary’s Island, near Udupi in Karnataka.
A Matter of Numbers
6 min read . Updated: 08 Apr 2021, 10:37 PM ISTDilip D’Souza

Forces acted on these rocks from the inside towards the outside, giving it a familiar shape

About 2km off the coast of Malpe Beach—near Udupi in Karnataka—is a tiny scrap of land known as St Mary’s Island. You can reach it only by boat from the mainland. Nobody lives there, but it has clean beaches with plenty of shells, swaying palms and striking views of the sea. There are a few structures, built only to give visiting tourists some shelter from the sun.

That’s right, tourists. St Mary’s Island gets plenty of them. Now they don’t make the trip to pick up seashells, nor to admire the palms—after all, they could do that on nearby Malpe Beach. So why do they pile into those boats and come?

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In three words: columnar rhyolitic lava. Really just volcanic rock, which isn’t unusual and is found the world over. That word “columnar" is the clue. There are rock formations all over the St Mary’s beach, as there are on many other beaches. But the ones you find here are startling. For instead of having no particular shape, they are actually hexagonal. I mean thousands of hexagonal columns with flat tops, joined together, rising up out of the sand and water.

Startling, because these are not human creations. They were formed naturally. And yet, every notion we have of nature tells us that regular shapes like these are near-impossible. To be sure, bees produce hexagons in their honeycombs. But hexagon-shaped volcanic rocks?

When lava flows cool, they can produce such formations. I’ll return to that. Geologically, the St Mary’s rocks tell the story that India and Madagascar were joined together millions of years ago in an ancient supercontinent called Pangaea. We know that because there are similar regular rock formations on Madagascar. There was a volcanic hotspot under that part of Pangaea—under what is now southern Madagascar—about 90 million years ago. Its lava produced what are really six-sided pillars fused together, the tallest rising about 6 metres.

The geological history and the connection to Madagascar is fascinating by itself. But I imagine anyone who visits St Mary’s Island will wonder: why the hexagonal shape?

Well, as you might have guessed, mathematicians have pursued a line of inquiry that sheds some light on that question.

To get an idea of what they did, start with a sheet of paper. Cut it with one straight slice so that you now have two pieces. Place one on the other and slice straight through the pair again, to get four pieces. Keep stacking, then slicing at random, a few times.

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When you’ve had enough, do this exercise: take each piece of paper you’ve produced and count the number of vertices and sides it has. Add these numbers and average them over all the pieces. Here’s a prediction: on average, you’ll end with an average of four vertices and four edges per piece. What familiar shape has four vertices and four edges connecting them? A rectangle. On average, you’ve produced pieces of paper that are like rectangles.

No, by no means are they all perfect rectangles, with 90-degree angles and opposing sides that are equal. In fact, more than likely none is like that, unless you made your cuts deliberately to produce rectangles. Also, no boomerangs or starfish, though you might have a few triangles and pentagons. Still, we’re talking about an “average" of sorts, and the average piece of paper you’ve produced will have four sides, four vertices, and a generally convex shape with no indentations.

One way to get some perspective on this is to think of two successive slices, which is how we get the vertices. The angle they make with each other can range all the way from 0 degrees to 180 degrees. On average, therefore, that angle is 90 degrees. A piece of paper with straight edges and angles averaging 90 degrees is ... four-sided. Like a rectangle.

Maybe you find all this only mildly interesting? Well, what if you sliced a three-dimensional object, instead of a sheet of paper? The story goes that the Russian Nobel Peace Prize winning physicist Andrei Sakharov was once chopping cabbages with his wife. The analogous question occurred to him: if you picked up the average piece of sliced cabbage, how many vertices would it have? How many faces?

Well, in a paper published last year, three Hungarian mathematicians and an American geophysicist argue that it would have eight vertices and six faces. (“Plato’s cube and the natural geometry of fragmentation", Gábor Domokos, Douglas J. Jerolmack, Ferenc Kun, and János Török, Proceedings of the National Academy of Sciences of the USA, 4 August 2020).

If you don’t slice into cabbage all that often, think of chopping an onion, or french beans, or a cucumber, into small pieces. Or think of smashing a rock with a hammer. What the paper suggests is that the pieces you end up with, whether stone or vegetable, have, on average, eight vertices and six faces.

They are, on average, cubes.

“In this study," Domokos and his colleagues write at the start of their paper, “we draw inspiration from an unlikely and ancient source: Plato, who proposed that the Earth is made of cubes because they may be tightly packed together. We demonstrate that this idea is essentially correct: Appropriately averaged properties of most natural 3D fragments reproduce the topological cube."

So in effect, when rocks are split by natural processes, we have two intriguing results. On the two-dimensional surface, they are essentially four-sided with four vertices: the rectangles mentioned above. Take the third dimension into account, and broadly, the fragments are cubes.

But hold on. What the scientists’ analysis additionally showed is that in two dimensions, while the fragments you get are usually quadrangles, there are exceptions. Take fields that dry up in the summer heat and then display characteristic and familiar wide cracks. Those individual cells of mud tend to have six sides: a hexagon. Fascinatingly, Earth’s tectonic plates—the pieces of the planet’s crust that make up the land masses we are so familiar with—are also on average hexagons. As the scientists write, “the geometry of the [tectonic plates] is compatible with either 1) an evolution consisting of episodes of brittle fracture and healing or 2) cracking via thermal expansion." And we know the Earth has been through both those processes.

Then there are rocks that are formed as the result of lava that cools, turning solid starting at the surface and going down. The theory is that forces acted on these rocks from the inside towards the outside, instead of the other way around (for example, think of that hammer that shatters the rock). “As a consequence of maximizing energy release", these rocks take on a familiar shape. Indeed: consider a horizontal section of these “columnar joints"—or simply look at their top surface—and you’ll find they are often ... hexagonal.

One well-known example of these hexagonal rocks is the Giant’s Causeway on the coast of Northern Ireland. Another is on our very own St Mary’s Island. All left behind as molten lava cooled after long-ago eruptions.

It’s a unique and remarkable sight, these thousands of columns of “rhyolitic lava" with their hexagonal surfaces. So go visit St Mary’s Island. While you’re gawking at the rocks, spend some time thinking of Madagascar. Of lava. Of Northern Ireland. Of hexagons. Of a world made of cubes.

Of the wonders of mathematics.

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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