There are lots of reasons to study these cubes. One is this very intriguing question: can a given number be written as the sum of three cubes?
What’s a columnist to do when he wants to write about that most natural of substances, faeces? Do I use that word? Or another that begins with “s"? Or the more acceptable kiddie one, “poop"? Or a word from an Indian language?
There’s a dilemma, because this is what I want to explore — only figuratively, ok — in this column. Well s***, I’ll go with “poop". Not any random poop, either. I’m fascinated by the excretions of an animal humans find cute indeed: the Australian marsupial known as the wombat. It’s not a small creature—about 20-30kg, like a medium-sized dog—but it’s furry, has tiny ears, beady eyes, a snub nose and short stubby legs: yes, very cute.
But cuteness is not why it figures here. Its poop is: the wombat excretes cube-shaped blocks. I am not making this up. While most poop you’re familiar with is rounded or elongated or pellet-shaped, the wombat produces cubes, up to 100 in a single night. Not perfect cubes, but close enough to make you wonder. They are like little blocks, with flat faces and edges that are close to right angles.
Why they do this is not so interesting. One reason is that the poop attracts potential mates. The other is that wombats use their poop to mark territory. They actually stack the turds to make a wall that deters wombat intruders. When you make such a wall, when it must attract mates, you don’t want individual bricks rolling away.
Sure enough: These wombat-produced cubes of poop, deposited on even a slight slope and then stacked, certainly don’t roll away. In other words, the cube-like shape is not a biological accident. It is, instead, a product of evolution.
The more interesting question, though, is how wombats generate cubes. After all, this is not a shape found easily in nature. Nor are flat faces and right angles. No other animal shapes its poop like the wombat does. And no, in case you were wondering, the wombat does not have a square anus. So whence the cubes?
As always, where there’s a scientific curiosity, there’s a scientist or three looking to understand it. In this case, Patricia Yang, a postdoc in mechanical engineering at the Georgia Institute of Technology in Atlanta, USA, and three colleagues have investigated this phenomenon. They offer an explanation in a paper they presented a year ago..
They observe that “in the built world"— our human world, they mean—“cubic structures are created by extrusion or injection moulding". Clearly wombats don’t lug around cubic moulds to poop into, so injection moulding is out. But like other animals, they do extrude poop. So the muscles involved in that extrusion must shape the poop into cubes before it emerges.
Yang’s team worked on two wombats who were hit by cars in Tasmania, Australia, and had to be euthanized. They found that food waste remained essentially liquid — and thus no good for making cubes — for much of its journey through the wombats’ intestines. It’s in the last 8% of their length that the poop turns into solid pieces — “separated cubes of length 2cm."
The scientists inserted long skinny balloons into the intestines. By inflating the balloons, they got a measure of the “stretchiness", or elasticity, of the walls of the intestine. After all, the walls of any intestine—wombat or yours—stretch somewhat when filled with poop. But what if some sections are less stretchy than others? To understand this, try inflating such a balloon yourself. You will produce a long sausage-like shape. Now thread the balloon through a small metal ring—or just use your index finger curled into an “o". As you can imagine, the balloon won’t inflate at the point where the ring, or your finger, is. That is, its ability to inflate — its stretchiness — is constrained right there. So instead of one long sausage, you’ll have two strung together.
That’s the sense in which Yang and her colleagues wanted to learn whether the wombat’s intestine constrained the inflation of the balloon. Sure enough, it did. No metal ring, but the elasticity of the intestinal wall does vary along its length. There were stretchy parts and stiffer parts. As they wrote: “We found that the local strain varies from 20% at the cube’s corners to 75% at its edges." Variation like that shapes the poop.
Certainly there are questions to be answered. If the shape is to be a cube, you’d think the circumference of the intestine would need to alternate stretchy and stiff parts, ideally four of each. The stiff parts would produce the flat faces, the stretchy parts the edges. Yang’s team found, as one report explained, “two distinct ravine-like grooves where the intestine is stretchier". How do just two such regions produce a cube? Is it possible that more regions become evident as the walls stretch—that is, as the intestine fills with even more poop and the wombat strains to excrete?
Plenty to investigate, clearly. Apparently Yang’s team is pushing ahead with this research, simulating wombat intestines with pantyhose. There’s good reason to keep at it. Humans “currently have only two methods to manufacture cubes", Yang explained to the National Geographic (on.natgeo.com/2L2L1DJ). She meant moulding, and extrusion through a square orifice. But “wombats have a third way"— and maybe we humans can profitably learn to use it.
Not, however, for a different kind of cube. These are not extruded or moulded. They just are. I’m referring to the cube of a number, the number multiplied by itself three times. For example:
13 = 1 x 1 x 1 = 1
23 = 2 x 2 x 2 = 8
53 = 5 x 5 x 5 = 125
and so on.
There are lots of reasons to study these cubes. One is this intriguing question: can a given number be written as the sum of three cubes? This is one of those mathematical confections that mathematicians love, but lay people are baffled by. Of what possible use is this knowledge, I can hear it asked. Perhaps none of any wide import. Yet here’s the thing: so many numbers can be broken down like this that it’s easy to think they all can be. Still, there are many numbers which we know cannot be, and there are others for which we don’t know either way. As with so many mathematical pursuits, the search for answers leads us deeper into the intricacies of mathematics. So for example, we know 4, 13 and 32 can’t be written as the sum of three cubes. 3, 29 and 34 can be, like this:
But for many years, there were 14 numbers less than 1000—33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975—for which we didn’t know the answer. Until 2016—enormous number alert—when Sander Huisman of the École Normale Supérieure de Lyon found 74 could be expressed this way:
So now the smallest number we don’t know how to break into three cubes is 114. I find it startling and sobering that even after the many millennia that mankind has used and manipulated numbers, we can learn new things about numbers as “ordinary" as 33, 42 and 74 …
… and well, about plain-vanilla 3, too. Also in September this year, Booker reported a third way of expressing it as the sum of three cubes: