Cool Stuff: Fractals

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2008

## Cutting up cubes

Once when I was little, we we given a test as school to measure how much we knew, and one of the questions was, “which regular shape do you get by cutting a cube in half?”

We were obviously supposed to write down “a square”, but the question bugged me and I started to think about it (which I don't think we were supposed to do!).

Turns out, while “a square” is obviously the "correct" answer, it's not the right answer. The right answer is, “Please reword the question, because there's actually more than one solution.” We don't only have the option of producing a square, we can also make a hexagon.

If you hold a cube by two opposing corners, and cut it exactly through the middle, perpendicular to the line that joins those two corners, then your cut takes a chunk off all six of the cube's faces, and if you do it right, you get a perfect hexagonal shape.

In fact, while there are only three ways to halve the cube to get a square, there are four ways to get a hexagon. Here they are:

I put this diagram together for the "Relativity in Curved Spacetime" book (page 303), to illustrate that idea that although we may be taught that a particular answer is right, and that answer might be provably valid, it doesn't mean that there aren't other answers. There are few things more obvious than the idea that you get a square by cutting a cube in two, so if someone claims to be able to get a hexagon, we're liable to consider them an idiot (or to think that they are only  talking about some silly argument that only works in four dimensions). When we've been over-trained to see the "obvious" answer to a question, it can give us a mental block that stops us from seeing alternative solutions. The answer that we already know has a way of jumping in and telling us not to look any further.

Anyway, after I'd done the diagram, something about it kept nagging at me while I tried to complete the book, and eventually I went back and looked at it and realised what it was.
The eight pieces in the exploded diagram of the four cut cubes could be reassembled into a single shape. The eight hexagonal faces point outwards, the five-sided faces all fuse together,  and the 24 remaining triangular facets join up in clusters of four produce six smaller square faces. All the sides are the same length.

The resulting fourteen-sided shape was pretty weird ... if you stared at it long enough, you realised that multiple copies of it would fit together perfectly in three dimensions.

It's a perfect space-filling solid.

How cool is that? :)

I made the usual model out of drinking straws and bits of wire to check that I wasn't imagining it, and yes, it worked.

The drinking-straw model “pinged” another idea. When I was at school we were told that here were only three forms of carbon: diamond, graphite and soot. It was supposed to be be geometrically proven that no other forms were possible. Carbon forms four covalent bonds, and there weren't supposed to be any other regular 3-D shapes that you could make from that configuration, apart from diamond. Okay, so then we discovered buckyballs and buckytubes, which were spherical and tubelike arrangements of carbon atoms ... but these still weren't continuous lattices.

So now we peer at our shape, and notice that each corner is linked to four others by identical-length lines, and a thought strikes us – could we make this shape as a crystal lattice, using carbon?

Unfortunately, the shape has a mix of different bond angles: three atoms that form the corner of a hexagon have a 120-degree angle, but those that form the corner of a square make an angle of 90 degrees. While 90-degree carbon bonds aren't unknown in Nature, carbon seems to be a bit uncomfortable making them. And even if this novel form of carbon can exist, that still doesn't tell us how we're supposed to make the stuff (we had enough trouble making diamond).

If we could make this material, its properties could be interesting. The "cage" arrangement means that small atoms and ions can percolate through the structure, and larger ones can be trapped inside, changing its properties. The smallest atom that we have is hydrogen, and it seems that hydrogen has an affinity for carbon lattices, so our mystery material might make a good “hydrogen sponge”.

Researchers are already looking at buckytubes and buckyballs as potential storage material for hydrogen-powered cars – these sorts of materials seem to suggest a decent lightweight way of storing compacted hydrogen.

... But our new hypothetical material has a rather special additional property compared to buckyballs. If we look at a conventional buckyball, its carbon cage is usually made of sixty atoms. There are also smaller, less regular and less stable versions, which use fewer atoms, and enclose less space. But for our carbon sponge we tend to want the hydrogen atoms to be in as intimate contact with the carbon framework as possible.

Now let's look at our new hypothetical material. It produces a smaller cage, with only twenty-four atoms surrounding each void. But because the cages are butted up against each other, the atoms that form each cage are shared ... in fact each atom acts as a "corner atom" for four separate cages.

So for the bulk material, the number of atoms required "per cage" is 24 ÷ 4 = ... six!

In other words, if we could produce this material, and it was reasonably stable, and if it still had the sort of hydrogen affinity that we'd expect from it, then this structure might be expected to be more efficient at storing hydrogen per unit weight then the “conventional” fullerene forms of carbon. You'd only need ~six carbon atoms per cage.

What this exercise shows is that asking stupid questions can have potential payoffs. You can start off with a really dumb question about what happens when you try to cut a cube in half, and end up thinking about ideas for engineering storage systems for hydrogen-powered cars. You never know quite where you're going to end up.

www.relativitybook.com