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 overtrained 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 fivesided 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 fourteensided 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
spacefilling 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 drinkingstraw
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
3D 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 identicallength
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 120degree angle, but those that form the corner of a square
make an angle of 90 degrees. While 90degree 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
hydrogenpowered 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 twentyfour 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 hydrogenpowered cars. You never know
quite where
you're going to end
up.
all original material
copyright © Eric Baird 2007/2008
