MATHEMATICIANS are used to playing around in many dimensions. They pretend
that the world is flat and think about how things behave in 2D. They imagine
four-dimensional spheres and 20-dimensional pyramids. And they can work out
answers to the most esoteric topological problems: if you tie a knot in four
dimensions, will it immediately unravel? Does a five-dimensional doughnut look
like a five-dimensional teacup? But for all this fancy mathematical footwork,
there is still a dimension that is proving unexpectedly tough to understand: the
third.
Hang on. The third dimension—the one we inhabit and have spent several
million years of evolution getting to know? By now, you might expect, 3D
geometry should hold few secrets. In fact, nothing could be further from the
truth. Three-dimensional space is a hot topic in maths. If anything, it’s
hotting up, for in certain respects we know less about the 3D space we live in
than we do about spaces with any other number of dimensions. And even those
problems in 3D that we do understand often succumbed long after they were worked
out in 5D, 6D or higher.
An extreme example of this is a little puzzle that has been bugging
mathematicians since 1904. It was posed by the great French mathematician Henri
Poincaré, arguably the founder of topology. Known as Poincaré’s
conjecture, it has been answered for spaces with every number of dimensions
except three. For the 3D case, mathematical opinion is seriously divided. The
one thing we have learned is that the geometry of three dimensions is entirely
different from any other—and far more interesting.
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Maybe that’s why we live in it. The inverse-square law of gravity is
especially fine-tuned in 3D, as it implies that any given body responds not just
to its neighbours, but to the total gravitational attraction of the Universe.
This is because the surface area of a 3D sphere grows as the square of its
radius, neatly compensating for the way gravity tails off as an inverse square
with distance. Stars that are farther away attract less, but there are more of
them. As a result, matter becomes more and more clumpy as time
passes—which is why we have a nice, solid planet to inhabit.
But although three dimensions provide a friendly environment for the likes of
us, they are decidedly hostile when it comes to solving topological conundrums
such as Poincaré’s. To get to grips with his conjecture, we must first
dig a little deeper into what is meant by “space”. One way of looking at space
is as an arena made up of different places. To specify a place in 2D—on a
road map for instance—you need two numbers. One identifies its location in
the east-west direction, and the other in the north-south direction.
The space we live in has three dimensions, because it takes three numbers to
say where a place is. For a place on the Earth, for instance, it’s not enough to
know its latitude and longitude. You also need its altitude.
The mathematician’s notion of space takes its cue from these practical
observations. Two-dimensional space is the set of all pairs of
numbers—east and north (with minus numbers for west and south).
Three-dimensional space is the set of all triples of numbers: east, north and
up. Taking a deep breath, we can extend this idea into realms that transcend the
limitations of the physical Universe. Four-dimensional space must be the set
of all quadruples of numbers, with the fourth somehow representing a new
direction—say east, north, up and surprise. Twenty-six-dimensional space
is the set of all 26-tuples of numbers, and so on.
Coffee and doughnuts
The feature that unites these different spaces is the notion of “near to”.
Two places are close together if the lists of numbers that represent them differ
by very small amounts. This notion of a space led to the creation of
topology—”rubber sheet geometry”. In topology, two spaces are considered
to be the same if it is possible to deform one into the other without making any
breaks or tears. To a topologist, for instance, a 3D coffee cup and a 3D ring
doughnut are the same. The hole in the doughnut corresponds to the hole between
the cup and its handle.
Just as you can ask in topology whether two shapes within the same dimension
are equivalent, you can also ask whether spaces within different dimensions are
equivalent. Is it possible, for instance, to deform a plane continuously into 3D
space? Agreed, that seems unlikely, but then to most normal human beings a
coffee cup is definitely not a doughnut, so it’s best not to jump to
conclusions.
The answer turns out to be no: 2D and 3D space really are topologically
different. Here’s one way to prove it. If you cut a closed curve in a sheet of
paper with a pair of scissors, a piece falls out. This is not what happens when
you do the same in 3D space: if you remove a flat closed curve, what’s left is a
single lump. To disconnect 3D space you have to scoop out a ball. Because the
plane has a topological property that does not hold for 3-space, they must be
different.
In 1904, Poincaré wanted to take the next step. Given two spaces, is
there some formal, logical test that will tell us whether or not they are
topologically the same? Poincaré started thinking about a particularly
convenient type of 3D space, known as a 3-sphere. It’s easiest to picture what a
3-sphere looks like by thinking first of its 2D analogue, the 2-sphere. Roughly
speaking, a 2-sphere is a disc that has been rolled up to form a hollow ball
with a hole in the top, after which you plug the hole with a new point, the
“north pole”. More simply, just pretend that the entire boundary of the disc is
a single point, as though you had pulled it tight like the drawstring on a
shoulder bag.
You can play the same game with a solid lump of three-dimensional space:
again, pretend that its boundary is a single point. In effect, this pulls the
whole thing together into a sort of ball, minus its north pole, and again we can
pop in a north pole to plug the gap. It’s very hard to imagine, but although
this ball has three dimensions not two, it’s not an ordinary solid ball. A solid
ball has a definite boundary, but the 3-sphere does not. A 3-sphere is hollow,
like an orange with a 3D “rind” but no fruit inside.
Why not stick to 3D space itself? The reason is that topology works best on
spaces that have no edges and don’t wander off towards infinity without ever
getting there. The easy way out is to get rid of edges and infinities before you
start. Choosing a finite lump of space and plugging the gap at the “north pole”
does the trick—all the points that would have marked an edge get squashed
together into the extra point.
The 3-sphere is a friendly sort of beast, familiar 3D space in a handy
pocket-sized version, a space so simple that we ought to know everything about
it. It may not sound that simple, but trust me: it makes the mathematics a lot
easier.
One thing about a 3-sphere is that it doesn’t have any interesting closed
loops. A doughnut has two kinds of closed loops: those that cut it in two, and
those that don’t. In fact, Poincaré was more interested in what happens
when you shrink a loop, rather than whether a given loop cuts the space into
separate bits, though it amounts to much the same thing. If you have a loop that
forms the edge of a disc, then you can shrink the loop down to a
point—rather like pulling on a slip knot until the loop that it makes
disappears. Any such loop always cuts the space into two disconnected bits. But
a loop that embraces some kind of topological obstacle—the doughnut’s
hole, for instance—is special. It cannot be shrunk to a point while
staying inside the space.
Shrinking loops
Every loop on a 2-sphere is shrinkable. So the doughnut is different from the
2-sphere. A more powerful result is that the 2-sphere is the only
two-dimensional space in which all loops are shrinkable—given a few
conditions about no edges, no wandering off to infinity but not getting there,
and so on.
Poincaré knew that every loop on a 3-sphere is shrinkable too. Early
on, he assumed without comment that the 3-sphere has to be the only such 3D
space: in other words, if every loop is shrinkable, you’ve got a 3-sphere. Soon
afterwards, he realised that such a statement needed proof, tried to find one,
and couldn’t. It was a problem so simple that surely there had to be an answer.
Poincaré never quite committed himself to what the answer was. Instead he
posed the question: “If all loops shrink, is it a 3-sphere?” and left it at
that. Within a few decades, however, the question had become known as the
Poincaré conjecture—and the conjectured answer was assumed by all
to be yes.
When mathematicians are stuck on a question, they usually try asking a more
general one. This helps to take attention away from special features of the
problem at hand that might be misleading. In this spirit, you can ask
Poincaré’s question in any number of dimensions. “If you have an
n-dimensional space in which all loops shrink, is it an n-sphere?”
Poincaré’s original question is the 3D case, n = 3. In his day
the answer was already known to be yes for n = 2, and for a long time,
that was all anyone could say.
As the methods of topology became more powerful, this more general question
began to yield up its secrets. In 1961, Stephen Smale from the University of
California at Berkeley developed a theory in which a space could be cut into
pieces called “handles”, and the way the handles glued together could be made to
yield useful information. One offshoot of his theory was a proof of the
Poincaré conjecture for seven or more dimensions. Then mathematicians
pushed Smale’s methods into six and five dimensions too, leaving only three and
four to be dealt with. But there everyone got stuck, and mathematicians began to
wonder whether spaces of three and four dimensions are different from the others
in some fundamental way. Perhaps the Poincaré conjecture is false. There
the matter stood until 1982, when mathematician Michael Freedman defied the
conventional wisdom by proving that the Poincaré conjecture is true in
four dimensions too.
After three-quarters of a century, mathematicians had answered
Poincaré’s question in every number of dimensions except the number for
which he posed it. And there we remain today. Every so often a proof of the 3D
Poincaré conjecture is announced, but to date all such announcements have
either turned out to be mistaken, or else the proposed proof has been so long
and obscure that its logical status remains impenetrable. About five years ago
there was a big surge of optimism when several radically new approaches to the
Poincaré conjecture emerged. The hope was that at least one would
succeed. So far, none has, but mathematicians are still trying, and a couple of
new ideas have breathed new life into the area.
Exotic geometry
The most significant research direction is what is known as the
geometrisation conjecture put forward by Bill Thurston of the University of
California at Davis, which he developed from a revival of non-Euclidean
geometry. A prominent concept in traditional Euclidean geometry is that of
parallel lines. In non-Euclidean geometry, parallels may not exist, or they may
behave strangely: roughly speaking, these exotic geometries correspond to
various kinds of curved space. Thurston found that in 3D there are seven key
types of non-Euclidean geometry, plus Euclid’s—eight in all. The
geometrisation conjecture says that every 3D space can be built up from these
eight geometries. If you could prove the geometrisation conjecture, the 3D
Poincaré conjecture would fall out from the conclusions. Unfortunately,
proof of the geometrisation conjecture remains tantalisingly out of reach.
Most other approaches involve “triangulating” 3D space—dividing it into
tetrahedra that abut each other face to face. A few brave souls are still trying
to tackle the Poincaré conjecture directly by studying how these
tetrahedra can fit together. It seems unlikely such an approach can succeed, but
you never know. One of the biggest recent advances is the proof by Hyam
Rubinstein and Abigail Thompson that there exists an algorithm—a computer
program guaranteed to give an answer—to decide whether a given space is or
is not a 3-sphere. This works in terms of a triangulation, and it means if
someone comes up with a space that looks like it might disprove the
Poincaré conjecture, we’ll be able to verify that claim.
But while all these efforts continue, here is a disturbing thought. Although
most topologists are convinced that the Poincaré conjecture in 3D is
true, a few are wondering if it could turn out to be unprovable. In the 1930s,
Kurt Gödel showed that certain questions in arithmetic can never be
answered within a formal logical system of mathematics, and not long afterwards
Alan Turing showed the same for fundamental questions in computer science.
Maybe the Poincaré conjecture is like that: perhaps it is truly
impossible to prove. Even then, it may take some radical new ideas to show that
no proof can exist. Poincaré’s innocent and simple question may yet stir
up a real mathematical wasp’s nest. It’s a sobering thought that while we
understand all sorts of astonishing things about space of thousands of
dimensions, we still can’t answer a straightforward question about the 3D space
that we live in.