In the late nineteenth
century physicists tried to see if they could detect a cosmic deviation
from Euclidean norms by measuring the angles between distant stars.
If the universe wasn’t flat then the angles between three
stars in a triangular configuration wouldn’t add up to 180¾.
As we saw in the previous section, if the universe was spherical,
the angles would sum to more than 180¾; if it was hyperbolic, they
would sum to less. All measurements, however, revealed the standard
180¾ and for most of the past century the evidence has pointed to
a Euclidean cosmos.
In the past two decades, however, a problem has arisen which suggests
that perhaps our universe may be hyperbolic. If that is so then
it will also be finite – not the infinite void of classic
The idea of a finite hyperbolic universe may sound bizarre, but
just as we can construct finite Euclidean forms from a piece of
paper, so we can construct finite hyperbolic spaces from hyperbolic
paper. Take a piece of regular paper (which is a Euclidean surface)
and wrap it into a cylinder. The object you are holding is still
a Euclidean surface – because every small section of it remains
mathematically flat. Now imagine that you wrapped the ends of the
cylinder around to connect together at the ends so the resulting
form was like a donut. If you actually try to do this, the paper
will crease and the surface will not stay flat. However if we allow
ourselves to make this move in a fourth dimension the resulting
donut would have a surface that was perfectly flat everywhere. This
donut or torus shape is an example of a 2 dimensional space that
is both finite and Euclidean.
Now we may ask: What would happen if we tried a similar construction
with hyperbolic paper? Again, we can turn to Dr Taimina’s
crochet models for help in visualizing the problem.