You probably think you are sitting in
Euclidean space right now.
But how can you be certain of this? Isn't it possible
that you are living in a
geometry that looks exactly like Euclidean space
inside a
sphere centred on the earth with some enormous radius
(let's say a few light years)
but outside that region is different? How could you tell?
In fact there
are nonEuclidean geometries that
locally look
like Euclidean space but
globally are not
Euclidean space. I would love to know how Euclid would react
to this suggestion.
But now the story is getting out
of order. It all starts with Euclidean
geometry. Euclid proposed various axioms
or postulates in his Elements
and starting from those
deduced the
familiar theorems
of Euclidean geometry. One of those
axioms, the parallel postulate,
seemed less basic than the others
and many people tried (unsucessfully) to show
that it could be deduced from the others.
Fast forward two thousand years
and Gauss solved the problem.
He showed that there was a very
interesting geometry,
called hyperbolic geometry or Lobachevski geometry,
which satisfies all of Euclid's axioms except
the
parallel postulate which he replaced by a variant.
The reason Lobachevski gets a mention here
(and also
János Bolyai) is that Gauss didn't publish his
results. Lobachevski published in 1826 and Bolyai
in 1829, ten years or so after Gauss' discoveries.
To go further than this we need
to make a precise mathematical definition of what a geometry is
and we need to see some examples to convince us that the notion is
worth pusuing.
Let's just say for now that
a geometry is a set equipped with a distance function that
has to satisfy certain natural properties. From this point of
view Euclidean space with the usual Pythagorean distance
is just an example of a geometry.
This is a modern way to think
about geometry, where we have properly divorced mathematics from
the real world. That doesn't mean that we must stop using our intuition to
guess what might be true, but we always remember that mathematics is
what follows from the axioms of set theory,
not something that we measure or observe.
Of course this point of view doesn't stop
mathematics from being useful for modelling the real
world!
Actually, it's worth pointing out that there are
some problems with Euclid's axioms and his logic. Despite this
the Elements is an incredible achievement
and we shouldn't be too harsh.
Euclid was a man of his time.
Let's consider spherical geometry.
First of all we have the natural
distance function
on the sphere, the distance between two points
is the shortest distance travelled by walking from one point
to the other along the sphere surface.
This is not some abstract nonsense,
this is how we measure distance on the earth!
Also, note that this is not the same thing as the Euclidean
distance between these points (travelling
along a straight line burrowing through
the earth is not very congenial
just to get from A to B).
For example, if the sphere has diameter d
then the distnace between the two poles in spherical geometry
is d.pi/2 whereas the Euclidean distance is d.
Once you have a distance function that satisfies
certain natural properties you are ready to
do some geometry.
What is the correct notion of line for this
geometry? It turns out that great circles
of the sphere are the analogues of lines.
Think about it.
This is a big difference from the usual Euclidean
geometry. Suddenly lines have finite length!
If you take two distinct points on the sphere that are
not antipodal then there is a unique
great circle passing through the points (just like
Euclidean geometry). But if
they are antipodal then there are infinitely
many great circles through the two points! So
this is quite different from Euclidean geometry
where two distinct points determine a unique straight line.
Another interesting difference is
that sum of the angles in a triangle.
turns out
to be greater than pi.
Enough nonEuclidean weirdness, it's time for the definition now.
Definition A geometry is a set X together with
a function d:XxX>R that satisfies
the following axioms for all a,b,c in X.

d(a,b)>=0 and d(a,b)=0 iff a=b.

d(a,b)=d(b,a).

d(a,c)<=d(a,b)+d(b,c).

Given any two points a,b in X and any
two positive real numbers d,e there exist
points p_{1},...,p_{n}
such that p_{1}=a and p_{n}=b,
for 1<=i<=n1, we have
d(p_{i},p_{i+1})<d
and
0<= d(p_{1},p_{2})+...+
d(p_{n1},p_{n})  d(a,b) < e.
The first three axioms are fairly natural for
a notion of distance, so let's just discuss the last
one. Intuitively you should think of this as saying
there is a curve in our geometry
joining
a and
b (approximated by
line segments
p_{i} to
p_{i+1} of the definition)
so that the distance along
the curve is the distance between
a and
b.
Here's another example a geometry on the torus.
This is a nice one. We start with a square strip

 B 
 
A A'
 
 
 
 B' 

and we imagine that the points on opposite
edges are identified. So, for example, we think of
the edge points
B and
B' as being equal
and likewise
A and
A' are equal.
Topologically what we get from this identification
is a
torus. Think about physically glueing the edge with
point
A on it to the edge with point
A' on it,
to make a cylinder. Then stretch and bend this cylinder around
to glue its two ends together.
Now we want to think about
distance. This is defined as follows, for
two points
a,b on the strip the distance
between them is just the usual Euclidean distance
except that if we get a shorter distance by using
the edge identification to zip from one edge to another
then we will always take
that.

 
 
 a b 
 
 
 a' b' 
 

For example the distance from
a to
b
for this geometry is the same as the usual Euclidean
distance but the distance between
a' and
b'
would be the sum of the Euclidean distances of
a'
to the leftmost edge and
b' to the rightmost edge.
It is tempting to think that this example is quite
similar to spherical geometry, we just replaced
a sphere by a torus, but this is wrong. When we made the
identifications to create the torus, the bending
and stretching we did distorted
distance, so the distance between two points on the torus
for this geometry is not just the distance
we get by walking along the surface of the torus.
One of the important ideas to grasp here is that we are thinking about
geometry from the perspective of an inhabitant of the geometry. We are
not thinking about our set as sitting inside some Euclidean space.
What we are doing here is intrinsic. This thinking, which is the
modern viewpoint, goes back to Gauss.