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 non-Euclidean 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 non-Euclidean 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.

  1. d(a,b)>=0 and d(a,b)=0 iff a=b.
  2. d(a,b)=d(b,a).
  3. d(a,c)<=d(a,b)+d(b,c).
  4. Given any two points a,b in X and any two positive real numbers d,e there exist points p1,...,pn such that p1=a and pn=b,
    for 1<=i<=n-1, we have d(pi,pi+1)<d
    0<= d(p1,p2)+...+ d(pn-1,pn) - 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 pi to pi+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 left-most edge and b' to the right-most 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.

In 300 BC Euclid invented euclidean geometry based on 5 postulates:

1. To draw a straight line from any point to any other.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to each other.

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

However it was not clear whether this was the minimal set of postulates. In particular the 5th postulate was thought to be derivable. But all attempts to derive it failed, for more than 2 thousand years.

In 1823 Bolyai began to realise that it was in fact an independent postulate that need not be true, and separately Lobachevsky published examples of non euclidean geometries from 1829 onwards, where this 5th postulate was denied.

Mathematicians were not receptive to this concept; as there was no proof of their consistency. Eventually a way of embedding 2 dimensional non-euclidean geometries in normal 3-space was found, and their consistency became obvious.

Therefore in the modern world, mathematicians are very comfortable with these geometries- for example the surface of the earth is non-euclidean due to its approximately spherical shape, and further, according to Einstein's General Theory of Relativity the very space around and in the earth is non-Euclidean, and therefore the total interior angles of a triangle differs from 180 degrees by a tiny amount due to the curvature induced by the gravity of the earth.

In fact, due to the space inside the Earth 'slumping' slightly, it is thought that the distance between the poles from the outside is lower than the distance if you were to drill a hole down through the Earth and measure the distance that way.

The basic principle for euclidian geometry was that for any given line and point there is exactly one line through the given point that is parallel to the given. Euclid was, however unable to prove this, yet felt that it must be true nonetheless. For millenia, mathematicians attempted to prove this postulate unsuccessfully.

In the early 19th century some mathematicians began to wonder if perhaps the lack of proof yet apparent truthfullness of Euclid's postulate meant that other truths existed. 2 possibilities presented them self readily, that either for any given point and given line, there was no line through the point parallel to the given. This relied on the concept that a line through the point begins intersecting the given line, and the point of intersection begins to move outward on the line in either direction approaching infinity yet never leaving the line, therefore appearing to be parallel but still at some point intersecting the line. This proved itself to lead to contradictions in other fundamental theorems, and so was thrown out.

Lobachevski, as well Gauss and Boylai at around the same time but seperately, devised another possibility that through any given point and given line, there were two lines parallel through the point to the given line. This was based yet again on a line through the point intersecting the given line. The point of intersection could be moved to infinity in either direction. At some infinite limit in one direction, the line becomes parallel, and in the other direction as well. He saw that as these lines went to opposite limits, they could be seen as two distinct lines. This postulate expanded into a new geometry with different theorems but none contradictary to each other.

A basic problem with this new geometry is that it is tough to visualize. This lack of visualizability (heh)only pertains to the basic axis of x and y at a right angle. The solution to this is hyperbolic geometry. picture this: draw a five sided figure. now draw more five sided figures such that at each vertex 4 of these meet, and that there is no space between five sided figures. the figures should get smaller and smaller and smaller from the central one. When drawn correctly, many "curves" can be seen. These curves are really illusions from the way the environment for them plays out, and can effectively be seen as lines. 2 curves very often intersect each other at a point and are parallel to another curve. hyperbolic geometry also leads to all manner of infinite patterns from a central figure and repeated smaller and smaller on all sides to infinity.

Lobachevski (1793-1856) was the first to publish his work on the subject in the book "The Theory of Parallels" in 1840. It is an excellent source for the theorems he found, and was translated and republished in 1914. Gauss and Boylai also found similars sets of theorems and mostly coincide with Lobachevski's.

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