Non Euclidean Geometry V – Pseudospheres and other amazing shapes
Non Euclidean geometry takes place on a number of weird and wonderful shapes. Remember, one of fundamental questions mathematicians investigating the parallel postulate were asking was how many degrees would a triangle have in that geometry- and it turns out that this question can be answered depending on something called Gaussian curvature.
Gaussian curvature measures the nature of the curvature of a a 3 dimensional shape. The way to calculate it is to take a point on a surface, draw a pair of lines at right angles to each other, and note the direction of their curvature. If both curve down or both curve up, then the surface has positive curvature. If one line curves up and the other down, then the surface has negative curvature. If at least one of the lines is flat then the surface has no curvature.
A sphere is an example of a shape with constant positive curvature – that means the curvature at every point is the same.
The pseudosphere is a shape which is in some respects the opposite of a sphere (hence the name pseudo-sphere). This shape has a constant negative curvature. It is formed by a surface of revolution of a called called a tractrix.
It might be surprising at first to find that the cylinder is a shape is one which is classified as having zero curvature. But one of the lines drawn on it will always be flat – hence we have zero curvature. We can think of the cylinder as analogous to the flat plane – because we could unravel the cylinder without bending or stretching it, and achieve a flat plane.
So, what is the difference between the geometries of the 3 types of shapes?
Firstly, given a line m and a point p not on m, how many lines parallel to m through p can be drawn on each type of shape?
A shape with positive curvature has no such lines – and so has no parallel lines. A shape with negative curvature has many such lines – and so has many parallel lines through the same point. A shape with no curvature follows our normal Euclidean rules – and has a single parallel line through a point.
Sums of angles in a triangle and other facts
Triangles on shapes with positive curvature have angles which add to more than 180 degrees. Triangles on shapes with negative curvature have angles which add to less than 180 degrees. Triangles on shapes with no curvature are our familiar 180 degree types. Pythagoras’ theorem no longer holds, and circles no longer have pi as a ratio of their circumference and diameter outside of non-curved space.
The torus is a really interesting mathematical shape – basically a donut shape, which has the property of of having variable Gaussian curvature. Some parts of the surface has positive curvature, others zero, others negative.
The blue parts of the torus above have positive curvature, the red parts negative and the top grey band has zero curvature. If our 3 dimensional space was like the surface areas of a 4 dimensional torus, then triangles would have different angle sums depending on where we were on the torus’ surface. This is actually one of the current theories as to the shape of the universe.
Mobius Strip and Klein Bottle
These are two more bizarre shapes with strange properties. The Mobius strip only has one side – if you start anywhere on its surface and follow the curvature round you will eventually return to the same place having travelled on every part of the surface.
The Klein bottle is in someways a 3D version of the Mobius strip – and even though it exists in 3 dimensions, to make a true one you need to “fold through” the 4th dimension.
The shape of the universe
OK, so this starts to get quite esoteric – why is knowing the geometry and mathematics of all these strange shapes actually useful? Can’t we just stick to good old flat-plane Euclidean geometry? Well, on a fundamental level non-Euclidean geometry is at the heart of one of the most important questions in mankind’s history – just what is the universe?
At the heart of understanding the universe is the question of the shape of the universe. Does it have positive curvature, negative curvature, or is it flat? Is it like a torus, a sphere, a saddle or something else completely? These questions will help determine if the universe is truly infinite – or perhaps a bounded loop – in which if you travelled far enough in one direction you would return to where you had set off from. It will also help determine what will happen to universe – will it keep expanding? Slow down and stop, or crunch back in on itself? You can read more on these questions here.