## Saturday, December 25, 2010

### Suppose the Universe is a 4-dimensional sphere

At school we learn Euclidean geometry, usually starting with a planar surface with x-y coordinates.
But Earth is not a planar surface, it is a spherical surface. We need a spherical geomety to more accurately describe movements on Earth.

In the same way maybe the Universe is not a x-y-z Euclidean orthogonal space. Maybe it is a 4-dimensional sphere

What happens in that case?

On Earth if you travel to any direction, eventually you will go round the entire planet and be back to the starting point.
In the same way in a 4-dimensional sphere if you travel to any direction, eventually you will go round the entire Universe and be back to the starting point from the opposite direction.

On Earth (which is a spherical surface) if you travel to any direction eventually you will reach the other extreme of the planet, the exact opposite point on Earth from the starting point.
In the same way in a 4-dimensional spherical space, if you travel from Earth to any direction in space, eventually you will reach the other extreme of the Universe, the exact opposite point in the Universe from Earth.

It means that no matter which direction you look in the sky, you are actually looking at the exact same point in the Universe, you are looking at the exact opposite point in the Universe from Earth.
But why the sky looks different at every direction? Because we are looking at the same point FROM different directions.

A cross-section of a sphere is a circle. In the same way the cross-section of a 4-dimensional sphere is probably a 3-dimensional sphere, I think....

One consequence of this is the concept of distance. What is the distance between 2 points? It is the lenght of a straight line between these 2 points. But on Earth you cannot have a straight line between 2 points, since we can only travel on the spherical surface of the Earth.
But what if you could travel in a straight line by drilling a giant tunnel that connects 2 points on the surface of the Earth directly? It would be a shorter route.

If the Universe is a 4-dimensional sphere it could be possible to drill a tunnel that connects 2 points directly, thus having a shorter route.

I am not talking about wormholes, black holes or anything like that. I am not talking about Einstein's theory of relativity in which time is considered the fourth dimension.

I am talking pure geometry. Just suppose the Universe is a 4-dimensional sphere. Period. What would happen then?

What is a 4-dimensional sphere? How do you build it?
Think about a sphere. How do you build a sphere from a planar surface? By folding the surface and connecting all the edges to a single point. In this case I mean a topological folding, i.e. you can stretch or compress the surface at will.

In the same way how do you build a 4-dimensional sphere from a 3-dimensional object, let's say a cube? By folding the cube and connecting all extenal faces and edges to a single point. This folding is only possible in the 4th dimension.