Physical space

In everyday experience, physical space is characterized by being three dimensional and Euclidean. The former characteristic refers to the fact that any point can be located by specifying three numbers. Alternatively, it means that any point can have three mutually perpendicular lines extending from it. The latter characteristic refers to the fact that distances follow the Pythagorean theorem and several equivalent characteristics such as the fact that the angles of a triangle sum to 180 degrees.

Although the physical space in everyday experience is structured this way, there is no mathematical reason why this must be so. It is perfectly mathematically possible for space to have four or forty dimensions, meaning that one would need four or forty numbers to specify a point and that from a given point there would be four or forty mutually perpendicular lines. Similarly there is no mathematical reason why the angles of triangles must some to 180 degrees or that the Pythagorean theorem must hold for actual points. Since the early-19th century, mathematicians have worked out in great detail what how these alternate worlds would behave. Mathematically speaking, the universe could act in these strange ways, but at the scales that we are familar with, they don't.

This opens the question of whether space behaves differently for situations that we aren't familiar with. To fit with our daily observations, physical theories must predict three dimensional Euclidean space for our everyday world. However it may be that our everyday observations may be an approximation for a more complex reality. Indeed, physics suggests that this is the case in several ways.

Space-time

First of all, in our everyday observations, it appears that space and time are separate. It also appears that distance is an invariant. Suppose you have a box and you label each of the points of the box with different (x, y, z) coordinates. You can choose what numbers you use to describe the box to be what ever you want provided that the distance between the points Delta X ^ 2 + Delta Y ^ 2 + Delta Z ^2 remains constant.

Curved space-time

Spacetime is the fabric in which everything happens and which everything has happened. Time and space are not just related, they are the same thing. To describe spacetime, it's necessary to invoke an analogy.

Imagine you're an ant, an ant which has no perception of up or down. You can only move in a plane. If you set off along a straight line, you can do two things. You can walk forever, or you can end up where you started. If you end up where you started, the only explanation is a third dimension, that you're on the surface of a sphere. The third dimension is inaccessible to you, but it has to exist to explain your findings.

Spacetime may be one of these states. It can be open or closed. Open spacetime would have you walking forever. Closed spacetime would have you returning to where you started. However, the geometry works such that a closed spacetime can be closed, yet not "wrap around". There is also a flat spacetime, which is a strange intermediate state.

Extra dimensions