The Dimension of Space - 3D or not 3D?

The Dimension of Space - 3D or not 3D?

We live in a 3 dimensional world. Or do we.

Consider this..

A simple way to think about the gravitational field of a point mass is to imagine a fixed number of "lines of force" that radiate from the mass evenly into space. The density of the lines at any given point in space represents the strength of the gravitational field at that point.

With this simple model in mind, we write down Newton's famous formula for the gravitational force f between two masses m and q at a distance r apart..


G is just a constant which goes by the impressive name "universal gravitational constant".

But what is this equation really saying? If we think about our lines of force model in a 1-dimensional space and then in a 2-dimensional space we realize that this equation is actually..


where n=number of dimensions of the space containing the mass m.

We can solve this equation for n..


The expression on the right is an experimentally measurable quantity. But it's the dimension of space. Newton's Law of Gravitation says that this expression is an integer, and in fact it's the integer 3. What's the probability of that!

Can we push our luck further? Hey, why not..

In physics we know that the log function shows up in discussions of information entropy. But we've expressed dimension in terms of the log function. So perhaps dimension is derived from the information content of a space. Of course, we need to define "information content", but once that is done we might have a concept more fundamental than dimension.

Now that would be a breakthrough!

Content written and posted by Ken Abbott