This is a thought that I came up with while I was building a giant fractal in minecraft.
If you have a space of some sort you can also create a coordinate system for it. At first I defined that as being a tupple of values that uniquely indicates each point in a space. Then I realized that this definition had a problem as you can uniquely identify a point within an $\mathbb{R}^2$ space by using a space filling curve. Which is not what you think of when you normally conceive of a co-ordinate system.
So I think there needs to be a further restriction. That if there are two points in the space then there should be a proportionate difference within the co-ordinates. With this restriction I think we get the requirement that for a 2D space you will need a tuple of two numbers and likewise for 3D and higher dimensions.
Now consider a space that is just points on the sierpinski carpet. You are going to require 2 real numbers to indicate where you are but because there are so many forbidden spaces there are quite a lot of forbidden spaces.
I think this will result in the numbers being compressible in a shannon information sense. Is my supposition correct and is there a better and more standard way of expressing this idea?