"Alexander Sheppard" <email@example.com> wrote in message news://firstname.lastname@example.org... > Are there any definitions for negative or complex dimensions? >
I've never seen anything involving complex dimensions but if you want you can *sort of* extend dimensions into the negatives, consider the following analogy:
If you take a point (0D) and extend it a finite distance in 1-space, you find a line segment. If you take a line segment and extend it a finite distance in 2-space, you find a square. If you take a square and extend it a finite distance in 3-space, you find a cube. If you take a cube and extend it a finite distance in 4-space, you find a tesseract.
This process can be repeated forever. What if we look at it in the opposite direction? In some negative dimensional space, there must (loosely using the word "must" here) exist some -1 dimensional that when extended a finite distance, you arrive at a point. This is almost impossible to visualize, even harder than high dimensions like a 15D hypercube. However, here's the way I would visualize it: think of a point as the basic unit for all zero and positive dimensions. You can form the analogy to the atom. Now, think of the electrons/protons/neutrons as the -1 dimensional objects (for the purposes here, I'll just call them -1-points). They're like little strings that when all hooked together we get a 0D point. To extend it into -2 dimensions, think of -2-points as little quark-like strings that hook together into a -1-point.
However, complex dimensions (as far as I can see) serve no real purpose. Maybe in the future there will be some field of math which grows around them.