What are "curled" dimensions?
I recently read the book The Elegant Universe by Brian Greene, which talks a lot about dimensions. In particular, it states that there could be curled up dimensions on microscopic scales. I don't quite understand what this means. It seems to me that every single point in space can be described by the three visible spatial dimensions -- left/right, forward/backward, and up/down. How can more dimensions exist, if every point in space can already be described with just three pieces of data?
We are sensitive only to three spatial dimensions and the dimension of time and that is the reason why we think that we are able to describe every point in space by three pieces of data. If there are extra dimensions and these are very small, then one cannot detect these except through certain very sensitive experiments.
A simple example of a curled dimension: Consider several small threads that are braided to form a thicker thread. When seen from far off, there appears to be only a single thread characterized by a radius and length. However, if one sees more closely, one is able to see that there are in reality several miniscule threads and that one needs more than two numbers to characterize this system. The extra numbers are analagous to extra dimensions that are curled up. (Note that this is just an analogy).
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