The New Scientist blog posted this nifty piece as a primer for thinking about multiple dimensions.
The article covers most of the first 10 dimensions, takes a pit stop at 1.5 dimensions. There are links which explain why thinking in terms of, for instance, 6 dimensional space. One of the quick things that I learned was that thinking in 5 dimensional space it is possible to prove that electromagnetism and gravity are two aspects of the same force.
http://www.newscientist.com/special/beyond-space-and-time
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