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According to wikipedia, atleast. there are four dimensions of space and one of time.
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Space has more than 3 dimensions. Going by Einstein and a whole bunch of other physicists, there are either 10 or 26 dimensions.
And I don't mean "deamon dimensions", I mean measurable parameters. Look up "string theory" for more info.
I've already seen 4-D PC mouses for sale and thought "...wtf?!".
Not counting time, a game with 4 dimensions would be slightly beyond the majority of human comprehension. We think in terms of length, width, depth. Any other axis of orientation would really mess up most people. I suppose you could use a colour-shift of some sort {i.e. you must be in the right area of a spectrum to do something} to simulate some kind of dimension, but I really can't see people understanding a 4-D game for a while.
Bullet Magnet, thank you SO MUCH for that link... although it really made my head hurt around dimension 7. To get to 26 dimensions, most people think of dimensions being smaller than the basic 3 {the analogy of an ant on a cable which appears 1 dimensional to us. The ant can rotate around the cable [or axis] in a very small dimension... although it might just be the other 2 axis}