Hello, I can graphically image square power as a surface and third power as volume but how can I image powers higher than third ?
Topology routinely considers spaces of dimension higher than three and some of those are much stranger than the "nice rectangular" spaces you might ordinarily think of. You might look at a few topology books and see if you can find one that meets your needs, but topology covers lots of other things and you might have difficulty finding just what you want. Steenrod, if I haven't gotten him confused with another name, wrote a book in the last twenty years that specifically described one method of visualizing 4 space. Unfortunately my copy is buried so deep that I'll likely never find it again. Charles Howard Hinton was an individual about a century or a bit more ago who wrote several books on the fourth dimension. http://en.wikipedia.org/wiki/Charles_howard_hinton Dover I think published a chopped up version of parts of this about 40 years ago, but some university libraries might have reprints of Hinton's originals. He had what was claimed to be an astonishing memory. In his book he described a set of colored cubes where the colors showed how the faces would be glued together to create a 4D solid if we just had 4 space to work in. It was claimed that if you just worked hard enough, and perhaps had good enough memory, that after a while you would begin to be able to "see" 4 space, either that or you just learned how to quickly correctly answer questions about 4 space as if you could see it. Unfortunately I have never been able to find a precise description of the coloring of the cubes or the complete original instructions so it would be possible to try this. Rudy Rucker in one of his books claims to have been able to reproduce a set of the cubes and followed the instructions, but I do not remember the outcome. There is a whole field called "wild topology" that is far outside what I expect you are starting with, but lens space is much tamer than that and can give you a peek into the simpler spaces that topology studies. http://en.wikipedia.org/wiki/Lens_space