What prerequisite material should I work through to understand four dimensional space?

Question

I understand (at least to a comfortable degree) dimensions which are less than or equal to 3. For the past several years, I have been hearing a lot about four dimensional space. I'm intrigued and would like to learn more but, I do not know where to start.

Any breadcrumbs would be appreciated.

Edit: What I mean by understanding 3-dimensional space is that I can comprehend concepts like l*w*h. I am not remotely in the ballpark of being a geometer (in fact I just learned the term) or topologist.

I've done coursework in statistics, pre-calculus, finite, and discrete; I imagine that stats and finite won't help me here. These courses were all several years ago. I'm not opposed to a long journey if that is necessary. I am just looking for a path to the material. I hope this makes sense. Please let me know if further clarification is needed; I'll try my best.

Answer 1

Given your comment "I've done coursework in statistics, pre-calculus, finite, and discrete", Sommerville's and Coxeter's books (J. M.'s suggestions) are almost certainly way too advanced. I recommend The Fourth Dimension Simply Explained edited by Henry Parker Manning. I'm rather surprised that no one has yet mentioned Manning's book (it was a well known Dover reprint in many school and public libraries when I was young), since it seems to be exactly what you're looking for. Also, it's freely available on-line, something I didn't know until just now, when I looked.

http://etext.virginia.edu/toc/modeng/public/ManFour.html

http://books.google.com/books?id=Y7cEAAAAMAAJ

Answer 2

Flatland.

Answer 3

You might find this discussion of higher dimensional cubes useful: http://york.cuny.edu/~malk/tidbits/n-cube-tidbit.html

Answer 4

Once you've perused Flatland, I will recommend looking at the classic An Introduction to the Geometry of $n$ Dimensions by Sommerville. You'll also want to search for books by H.S.M. Coxeter, like his Regular Polytopes.

Answer 5

You might want to check out "The Shape of Space" by Jeffrey Weeks. One of the topics discussed is visualizing more than three dimensions, but the whole book is a fascinating read, touching on a nice array of topics in low-dimensional topology and aiming to get you to understand things at the gut level.