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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.

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You "understand" three dimensional space? That's more than any serious geometer or topologist would say of himself. (You do realise that one of the Clay 1 Million Dollar problems was about three-dimensional objects?) Seriously though, it's not clear at all what you know and what prerequisites you have, and without that information answerers will have to guess. – Alex B. Oct 19 '11 at 4:55
The add comment button didn't show up before I edited my question. – somehume Oct 19 '11 at 14:17
@Somehume: do edit the question to include additional information (esp. those requested by commenters) instead of leaving them buried in the comments. Your response to Alex B's query is one that will be useful in guiding other users compose their answers, so deserves to be prominently displayed in the question statement itself. – Willie Wong Oct 19 '11 at 15:27
up vote 2 down vote accepted

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.

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The Virginia link works, and Google won't let me peek, so here is another link. – J. M. Oct 19 '11 at 15:03


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More lengthily: if you can relate to A. Square's situation after reading the tale, you're ready. – J. M. Oct 19 '11 at 1:50
There's also a wonderful perspective of viewing time as just another spatial dimension in Vonnegut's Slaughterhouse Five. The Tralfamadorians can "see four dimensions" – Shaun Ault Oct 19 '11 at 2:40
@somehume: (slightly off-topic: that's because Flatland is out of copyright and the digital edition came from Project Gutenberg, so Nook users needn't be left out) One shouldn't think of Flatland as a true novella. It should be thought of as a mathematics textbook disguised as a novella. – Willie Wong Oct 19 '11 at 15:12
Flatland was intended as a satire on Victorian society. We are far enough removed to ignore, or be oblivious to, the satire, and to read it as an entertaining math book. There's also the sequel, Flatterland (, by Ian Stewart. – Gerry Myerson Oct 19 '11 at 22:23
somehume, are you asking which answer to accept if you find more than one answer useful? It doesn't matter! Pick any useful answer, and accept it. Pick the longest one, or the shortest one, or the one most in need of a boost in reputation, or toss a coin. No one but you will ever know what method you used, so don't worry about it. Leave nice messages at the useful ones you didn't accept, and everyone will be happy. – Gerry Myerson Oct 23 '11 at 12:01

You might find this discussion of higher dimensional cubes useful:

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Those who found Joseph Malkevitch's essay interesting may also be interested in the comments I posted at See also the 2005 Bull. Amer. Math. Soc. survey article at – Dave L. Renfro Oct 19 '11 at 14:45
@Dave: I assume you mean in particular this article from that issue? – Willie Wong Oct 19 '11 at 15:13
@Willie Wong: Yes, that's the article. Obviously, it's much too advanced for the original poster (and mostly too advanced for me as well), but it's certainly something that anyone remotely interested in this topic will want to look over. – Dave L. Renfro Oct 19 '11 at 15:22

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.

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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.

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