# Explaining multiple dimensions **URL:** \ **Category:** Factual Questions\ **Created:** [February 7, 2007, 1:28am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164 "2007-02-07T01:28:40Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [February 7, 2007, 1:28am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/1 "2007-02-07T01:28:40Z")
On another board This site came up: > **[Imagining the Tenth Dimension - A Book by Rob Bryanton](http://tenthdimension.com/medialinks.php)** > > Imagining the Tenth Dimension, a new way of thinking about time, space, and string theory, a book by Rob Bryanton Is this a valid way of describing multiple dimensions? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 7, 2007, 1:49am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/2 "2007-02-07T01:49:51Z")
In a word, nope. That site is all guesswork, which wouldn’t be so bad, except it’s uninformed guesswork. ---
**Author:** ![desdinova](https://avatars.discourse-cdn.com/v4/letter/d/8baadc/32.png) [@desdinova](https://boards.straightdope.com/u/desdinova)\ **Post date:** [February 7, 2007, 2:52am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/3 "2007-02-07T02:52:31Z")
For half the price of that book, I’ll sell you an Oscillation Overthruster that will show you everything you want to know about the 8th dimension. ---
**Author:** ![aerodave](https://avatars.discourse-cdn.com/v4/letter/a/f08c70/32.png) [@aerodave](https://boards.straightdope.com/u/aerodave)\ **Post date:** [February 7, 2007, 4:45am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/4 "2007-02-07T04:45:17Z")
Wow…worst ending ever. The guy basically said “Oh shit…I ran out of road.” Followed by “Hey, look over there!” as he snuck out the back door. ---
**Author:** ![BACI](https://avatars.discourse-cdn.com/v4/letter/b/ce73a5/32.png) [@BACI](https://boards.straightdope.com/u/BACI)\ **Post date:** [February 7, 2007, 6:52am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/5 "2007-02-07T06:52:06Z")
Agreed. So how do we intuitate (!) dimensions 5 and up? ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [February 7, 2007, 7:31am UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/6 "2007-02-07T07:31:19Z")
[QUOTE=BACI] Agreed. So how do we intuitate (!) dimensions 5 and up? [/QUOTE] With extreme difficulty. Our mind’s eye is simply not trained to visualize higher dimensions spatially (as opposed to mathematically…coordinate geometry can be thought of a non-visual way of reasoning about higher spatial dimensions). There are tricks, but they’re very limited. The simplest trick to to just to ignore dimensions that you don’t need to worry about at the time. If you’re learning about Minkowski spacetime, which is four-dimensional (three dimensions of space and one of time), you rarely need to worry about all three spatial dimensions just to understand what’s going on. In fact most of the time you can just ignore two of the spatial dimensions and focus on the remaining spatial dimension and the time dimensions, effectively “compressing” four dimensions to two. Another trick, a personal favourite of mine, is to take advantage of the fact that we’re trained to imagine that television images are three-dimensional even though they’re only two-dimensional. So if I want to visualize four dimensions, I can imagine a stack of transparent glass plates, each one acting like a TV screen to “contain” three dimensions in only two. But this trick doesn’t scale well to five dimensions. I’ve tried to imagine a two-dimensional array of tiny glass cubes, each cube acting like a tiny television to “contain” three dimensions in a point, thus visualizing five dimensions…but it doesn’t work very well. Much like that video, which seems to be a classic example of stretching an analogy well past the breaking point. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [February 7, 2007, 1:23pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/7 "2007-02-07T13:23:21Z")
> [@](#): > > Agreed. So how do we intuitate (!) dimensions 5 and up? There are other ways, too. Imagine you’re a two-dimensional being trying to comprehend a three-D world. Besides the possibilities of only looking at two dimensions of that 3D world at once, or of imagining the 3D world as a succession of 2-D slices, there are at least two other ways to think about 3D objects: 1.) Imagine the 3D object projected onto a 2D surface. You can thin k of this as a mathematical projection, or as the shadow cast by a 3D object on a 2D surface (of course, a 2D being can’t do the latter, but you see what I mean). A 3D cube can cast a square shadow, a rectangular shadow, a regular hexagonal shadow, or irregular polygonal shadows. The particular shadows it casts and the way it transitions between them is characteristic of the 3D shape. (So you can’t get, for instance, a triangular shadow from a cube). Similarly, 4D objects casts 3-Dimensional “shape shadows” in our 3D world. I understand that higher-order atomic orbits are are like. We clearly see the symmetry of _s_ and _p_ orbitals, which only require rotation in the 3D world, but you can apparently generate the higher orbitals, like _d_, by rotation and projection of the same 4D form. 2.) You can “unfold” or “flatten” the 4D shape into 3D form, the same way you can cut a cube along several of its edges, leaving some intact, and unfolding the rsult into a 2D shape. There are several ways to “unfold” a tesseract into 3D space in this way (see salvador Dali’s _Crucifiction_ to see Chrrist crucified on an unfolded tesseract. Heinlein used this method to “unfold” a tesseract in his story “And he Built a Crooked House”. Another method of “unfolding” a tesseract uses distorted and nonuniform basis cubes – you place a small cube inside a larger one, and connect the nearest corners of the interior one with the nearest corners of the exterior one. The inner and outer cube, along with the very distorted six-sided figures you generated by joinin the corners all represent the basis cubes forming the hypersides of the tesseract. There are other ways of unfolding the tesseract, as well, but you get the idea. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [February 7, 2007, 4:14pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/8 "2007-02-07T16:14:13Z")
[QUOTE=CalMeacham] Another method of “unfolding” a tesseract uses distorted and nonuniform basis cubes – you place a small cube inside a larger one, and connect the nearest corners of the interior one with the nearest corners of the exterior one. The inner and outer cube, along with the very distorted six-sided figures you generated by joinin the corners all represent the basis cubes forming the hypersides of the tesseract. There are other ways of unfolding the tesseract, as well, but you get the idea. [/QUOTE] Note that this is just the projection of a tesseract into three dimensions, so it might be better to think of it as an example of your first visualization technique, not your second. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [February 7, 2007, 4:57pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/9 "2007-02-07T16:57:52Z")
> [@](#): > > Note that this is just the projection of a tesseract into three dimensions, so it might be better to think of it as an example of your first visualization technique, not your second. I don’t believe that it is ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [February 7, 2007, 5:04pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/10 "2007-02-07T17:04:56Z")
If you’ll forgive the slight hijack, n-dimensional spaces can be used to model some surprising things. In the 1970s, a computer scientist names Salton came up with such a model for a subject indexing and citation system, in which the dimensions were the thousands of subject terms and the similarity between queries and citations was determined by cosine relationships. Apparently trig works in n-dimensional space. ---
**Author:** ![Diceman](https://avatars.discourse-cdn.com/v4/letter/d/22d042/32.png) [@Diceman](https://boards.straightdope.com/u/Diceman)\ **Post date:** [February 7, 2007, 5:50pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/11 "2007-02-07T17:50:20Z")
I was impressed by that site, until I realized how totally asinine it is. He defines the third dimension in terms of warping a 2-dimensional plane so that you can jump from one point on the plane to another without traveling through the space in between. You might have asked yourself: “Is there some reason he can’t just define the third dimension as ‘up & down’?” There is: because his model won’t work that way. Basically, he’s trying to stretch the universe to fit his model, rather than fitting his model to the universe. It’s an interesting thought experiment, but if that’s an accurate description of string theory, then my respect for theoretical physics has fallen a notch or two. ---
**Author:** ![BJMoose](https://avatars.discourse-cdn.com/v4/letter/b/5fc32e/32.png) [@BJMoose](https://boards.straightdope.com/u/BJMoose)\ **Post date:** [February 7, 2007, 6:06pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/12 "2007-02-07T18:06:48Z")
Just wondering. Since computers have no problem with multi-dimensional arrays, and computers can print out graphical representions of 3-D objects (I’m thinking of those printouts with curved graph lines or whatever they are), could we get a computer to print out a representation of something in, say, five dimensions, or do we have to first know what such an “object” would look like and tell the computer how to do it? That’s probably not the clearest statement I’ve ever made. This is something in which I’ve had no practical experience myself. ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [February 7, 2007, 6:34pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/13 "2007-02-07T18:34:54Z")
[QUOTE=Enola Straight] On another board This site came up: > **[Imagining the Tenth Dimension - A Book by Rob Bryanton](http://tenthdimension.com/medialinks.php)** > > Imagining the Tenth Dimension, a new way of thinking about time, space, and string theory, a book by Rob Bryanton Is this a valid way of describing multiple dimensions? [/QUOTE] This has been posted here once before – if you do a search, you might be able to find my previous comments on it. In short: - No, it’s not valid, it’s not in any way connected to science, it’s complete crap. - To elaborate: The number 10 is obviously a reference to superstring theory, with its proposed ten dimensions. But the explanation they give is completely unrelated to string theory, as well as not being based on any sort of science or mathematics or anything more than some idiot talking out of his ass. - The proposed six extra dimensions of string theory are all spatial dimensions – any of this alternate possibilities and divergent timelines gibberish that site goes on about has nothing to do with it. - The proposed extra dimensions of string theory are “compactified”, i.e. wound up so tightly that they have basically no length. Again, the site’s comments on infinite alternate timelines are clearly unrelated. - I’d say that the author is confusing the extra dimensions of string theory with the (unrelated) “many worlds” interpretation of quantum mechanics, but that would be giving him too much credit. And anyway, the many worlds interpretation doesn’t say anything about moving between alternate timelines – that’s straight out of science fiction. - The number “ten” in string theory has to do with canceling certain anomalies in the theory – basically, making the math work out. It doesn’t have anything to do with this nonsensical talk about other sets of possible possibilities, or whatever. - I don’t believe it is possible for anyone to read even one book on string theory (even a book for non-scientists) and misunderstand it this badly. Even if they understood nothing, they couldn’t _believe_ they understood anything while simultaneously being so wrong in every possible way. I can only conclude that the author deliberately published nonsense in order to deceive. To sell books, I guess. - So basically, it’s a scam. - In fact, it’s not even a well constructed scam. It’s a scam that displays such an ignorance of the topic that anyone with any familiarity with the subject could see through it. I’m a grad student in physics, but I’m not studying string theory. My knowledge of the subject is no more than any technically minded person could gain in their free time, without even taking a class. And yet it was painfully obvious to me that that site was utter crap. - Obviously, the guy didn’t even bother to read a book about string theory. Otherwise, he could lie about extra dimensions much more convincingly. He seems to have looked at the jacket of “An Elegant Universe” or some such popular book, read the phrase “10 dimensions”, watched an episode of “Sliders” and than rattled off the first B.S. he could think of. - Either that, or he’s just nuts. It’s one thing to be factually inaccurate. It’s another thing to by a liar. In attempting to assert that the contents of that site have _any_ relation to modern science whatsoever (beyond simply appropriating catchy phrases like “ten dimensions”), the author is actively deceiving his audience. I hope this isn’t too harsh for GQ, but phoney science like this site or the movie “What the BLEEP do why know?” just makes me sick. As if there isn’t enough scientific ignorance in the world already. :mad: ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [February 7, 2007, 6:37pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/14 "2007-02-07T18:37:24Z")
[QUOTE=BJMoose] Just wondering. Since computers have no problem with multi-dimensional arrays, and computers can print out graphical representions of 3-D objects (I’m thinking of those printouts with curved graph lines or whatever they are), could we get a computer to print out a representation of something in, say, five dimensions, or do we have to first know what such an “object” would look like and tell the computer how to do it? That’s probably not the clearest statement I’ve ever made. This is something in which I’ve had no practical experience myself. [/QUOTE] Yes it is done. Look at the hyper cube. That is a 4 d cube projected onto a 2 d space. [http://local.wasp.uwa.edu.au/~pbourke/geometry/hyperspace/](http://local.wasp.uwa.edu.au/~pbourke/geometry/hyperspace/) ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [February 7, 2007, 6:48pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/15 "2007-02-07T18:48:06Z")
Now that I’ve filled the thread with negativity (albeit justified negativity, IMO), I feel the need to contribute something more positive. For those asking about visualizing higher spatial dimensions, I find the graphics in Mathworld’s [hypercube](http://mathworld.wolfram.com/Hypercube.html) article rather enjoyable. Blast, I see **gazpacho** kind of beat me to the punch. ---
**Author:** ![4.66](https://avatars.discourse-cdn.com/v4/letter/4/2bfe46/32.png) [@4.66](https://boards.straightdope.com/u/4.66)\ **Post date:** [February 7, 2007, 6:52pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/16 "2007-02-07T18:52:40Z")
[QUOTE=BACI] Agreed. So how do we intuitate (!) dimensions 5 and up? [/QUOTE] There have been some helpful suggestions above. Use them all to get the best intuition for this kind of thing. I prefer the following way of looking higher-dimensional space: A point in space (ie in _ **R** [sup]3[/sup]_) is a point determined by 3 real numbers. Such a point is usually imagined as the combination of three points on three real lines, where the three lines are placed in such a way that the three lines are perpendicular. Well, that is a convenient way to place them, but by no means the only way. Try placing them parallel to each other (vertically, in the same plane). That makes it possible to view such a point as a function, where each variable (coordinate) yields a real number. So _ **R** [sup]3[/sup]_ kan been seen as the set of functions _f_: _{x, y, z} -\> **R** _. This way of viewing _ **R** [sup]3[/sup]_ is a bit less intuitive, mainly due to the fact that it’s hard to see what it means for to points/vectors to be perpendicular to each other. This is however easily answered by mathematical formulas; two vectors _f_ and _g_ are perpendicular if their inproduct equals zero, where the inproduct is given by _\=Σ[sub]i[/sub]f(x[sub]i[/sub]) g(x[sub]i[/sub])_. Together with the intuitive assumptions that there are no gaps in _ **R** [sup]3[/sup]_ (formalized into saying that _ **R** [sup]3[/sup]_ must be complete), this gives an alternative way of viewing _ **R** [sup]3[/sup]_. This way of looking at the world is hardly convenient when dealing with three-dimensional worlds, but it’s a way of looking at it that is very easily used for high-dimensional space. Just replace _{x,y,z}_ by _{x[sub]1[/sub], x[sub]2[/sub], x[sub]3[/sub], x[sub]4[/sub], x[sub]5[/sub], x[sub]6[/sub], x[sub]7[/sub] }_ to obtain a seven-dimensional space. There’s not even a problem trying to imagine a infinite-dimensional space. [QUOTE=Spectre of Pithecanthropus] Apparently trig works in n-dimensional space. [/QUOTE] Yes. Using cosines and the like means your’e looking a the angle between two vectors. Now whatever the dimension of the space may be, those two vectors determine[sup]\*[/sup] a two-dimensional subspace. So do the math in that subspace. Or rembember the rule _\_ = cos \*ß \* |\*v \*| · |_w_| for inproducts in 2 and 3 dimensional space and use it as definition for the higher-dimensional analogon: cos _ß_ = \*\ / \*|\*v \*| ·|_w_| [sup]\*[/sup] Assuming the vectors aren’t parallel and not equal to the nullvector. ---
**Author:** ![BJMoose](https://avatars.discourse-cdn.com/v4/letter/b/5fc32e/32.png) [@BJMoose](https://boards.straightdope.com/u/BJMoose)\ **Post date:** [February 7, 2007, 6:54pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/17 "2007-02-07T18:54:30Z")
[QUOTE=gazpacho] Yes it is done. Look at the hyper cube. That is a 4 d cube projected onto a 2 d space. [http://local.wasp.uwa.edu.au/~pbourke/geometry/hyperspace/](http://local.wasp.uwa.edu.au/~pbourke/geometry/hyperspace/) [/QUOTE] Cool! Many thanx. ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [February 7, 2007, 7:18pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/18 "2007-02-07T19:18:55Z")
[QUOTE=4.66] Yes. Using cosines and the like means your’e looking a the angle between two vectors. Now whatever the dimension of the space may be, those two vectors determine[sup]\*[/sup] a two-dimensional subspace. So do the math in that subspace. Or rembember the rule _\_ = cos \*ß \* |\*v \*| · |_w_| for inproducts in 2 and 3 dimensional space and use it as definition for the higher-dimensional analogon: cos _ß_ = \*\ / \*|\*v \*| ·|_w_| [/QUOTE] Right–I’m no mathematician, but even I could see how the the formula would work as a basic measure of similarity. One thing that impressed me in particular was that the more terms, or dimensions, you had, the less need you had for weighting, and in some circumstances all the terms could be represented by bit values. However, I don’t know if this was ever actually implemented or not. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [February 7, 2007, 7:20pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/19 "2007-02-07T19:20:55Z")
[QUOTE=CalMeacham] > [@MikeS](#): > > Note that this is just the projection of a tesseract into three dimensions, so it might be better to think of it as an example of your first visualization technique, not your second. I don’t believe that it is [/QUOTE] It’s a projection from a point, not a parallel projection, but it is a projection. The big fancy name for it is a [Schlegel diagram](http://en.wikipedia.org/wiki/Schlegel_diagram). ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [February 7, 2007, 7:27pm UTC](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164/20 "2007-02-07T19:27:05Z")
> [@](#): > > It’s a projection from a point, not a parallel projection, but it is a projection Ah, that makes more sense. [Next page](https://boards.straightdope.com/t/explaining-multiple-dimensions/391164.md?page=2)