Talk:Four-dimensional space/Archive 2

Archive 1

Archive 2

Archive 3

This article needs a rewrite

This article appears to me to be a sprawling mess; it is confusing. Could someone please fix it?

Having said that, I think it should be mentioned that Einstein's originally published SR papers employed the construct of 3 spacial dimensions (vectors) plus time (a scalar). When Minkowski reformulated SR into the 4-D construct, he described the four dimensions to be x, y, z, and ct. The "fourth dimension" was not t, it was ct, which is a distance just like the other three dimensions. The Minkowskian reformulation of Einstein's SR was a mathematical convenience -- it was indeed a powerful computational aid. But this does not, philosophically speaking, necessarily mean that there actually "is" a fourth dimension. Minkowsky's reformulation is a powerful mathematical convenience. And, again, Minkowsky's fourth dimension is not time, it is the distance that light would travel in a vacuum in time t, namely ct. Worldrimroamer (talk) 14:41, 13 January 2010 (UTC)

It is both. I.e. you can think of it as 3 separate space and 1 time dimension, or of 4 dimensions in spacetime. It's not just a convenience. Einstein showed in particular that there's nothing special about our local space and time. An observer in another frame of reference, i.e. moving relative to the first, has her own set of space and time axes. They are connected by a Lorentz transformation, so in particular the time axes they experience are different, connected by a rotation in 4D. If there's no special absolute time then it does not make sense to treat it as something separate - though we tend to do it in our local version of spacetime where velocities are so low that relativistic effects are negligible.

I appreciate your taking the time to make a response, but I don't need to have these things explained to me because I already understand all of that, and more. (I made A's in the undergraduate and graduate physics courses.) I know about the rotations in 4D -- it is a very powerful mathematical construct. I was just pointing out that you can do SR with three vector dimensions plus the scalar time. It's not convenient to do it that way, but it can certainly be done. (Incidentally, I'm not sure, but I'm fairly sure that GR can be done, too, with 3D + scalar time, but then it really, really gets messy without the 4D construct.) And again, I point out, "time" is not the fourth dimension in Minkowski 4D "space-time". The fourth dimension is ct, not t. I'm aware of all the counterarguments as to why I'm just not "getting it". But I do get it. I just don't agree with the philosophical semantics, if you will, which pervades the textbooks.

I don't intend to modify the article, because I would be mugged by a host of people who believe that their equations are what is real, rather than believing that reality is what is real. I won't go there. Worldrimroamer (talk) 14:41, 13 January 2010 (UTC)

As for the article, it's already tagged as in need of expert attention. You could try and improve it yourself, or say in more detail what's wrong with it and how it could be improved.--JohnBlackburne^words_deeds 09:15, 13 January 2010 (UTC)

I think you were probably looking for the article of spacetime or Minkowski space. This is about a space of four spatial dimensions. Minkowski's fourth dimension is time-like rather than space-like in physics terms. If you lived in 4 dimensions in the terms of this article you would be able to point your finger in the fourth dimension. Dmcq (talk) 09:23, 13 January 2010 (UTC)

ps as to a rewrite, I'd advise a new editor to start on a less popular and for more glaringly in need article. There's lots of them. Dmcq (talk) 09:26, 13 January 2010 (UTC)

Good suggestion. But I'm very hesitant to make more than minor changes to articles. A major rewrite is a scary thing to do amongst so many knowledgeable and strong-opinioned experts. It can easily lead to warfare. Again, thanks for your input. Worldrimroamer (talk) 14:41, 13 January 2010 (UTC)

I find it convenient to think of the extra dimension as ict ... —Tamfang (talk) 21:19, 14 January 2010 (UTC)

I've made a start on improving it, so far just re-writing the lede to make it a better introduction. I've some ideas what else I could do, but want to be cautious as it's a long established article, and I don't think there's anything fundamentally wrong with the content. It's just a bit disjointed, a bit difficult to follow, and has not enough maths relative to the dimensional analogy section. Anyone else is free to help out, or comment here, whichever they think best.--JohnBlackburne^words_deeds 20:07, 1 February 2010 (UTC)

The mass of the tesseract

What is the 4-D equivalent of mass? What proportion of the 4-D mass-equivalent would be measurable in each 3-D 'manifestation' (eg for the 4-D version of a rectangle the eight 'shoeboxes')? Jackiespeel (talk) 16:45, 14 January 2010 (UTC)

The answer to that question depends, I suppose, on what 3D mass "really is". If it has to do with particles, which have no dimension, then mass is mass in any dimension. If you fill a tesseract box with stuff of constant density, the mass is proportional to the hypervolume of the box; the eight cubes are boundaries, not containers.

If this doesn't answer your questions, please paraphrase. —Tamfang (talk) 21:26, 14 January 2010 (UTC)

Properties of n+m-dimensional spacetimes

My own thought is there really is no definition, i.e. no straightforward or easy way to extend mass as we know it in 3D to four dimensions. You can just say that mass is volume × density, and certainly volume is defined, but density isn't. One problem is that mass is so tied into our description of our universe which has only three "normal" dimensions (+ one of time). E.g. in E = mc² mass is the ratio of energy, a pretty basic quantity, speed, another one. Energy is closely related to our understanding of e.g. particles, radiation etc. Speed is simply the ratio of two of our dimensions and is an intrinsic property of anything relative to anything else.

More fundamentally a lot of our theory only works in 3D. What the image to the right says is that in any other dimensions other than our own (the white square) the laws of physics break down. You don't get galaxies and stars. You don't even get atoms - stable atoms, or at ones like we know, cannot form. Lots more about it here: Spacetime#Privileged character of 3+1 spacetime. So matter and mass, which is made up of atoms, cannot exist, or at least not as we know it, in 4 spatial dimensions.--JohnBlackburne^words_deeds 21:56, 14 January 2010 (UTC)

Most people with at least a basic knowledge of mass could follow the geometrics for 4-D space (show parallel examples of 2D, 3D, 4D... 'calculating angles, areas, volumes etc.' With such 'basic maths' it is easy to see that the 'geometrical appearance' of a higher dimensional object in the various lower dimensions can differ according to the various planes (the 3-D shoebox will have different lines on x, y and z, planes, and different faces x-y, x-z, and y-z, and the 4-D will have different 3-D shoeboxes across the different sets of w, x, y, and z etc).

It 'appears to be logical' that the 4-D objects will have an equivalent of mass (and I know the difference between weight and mass) - and, for a tessaract each 3-D component will have the same mass - but will the shoeboxes, which will differ in shape, as they do not 'show' the whole of the 4-D shape? Jackiespeel (talk) 22:29, 16 January 2010 (UTC)

If by "shoebox" you mean the cubic boundary of the tesseract then you can suppose they will have a finite thickness in a model of a tesseract. So if the tesseract has side length l and the thickness of the boundary is t the hypervolume is l ⁴ and the corresponding volume of the boundary, ignoring overlap, is 8tl ³ (more precisely not ignoring overlap it's l ⁴ - (l - 2t) ⁴) That's straightforward geometry. To get the mass you need to postulate a density and multiply everything by it, using whatever units you create for the purpose. But in doing so you're moving beyond the realms of the known laws of the universe.--JohnBlackburne^words_deeds 12:01, 17 January 2010 (UTC)

Sparked off by Dr Who/Tennant/Tardis blurbs on BBC and wondering how the mass of the Tardis would be 'distributed' given that I could deduce the mathematics/geometrics as above.

If anyone wishes to develop the thought exercise and link up the OR piece 'placed somewhere' (with sufficient 'descriptive text' to clarify the 'symbol soup') to the article feel free (g). 22:15, 17 January 2010 (UTC)

The Tardis ??? Apart from being from sci-fi where anything goes (just watched Avatar and loved how they called their mineral unobtainium) the Tardis is a collection of contradictions. It's a space-time machine for one, very powerful, sometimes sentient, and bigger on the inside than out. And it changes internally every few years: reportedly the next iteration will have stairs (again?). So anything you calculate about it is bound to be true for some perfectly reasonable assumptions by the show's standards. --JohnBlackburne^words_deeds 22:32, 17 January 2010 (UTC)

Why not use fictional objects (which can be bought or stolen in certain multiplayer games) as a starting point? As the 'geometric 4-D construct exists' can consider the practical aspects.

Might as well see if "the proverbial someone" (who has made much money from "round toits") has come up with a possible answer to an obscure question. Jackiespeel (talk) 15:26, 18 January 2010 (UTC)

Latest changes

My instinct is to revert this change to the article, as it has the following problems.

• Confusing introduction which is seeming to describe spacetime, not the subject of this article.
• The rest is about visualising 4D, but in this it largely duplicates the section above. That section is already overlong and unencyclopaedic, in the WP:NOTGUIDE sense.
• Very poor images: ugly and crude red on black and too large. The first two are simply much too big, and sit one above the other on my display as they're too wide to go side by side. The others are too large as they are animated gifs, far too many of them. All have borders too big and some of the animaations simply look wrong (the lowest middle one for example).
• Poor image layout, not properly inline with the text
• No captions or ALT text
• Pointless extra blank lines and so spaces

I.e. there's a lot wrong with it. Possibly fixable if someone were to fix all the images, rationalise the number and properly caption them and integrate them into the text so they can be followed with the article. But that would be a lot of work and I don't know it's needed as the article already has a long section on visualising the tesseract in 4D, too long already I think. What do other editors think?--JohnBlackburne^words_deeds 22:27, 25 February 2010 (UTC)

Agree with all that I'm afraid. There is one thing there which the rest of the article is missing and that is visualization via sections as in Hinton's original romance where the sphere moved through the plane. It should be added as well as the current visualization by projection and by shadows. It needn't involve so many images though to illustrate the idea. Dmcq (talk) 22:44, 25 February 2010 (UTC)

I've removed it. I agree there's scope for adding something along those lines, but it's difficult to see that could be used as a basis. There's already an overlong/image heavy dimensional analogy section, so the ideal thing would be for that to be rewritten to incorporate some of the ideas from the last added section, resulting in a shorter and more encyclopaedic section.--JohnBlackburne^words_deeds 10:17, 1 March 2010 (UTC)

Model

The first model of the cube, you know, the blue one that spins? I don't really "understand" it, I guess. Can you do a 4d model of an everyday object like a boul of ramen noodles or a mug? --BrandiAlwaysSmiles (talk) 19:07, 7 April 2010 (UTC)

I don't know how that would help: such things do not exist in four dimensions, the only way to make one would be to take the 3D object and "extend" it into 4D but there's no single way to do that.

The 4D shape is a tesseract, and you can go to its page to find out more about it. Also polychoron for other shapes in four dimensions. None of them are especially simple: that's the nature of 4D, but also one of it's attractions.--JohnBlackburne^words_deeds 19:13, 7 April 2010 (UTC)

Hologram shadow

If an actual 4d tesseract were to cast a shadow in our world would it appear as a 3d hologram type image or a normal 2d shadow? 67.243.159.27 (talk) 12:52, 12 May 2010 (UTC)

The best way of going round hypotheticals like this is to think of the 3D to 2D analogue. There is noting relevant to holograms about your shadow on the ground. It is a dasrker area of the ground. Analogously the 4D object to 3D shadow shold give a darkened volume of 3D whatever darkened in that respect would mean to us. Dmcq (talk) 13:06, 12 May 2010 (UTC)

I think I have a problem with this logic... If you take a picture , it is on a 2d plane , yet you can tell that the picture represents a 3d plane because of the shading. If that's not how "shadows" should be taken then I still offer this argument. From your perspective at any given moment you can only see one side of an object Creating a 2d image until you move and see the rest of it. This is the same with light so in theory a 2d image is creating a 2d shadow. And when looking at the 4d cube I saw one side of meaning the shadow created by something would be 2d because from my perspective and from any beam of lights perspective the object is 2d until one were to move. 6:15pm (18:13) EDT June 13 2011 — Preceding unsigned comment added by 71.206.162.23 (talk) 22:22, 13 June 2011 (UTC)

But the question was about shadows. When a 3d object casts a shadow on a 2d surface, there is no "shading" (of the kind you mean; there may be variation in opacity) and no front or back. —Tamfang (talk) 19:19, 14 June 2011 (UTC)

Yes but the only way we can tell a shadow of a 2d object like a circle , from the shadow of a Sphere (when both the circle is facing directly at the light source) is because the sphere will have shading on it giving it the Z coordinate of shadows. So if a 4d object were to have a 3d shadow , the shadow itself would need to have Depth and because a shadow is a lack of light this would not be possible June 16, 2011 EDT — Preceding unsigned comment added by 71.206.162.23 (talk) 17:24, 16 June 2011 (UTC)

The shadow of a sphere looks exactly like the shadow of a disc (appropriately placed), and yes, sometimes that's a problem. A shadow is a projection but not all projections are shadows. Sometimes color is used to represent the extra dimension (as in at least one xscreensaver module). —Tamfang (talk) 02:55, 17 June 2011 (UTC)

A shadow is a LACK of light because the light CANNOT pass through an object. For a shadow to be 3D so that we could stick our hand in the shadow the light would have to start and stop being able to pass through the air. June 16 11:42 pm ( 23:42) EDT

Yeah, so?

By "3D shadow" we mean the shape of a shadow that would be cast on a 3D hyperplane in 4D space. A model of that figure in 3D space is not really a shadow, but that's a convenient term for it. Do you find it misleading? —Tamfang (talk) 03:19, 18 June 2011 (UTC)

Exactly. Light in the 4D world would have no correspondence with something called light by any beings living in a 3D hyperplane. The shadow for instance could correspond to a green area with a high pitched sound and cold rough feel though transparent grey is probably better for naturalness and seeing the shape when thinking about it. Dmcq (talk) 07:22, 18 June 2011 (UTC)

I find this VERY misleading. Without the second post I would have never understood it ( It is still sinking in). That should be explained on the actual topic itself June 21 6:43 pm (18:43) — Preceding unsigned comment added by 71.206.162.23 (talk) 22:44, 21 June 2011 (UTC)

We can only really go by citations and I don't know of anyone who spells it out in detail. We can only say it in the level of fuzziness that's there, more and we'd be going in for original research. From the 4D viewpoint looking at the 3D hyperplane it is a grey volume, i.e. one that is not so illuminated just like in 3D, it is just how that would be interpreted by a being living in the 3D I was talking about. Dmcq (talk) 23:33, 21 June 2011 (UTC)

Comprehension

Is it possible to, or has anyone claim to be able to comprehend what a tesseract may look like in its 4D world by using any or all of the geometrical and mathematical aids available to us? 67.243.159.27 (talk) 18:06, 12 May 2010 (UTC)

Yes and with improved representations and aids it should be possible for more people to do so. 3D television and better sensory feedback should help. People differ greatly in their ability to visualize things even in 3D, for instance many people get simple problems wrong like what is the shape impressed in some plasticine like if you indent it with the corner of a cube? By the way please try avoiding putting a space at the start of your lines as it then sticks the sentence in a box and doesn't wrap the text. Dmcq (talk) 18:22, 12 May 2010 (UTC)

4D BEING ALL SEEING?

A 4d being can see all six sides (inside and out) of a non transparent cube simultaneously, but can he see all six sides of all of the eight cubes of a non transparent tesseract simultaneously? Or is it partially obscured as a cube would be to us, or a square to a 2d being? Yes or no, please explain (in layman terms if possible) thanks. By the way, late to be asking, but is it okay to ask these questions here or should I be on another site?67.243.159.27 (talk) 16:22, 16 May 2010 (UTC)

In Flatland you'd see (up to) 2 sides of a square. In 3space you see (up to) 3 square faces of a cube, including all four edges of each square. In 4space you'd see (up to) 4 cubical "facets" of a tesseract, including all six of each cube's square "ridges"; none of the squares is interior to the tesseract. —Tamfang (talk) 19:16, 16 May 2010 (UTC)

Yes, if we assume eyes in 4D are analogous to ours in 3D, so work by projection, then the section of the article on projection shows you the 3D figures that are projections of the 4D shape, containing up to four cubes, though distorted by the projection. Of course it's a bold assumption to make that eyes work the same in 4D - the laws of physics as we know them only work in 3D, i.e. with three space dimensions alongside one of time.--JohnBlackburne^words_deeds 19:53, 16 May 2010 (UTC)

It's an even bolder assumption to postulate the existence of 4D beings. But it is helpful to make such generalizations, even if fictitious, as an aid to our understanding of 4-space.—Tetracube (talk) 22:09, 16 May 2010 (UTC)

Some hints how a 4-dimensional being might see its surroundings may be given by the 4D Maze game that has been used in the maze study (Fourth dimension#Cognition). In general, the all-vision is only present for objects with three or fewer dimensions, just as we can overview objects up to two dimensions (e.g. the screen in front of you). 4-D objects may obstruct other objects since vision, as we know it, is always a projection (e.g. of the 3-D world to the retina which can be treated as a 2-D manifold). The obstructions in 4-D have been accounted for in the maze game. In contrast, the tesseract at the top if our article seems to transparent /a wire model. It would be nice to have an opaque version, too. An face-on opaque tessaract would appear as a transparent cube (actually the front face), and a rotating one should look like interchanging and distorted cubes (like a rotating cube appears like interchanging distorted squares).--SiriusB (talk) 15:49, 8 October 2010 (UTC)

Just one eye

Pretending that beings exist in these dimensions, the being in 2D world can look at the line in 1D world (who's direction is only forward and back) and by standing to the left or right, perpendicular to the line he can clearly see the whole line and both ends of the line with just one eye, the guy in 1D world probably thinks that's impossible. Standing up above or down below, perpendicular to the square in 2D world, we can clearly see the whole square, all four sides, inside and out with just one eye, the guy in 2D world thinks that because the edges and corners are sealed the square is completely concealed, no one can enter or see inside of it, he doesn't understand that there is a gigantic opening on the top and bottom, because it's hard for him to comprehend up and down. When the being in 4D world looks at our cube while standing in the new direction in his dimension (Ana and Kata maybe), perpendicular to the cube he can clearly see the whole cube, all six sides of it, inside and out with just one eye, he doesn't need x-ray vision, the cube is not transparent nor does it change it's shape, 4D guy has a new point of view that is hard for us to comprehend,(Ana and Kata) to us is just as hard as up and down is to 2D world beings.Is any of this right, what do you think?67.243.159.27 (talk) 17:27, 27 May 2010 (UTC)

Yes that would be a reasonable extrapolation for eyes like ours. Dmcq (talk) 17:53, 27 May 2010 (UTC)

This and many other similar ideas have already been explored before. See, for example, [1].—Tetracube (talk) 23:37, 27 May 2010 (UTC)

We're all just shadows of our 4dimensional selves!!

/cynic —Preceding unsigned comment added by 81.191.92.230 (talk) 05:11, 14 November 2010 (UTC)

Time?

Hogswash. It's all hogswash. —Preceding unsigned comment added by 134.7.248.131 (talk) 06:17, 26 June 2010 (UTC)

This article really shouldn't be in the "Time" template – it is about four spatial dimensions, not spacetime which has its own article. I'd remove it myself, but I want to hear any possible opinions on this first. ~ Keiji (iNVERTED) (Talk) 11:39, 31 May 2010 (UTC)

Which time template? I can't find any on this article.—Tetracube (talk) 16:39, 31 May 2010 (UTC)

The nav template at the bottom. I agree it seems out of place there. It's protected, so you can't edit it, but you can use {{editprotected}} on it's talk page to get an admin to do it.--JohnBlackburne^words_deeds 16:54, 31 May 2010 (UTC)

Yep, I agree too. It's a different issue entirely. This article is about 4D spatial geometry, whatever that is supposed to be. The template should be removed as being a wrong classification. David Tombe (talk) 18:49, 31 May 2010 (UTC)

"Whatever that is supposed to be", is simply Euclidean geometry in 4 dimensions. :-) 4-dimensional space-time isn't Euclidean.—Tetracube (talk) 23:20, 31 May 2010 (UTC)

Tetracube, I agree, but I'm not so sure that the 4D spatial geometry is Euclidean either. Did Euclid ever write about 4D geometry? David Tombe (talk) 00:07, 1 June 2010 (UTC)

Any geometry in which Euclid's Five Postulates are true is Euclidean (or should we use another word?). The number of dimensions is not part of the definition. —Tamfang (talk) 01:30, 1 June 2010 (UTC)

In modern times, "n-dimensional Euclidean geometry" is understood to mean Euclid's geometry with the requirement on the number of dimensions relaxed to allow any integer n. Coxeter, for example, writes in Regular Polytopes (p.118): "There are three ways of approaching the Euclidean geometry of four or more dimensions: ..." (emphasis mine). I do not think he is ignorant of the fact that Euclid only considered plane geometry and 3-dimensional geometry. Further on he elaborates (p.119): "We merely choose to enlarge the scope of Euclidean geometry by denying one of the usual axioms ('Two planes which have a common point have another'), ...". It is clear from this what exactly is meant by n-dimensional "Euclidean" geometry. One may argue, of course, that this should be termed "non-Euclidean geometry" since, after all, we are denying one of Euclid's axioms; however, the term "non-Euclidean geometry" is generally understood to refer rather to geometries arising from denying a different axiom of Euclid's (namely, that parallel lines never meet). Geometries that preserve the parallel axiom are generally referred to as "Euclidean", even if they are really extensions of Euclid's original system. Such is the phenomenon of historical accident in nomenclature.—Tetracube (talk) 01:40, 1 June 2010 (UTC)

Tetracube, Thank you very much for clarifying this for me. David Tombe (talk) 08:20, 1 June 2010 (UTC)

Being chased by a 4th dimensional lifeform

i found this interesting gameplay on 4th dimensional space. fourth-dimensional-lifeform Also, the wii game "super paper mario" allows mario to flip between 2d and 3d space. Imagine if mario were a man-eating monster/hunter, how would you prevent him from eating you? this is a very great visualization of how the i imagined the novel flatland.24.56.227.108 (talk) 04:29, 4 August 2010 (UTC)

maze study

In a another study the ability of humans to orient oneself in 2-D, 3-D and 4-D mazes consisting of four path segments of random length and connected with orthogonal random bends.

So far, this sentence has a subject: the ability. Would someone like to give it a verb? —Tamfang (talk) 00:40, 7 October 2010 (UTC)

Oops, this can sometimes happen with long sentences (even if they are only one-dimensional). I have added the missing verb and slightly expanded that section.--SiriusB (talk) 15:37, 8 October 2010 (UTC)

Vision within 3D forms

I was reading the section on visual scope and I could follow everything up until where it is mentioned that a 4D creature can see within a 3D form as we see *into* a square. How does the ability to process visual images in 4D give the ability of what sounds like x-ray vision as well? If all planes creating the cube are opaque and completely enclosed, how would the 4D eye penetrate that? Wouldn't it need to have an additional range of sight that focuses down to a microscopic level, affording such vision the ability to see past the particles that create the opacity? 84.215.54.24 (talk) 13:22, 19 October 2010 (UTC)

If you consider the analogue one dimension down, a 2D being living in a sheet of paper wouldn't be able to see through the side of a square into the interior. We are 3D and living in a 3D world and unable to see in to the interior of cubes. A 4D being would be able to look at the 3D cube from the side like we can look at a square from the side. We can see the inside of a square, a 4D being could see all the inside of a cube - even a solid cube where solid means 3D solid but not extending into the 4th dimension. Dmcq (talk) 23:43, 21 October 2010 (UTC)

In 4D, the extra dimension affords the possibility of paths from the 4D being's eye to any point in the interior of the cube without intersecting with the outer parts of the cube. No x-ray vision is involved; the 4D being is simply looking from a point outside of the hyperplane that the cube lies in, so the line-of-sight to every interior point is unobstructed. A truly 4D object, however, such as the tesseract, would be opaque to the 4D being, because its 3D bounding surfaces obstruct the line-of-sight to the tesseract's interior points.—Tetracube (talk) 05:08, 9 November 2010 (UTC)

A 3D being can see the interior of a 3D cube, if the cube were made of glass. I would entertain then: What would a 4D being see as the interior of a 4D cube, assuming that the cube is composed of a material that is equivalent to our 3D transparent glass? The interior of an interior? Or is this just an iteration of the wireframe? — Preceding unsigned comment added by 2601:9:8400:4D3:FCE6:6EC6:44C3:62B8 (talk) 06:12, 9 May 2014 (UTC)

A 4D being could see inside a 3D cube whether or not it was made of a material that was transparent in 3D. If the box was transparent in 4D a 4D being would see other 4D objects through the 3D box, it would be like us looking through a window which is essentially a 2D square which is transparent in 3D. Properties like the transparency and colour of an oobject don't have to be the same in different dimensions and they don't have to be the same for a 3D perspective of 4D. In fact I think it works easiest to consider the 3D perspective image of a 4D imageas being solid where there is abrupt changes and a light transparent material where the image doesn't change rapidly. Dmcq (talk) 19:31, 9 May 2014 (UTC)

Fitting in

In 2space although a square is made up of just 4 lines of equal length you can fit 100s of lines of that same length inside of it, in 3space you can fit 100s of squares of a given size inside of a cube that is made up of just 6 squares of that same given size, using this analogy does that mean you can fit 100s of cubes of a given size inside of a 4space tesseract that is made up of just 8 cubes of that same given size? 67.243.159.27 (talk) 19:28, 8 November 2010 (UTC) 67.243.159.27 (talk) 19:18, 8 November 2010 (UTC)

Yes Dmcq (talk) 19:22, 8 November 2010 (UTC)

Dimensional barrier?

It seems like 4space is close to us if not touching us, what form of barrier is preventing us from moving or falling towards 4space?67.243.159.27 (talk) 21:23, 10 November 2010 (UTC)

If 4space is "close to us" in any meaningful sense, we're already in it. —Tamfang (talk) 22:45, 10 November 2010 (UTC)

We move up,down,left,right,back,and forward. What stops us from moving towards ana and kata in a way where we would be able to see in real 3D like a 4space being. Or what prevents compressed air in a airtight container from escaping into 4space?67.243.159.27 (talk) 02:14, 11 November 2010 (UTC)

Most likely, our universe has only three (macroscopic) spatial dimensions. Is that a problem? —Tamfang (talk) 03:58, 11 November 2010 (UTC)

Not at all, I'm just trying to figure this thing out. 67.243.159.27 (talk) 07:11, 11 November 2010 (UTC)

4d beings

why do people talk of 4d 'beings', when 4space only exists in geometry? -- as far as i know there are no organic 'beings' that exist in geometry. By placing hypothetical organisms into places where they should not be leads to unnecessary obfuscation. ie is the article adressing mathematic theory or something that is realistically possible in the space we live in (ie reality). Perhaps it would be better to put the hypothetical stuff (much of the analogy) in a neat section of its own, or at least clearly label it as something that's clearly not possible in our reality. —Preceding unsigned comment added by 81.191.92.230 (talk) 04:56, 14 November 2010 (UTC)

You have evidence that there are no 4d beings? ;-) There's many people who believe we live in a all sorts of dimensions from 10 to 23. Reality is a strange thing. Dmcq (talk) 12:15, 14 November 2010 (UTC)

Plus there's some who think we may actually live in a 2d world are that the whole business of dimensions is a construct of something else. Dmcq (talk) 12:17, 14 November 2010 (UTC)

I'm pretty sure there are no organic beings in geometry. And how about we stick to the common concensus for explaining our reality based on the evidence we have -- sound fair? —Preceding unsigned comment added by 81.191.92.230 (talk) 16:53, 15 November 2010 (UTC)

No not fair enough, that is just your idea not a consensus, plus it works out better for explaining things. If you think the whole business is hypothetical why are you worried about hypothetical 4d beings going about in a hypothetical 4d space? Are we not supposed to talk about hypothetical aliens as beings because we don't know if aliens exist? Dmcq (talk) 17:59, 15 November 2010 (UTC)

geometry is a field of math, a theoretical construct - no organic beings exist in a theoretical construct, by definition. As I pointed out, it does not help explain things at all - it only leads to obfuscation about whether we're talking about the possibility of 4 phyiscal dimensions in our 3d world, or geometry theory (eg see section above this). All I asked was for the hypothetical crap to be placed somewhere separate OR AT LEAST made more clear by labelling, for ease of understanding. —Preceding unsigned comment added by 81.191.92.230 (talk) 02:08, 16 November 2010 (UTC)

And I believe you are wrong about it harming understanding, I believe it helps. Many books try to make the subject easier by talking about 4d beings. By the way no-one said any such hypothetical beings were or were not organic. Dmcq (talk) 18:07, 16 November 2010 (UTC)

"And I believe you are wrong": are you trying to say that your opinion over-rules mine? or that our opinions have level-footing? If the former, can you please explain why; if the later, is it then my prerogative to edit this page however I want?

At least I have an evidence to support my idea (see section above).

"no-one said any such hypothetical beings were or were not organic": so i suppose reference to said being having an eye with organic properties (rods and cones) isn't implying that the being is organic at all... —Preceding unsigned comment added by 81.191.92.230 (talk) 21:20, 18 November 2010 (UTC)

Was a passage about rods and cones recently removed? I missed it. —Tamfang (talk) 22:47, 18 November 2010 (UTC)

My statement that I believed you were wrong and saying 'beings' helped was explained in the following sentence 'Many books try to make the subject easier by talking about 4d beings'. I was not just relying on my own judgement. Dmcq (talk) 00:24, 19 November 2010 (UTC)

'Many books try to make the subject easier by talking about 4d beings': isn't that just a bunch of other peoples opinions though? As far as I know, you're not a fan of concensus (i.e., you don't think it has any relevance whatsoever); referring to a comment I made about a general view held by many people (a concecus; admittedley I did not provide references, but neither have you), you responded: 'that is just your idea not a consensus'.

@ Tamfang: you're right, rods and cones aren't mentioned. I must've been confused when the article started talking about eyes and the eyes retina and receptors - you must admit that it appears to be making a direct reference to a biological eye. Perhaps reference to a camera would be better??? —Preceding unsigned comment added by 81.191.92.230 (talk) 14:34, 19 November 2010 (UTC)

The reference to "4D beings" is merely a device for making the subject more accessible to a lay audience. They are no more and no less real than the physical existence of 4D polytopes—the latter is physically impossible according to our present understanding of the universe (even if you subscribe to string theory, you'd have to admit that dimensions greater than 3D are only accessible in scales far smaller than the Planck scale, which then begs the question, what would you construct these things out of?). Surely it must be eminently obvious that we're talking about hypothetical objects here, in the sense that they do not physically exist. Even mathematically, higher dimensional polytopes are merely mathematical structures arising from a particular set of rules (specifically, the axioms of Euclidean geometry suitably extended to allow spaces with higher dimension than 3D); treating them as actual objects existing "somewhere" is no more than a mental tool to help one grasp the mathematical structure. But just because they have no physical existence does not mean that drawing an analogy with physical objects is unhelpful—in fact, it greatly aids the human mind in grasping the structure of the underlying mathematics. This applies not just here, but in science in general—when was the last time you actually saw an atom? Yet physics and chemistry textbooks abound with illustrations of atoms as spheres or as electron clouds surrounding little raspberries with red protons and grey neutrons, and molecules as little balls connected by sticks. In reality, electrons are not clouds and nuclei are not raspberries, neither are molecules little balls connected by sticks. Protons aren't red, and neutrons definitely aren't grey; but it would be rather difficult to do any chemistry whatsoever if you have no visualization aid whatsoever and had to work directly with Schroedinger wave equations (most of which have no known analytical solution, by the way). It can't be that hard to understand that "4d beings" are merely an illustration, not an actuality. In fact, the article specifically refers to hypothetical 4D beings, which should make this point eminently clear.—Tetracube (talk) 16:23, 19 November 2010 (UTC)

As to 'isn't that just a bunch of other peoples opinions though?'. That is true - but how their opinions differ is that they have sold their books or had papers reviewed. This is referred to as being a WP:reliable source on Wikipedia whereas Wikipedia editors own opinions are called WP:original research and that is a very bad thing as far as Wikipedia is concerned. Dmcq (talk) 17:37, 19 November 2010 (UTC)

Existence of dimensions?

What causes dimensions in space, space, or things in space, or what? 67.243.159.27 (talk) 09:08, 15 November 2010 (UTC)

Nothing that we know of causes them: they just are. Our universe has been like this since the Big Bang, but we don't know what triggered that. What we do know is that there must be three space dimensions (and one of time) otherwise we would not exist. The physical reasoning for this is given in detail at spacetime#Privileged character of 3+1 spacetime. The more philosophical justification is the anthropic principle.--JohnBlackburne^words_deeds 09:24, 15 November 2010 (UTC)

67.243.159.27 (talk) 07:14, 18 November 2010 (UTC)

Dimensions may in a sense be illusory. See holographic principle for an argument out three dimensional world is really 2 dimensional. Dmcq (talk) 12:21, 18 November 2010 (UTC)

Re: “We do know that there must be three space dimensions…otherwise we would not exist.” Let’s assume that we know there must be at least three dimensions of space. How do we know there aren’t more? I don’t (necessarily) mean tiny curled extra dimensions as in string theory; I mean extra dimensions equivalent to the three. This is a thing I would like the article to discuss. It’s all very well to treat the fourth dimension as a mathematical or hypothetical construction, but surely at some point its actual existence or non-existence has been seriously considered. TheScotch (talk) 17:25, 16 June 2021 (UTC)

A question about whether something actually exists in our universe lies outside the purview of math. It would seem to be a question for physics. But I'm not sure that the existence of extra dimensions is actually a physical question. It may rather be a question about the mathematical models that physics finds useful. Hmm.

In any event, the question is not ours to answer. If you would like to add such material to this article, then you know what to do: Find reliable sources and summarize what they say. :)

The only sources that I know of are highly mathematical, and they discuss these dimensions as features of mathematical models rather than of the universe itself. Mgnbar (talk) 17:46, 16 June 2021 (UTC)

Re: “[The] question…lies outside the purview of math.”: Is this article strictly about math then? That was not my impression. If it is strictly about math, then it should specify so, and the question should be considered in another Wikipedia article—but I don’t think it is strictly about math. Re: “The question is not ours to answer”: Who is supposed to be “we” in this sentence? Wikipedians? I’m not suggesting that an editor just fabricate an argument. I’m suggesting that editors find sources that discuss it. I think I said that. Re: “If you would to add such material…”: If I had “such material” to add, I wouldn’t ask your damn permission, I’d just add it. I came here looking for “such material”. If I sound annoyed with your response, that’s because I am. You don't address the matter; you dredge up three irrelevant and trite excuses, each of which contradicts the other two, to officiously dismiss the matter out of hand. TheScotch (talk) 19:25, 16 June 2021 (UTC)

I'm sorry to have irritated you. I didn't mean to.

You are correct that this article is not solely about math. I didn't intend to suggest that it was. Yes, by "we" I meant Wikipedia editors.

I was trying to say this: Can math answer your question? No, because it's a question about the physical world. Can physics answer your question? Well, all of the physics sources, that I have seen, treat it as a question about their mathematical models, and therefore do not really answer your question either.

You are asking such a deep question, that its answer is unknown — and perhaps unknowable — to our civilization. So I am pessimistic that this article will address your question in the near future. But of course I am just one editor, and other editors know of sources that I do not. Best wishes. Mgnbar (talk) 21:19, 16 June 2021 (UTC)

Antimatter

In the article "Forth Dimension: Tetraspace (under "Rotation"), it basically said hypothetically if you put a book from 3space into 4space, flip it around in the 4th direction and put it back into 3space, the entire book would be backwards like a mirror image of its self. Would this also change the matter that the book is made of into antimatter, and if you did the same to a magnet, how would the magnet be effected? 67.243.159.27 (talk) 20:09, 26 November 2010 (UTC)

This article is about the purely mathematical construct, the fourth dimension. As such it doesn't really have much to say about the laws of physics or how they would be affected. The laws are really only defined in three spacial dimensions, extended to four if you include time but rotations as you describe are not possible using that dimension. Rotations using that dimension are Lorentz transformations and are limited by special relativity so things can't be "flipped".--JohnBlackburne^words_deeds 20:18, 26 November 2010 (UTC)

Bogus projections

Did anyone notice that the projections are bollocks? The inner smaller cube is the outer most face of the tesseract in the fourth dimension, yet it is placed in the 3D middle of the front face (the big cube). That makes absolutely no sense at all. Objects in the front always overlap the objects behind them, no matter how many dimensions you have.

Most of those are transparent wireframe projections, that smaller cube is the back cube of the tesseract. Dmcq (talk) 14:30, 19 December 2010 (UTC)

In four dimensions, all faces of the tesseract are "outermost". —Tamfang (talk) 17:34, 19 December 2010 (UTC)

4space point of view

Is it possible for us as 3d beings to create an image in our minds of an opaque 3d cube the way it would look if one were able to view it from the side in 4space, where all points of the cube can be seen simultaneously? And if any being (weather its us or not) could see it, would all edges appear to be equal length and perpendicular at the vertices, not skewed, like a square is not skewed when we look at it from the side. 67.243.159.27 (talk) 10:49, 20 January 2011 (UTC). I asked a simular question under the heading "comprehension", "is it possible to comprehend what a tesseract may look like in its 4d world?", the answer was yes. I thought it would be easier if I start off with just one cube and imagine what it would look like moving around in a hypothetical 4space world, I've been trying for a long time, it seems impossible, if it is I don't want to keep trying. Any confusion I get from this information I'm sure is due to my own misunderstanding. You guys are great, I appreciate the answers. 67.243.159.27 (talk) 08:34, 22 January 2011 (UTC)

An image of a 3D cube in 4D would simply be a cube a bit squashed in some way. Personally I imaging things like that not by visualizing them as seeing but as inside myself by feeling so the cube is in 3D rather than being a 2D image of a 3D object viewed from a side. I don't have colours but textures do just as well including a haziness feeling for volumes I don't know about as for example when 'visualizing' a building.

The image of a square seen by us looks skewed but we know the original is not skewed unless someone is playing a trick on us. SO basically you're asking if one had a good enough experience of 4D whether one would automatically assume and imagine the correct shape when seeing a 3D image that would be a reasonable image of such an object. I believe the answer to that is yes that even some people could do that with a reasonable expreience of 'seeing' and 'handling' 4D objects via some way of simulating that.

I think you're better off imaging a tesseract rather than just a cube as the cube just warps a bit. An interesting project would be to design suitable controls so a person could feel in 4D and then one could feel it was still a cube or tesseract as the case may be. Dmcq (talk) 13:27, 6 February 2011 (UTC)

Title?

Isn't this article mainly about four-dimensional space as opposed to the fourth dimension per se? Lanthanum-138 (talk) 12:19, 5 March 2011 (UTC)

It's about the mathematical Euclidian space. It doesn't have a special name as far as I know, and both "fourth dimension" and "four-dimensional space" are used for it. It's notably not the same as the space obtained by adding time to the three spatial dimensions. That is non-Euclidian, and is called spacetime or Minkowski space.--JohnBlackburne^words_deeds 12:49, 5 March 2011 (UTC)

Four-dimensional space redirects here, but I reckon it ought to be the other way 'round. —Tamfang (talk) 17:04, 5 March 2011 (UTC)

Definitely "forth dimension" and "four-dimensional space" are not the same thing. This article mostly talks about the later, which is fine. However, the introductory line "In mathematics, the fourth dimension, or a four-dimensional space, is an abstract ..." absolutely makes no sense! I am re-phrasing it and describing the "fourth dimension" as the codimension of any 3 dimensional subspace of the the 4-dimensional space. - Subh83 (talk | contribs) 20:47, 17 March 2011 (UTC)

They are the same thing, or they refer to the same thing here. 'the fourth dimension' is an inexact, non-mathematical name for what mathematics calls 'four-dimensional space'. In some contexts the two are used interchangeably, in some one or the other is preferred. It is not just the extra dimension on top of the familiar three, as that breaks down in higher dimensions: Five-dimensional space is also known as the 'fifth dimension' as described in the section The "fifth dimension" in popular culture.--JohnBlackburne^words_deeds 21:21, 17 March 2011 (UTC)

1. I am not sure about popular culture, but in mathematics dimension is a number associated with a metric space. There are precise mathematical definition of dimensions (see Hausdorff dimension for example). Dimensions are numbers, which is different from the space itself.
2. 4 dimensional space need not be Euclidean. Euclidean spaces are metric spaces with flat metric. So even your statement "It's about the mathematical Euclidian space" is not correct. There can be curved 4 dimensional space (e.g. the 4-sphere), 4 dimensional space with pseudo-metric (see Lorentzian manifold), etc.

- Subh83 (talk | contribs) 21:32, 17 March 2011 (UTC)

Nevertheless that is what "fourth dimension" means here, and it is the common name. Again compare Five-dimensional space, which used to be at Fifth dimension (now a dab page). They also are two names for the same thing and 'fifth dimension' clearly does not refer to the extra dimension as there are two of them.

But the space described here is the Euclidian one. Spacetime is mentioned in the introduction, as it's perhaps a common misconception that it is the same, and the See Also section links to other generalisations.--JohnBlackburne^words_deeds 21:51, 17 March 2011 (UTC)

I am not trying to make this technical, but equating "n-th dimension" with "n-dimensional space" is factually incorrect. At no place in the Five-dimensional space article they have been equated. The redirect from fifth dimension only implies that the concepts are related, but they are not the one. You can keep it simple, but the statement "...the fourth dimension, or a four-dimensional space..." (which equates the two concepts) is absolutely incorrect. It should be re-phrased. If simplicity is a criteria of this article, I would suggest at least the removal of the term "...the fourth dimension, or..." from the first sentence of the article. - Subh83 (talk | contribs) 22:11, 17 March 2011 (UTC)

The article is called 'Fourth dimension' as that is the common name for the concept discussed here. As it says at WP:BEGINNING the article topic is usually the topic of the first sentence. It certainly should not be removed.--JohnBlackburne^words_deeds 22:35, 17 March 2011 (UTC)

Not at the cost of factual or logical inaccuracy. I have added WP:DISPUTED. Do not remove until WP:CONS. - Subh83 (talk | contribs) 22:53, 17 March 2011 (UTC)

Please read WP:BEGINNING. It states clearly that the topic of the article, the fourth dimension, should be the subject of the article. That is the common name, as shown by e.g. the titles of three of the references which all mention it. It's notable that those are non-scientific works (Hinton's writings are more philosophical than scientific), and that is where 'fourth dimension' most often occurs. Modern mathematics prefers 'four-dimensional space'.--JohnBlackburne^words_deeds 23:40, 17 March 2011 (UTC)

A fourth dimension is what must be added to familiar space to create 4-space; it is the strange and distinctive feature of 4-space, and as such a reasonable title for a book about 4-space, but not a synonym for 4-space. —Tamfang (talk) 04:43, 18 March 2011 (UTC)

Titles do not have 'factual accuracy', they are simply search keys for which the main rule is choose a name which is commonly used to describe the subject, see WP:TITLE. Truth or accuracy of the title does not come into it. The title is not the article, it is a way of finding the article. We can discuss whether it is a good title for the article but disputing its accuracy or requiring citations proving the title is correct is beside the point, unnecessary and irrelevant. Dmcq (talk) 10:26, 18 March 2011 (UTC)

It's not the title, it's about the first sentence that reads "...the fourth dimension, or a four-dimensional space...". Please read the above discussions or visit the article page to see the location of WP:DISPUTED.- Subh83 (talk | contribs) 15:49, 18 March 2011 (UTC)

As far as that is concerned it is just wrong as well I believe. Space is used without a qualifier and when used that way means either physical space or a euclidean space, in a maths context it'll just mean euclidean space. If you want anything more complicated you say something like a metric space or a manifold or in popular literature a curved space. Dmcq (talk) 16:55, 18 March 2011 (UTC)

well, what you said did not make much sense to me. Space without a quantifier does not make sense, especially for >3 dimensions. And after all, Euclidean space IS a metric space. The metric is a flat metric (the metric tensor's matrix representation being the identity). Whereas a 4-dimensional space can be ANY metric space, not just Euclidean. For example, the 4-dimensional space-time is Minkowski. There is nothing complicated about it. It just need to be correct/consistent according to the general conventions in mathematics and physics. - Subh83 (talk | contribs) 17:22, 18 March 2011 (UTC)

Euclidean space is a metric space does not imply metric space is a Euclidean space. 4-dimensional space without any qualifier does not normally mean anything except a 4 dimensional Euclidean space. Space without a qualifier does make sense as I pointed out above. In physics it means physical space and in mathematics it means Euclidean space. There is no requirement that it include all variations which can be got by adding an adjective. Have a look at the article cat for instance, it does not include big cats which means lions and tigers and suchlike. Dmcq (talk) 18:19, 18 March 2011 (UTC)

The statement "In mathematics, the fourth dimension, or a four-dimensional ("4D") space..." is still disputed as it is mathematically incorrect. No consensus has been reached more discussion from more people is required is required.- Subh83 (talk | contribs) 16:25, 22 March 2011 (UTC)

Do not remove tags.- Subh83 (talk | contribs) 16:26, 22 March 2011 (UTC)

You are the only one who seems to object to the current wording which is clear to all other editors, so there is a consensus for leaving it as it is. I also agree with Dmcq over the 'expert' tag, though you may have reverted that by mistake (you should always give a reason if you undo another editor's contributions). If you are still unhappy with it you should perhaps propose an alternate wording for the introduction which deals with your concerns.--JohnBlackburne^words_deeds 16:38, 22 March 2011 (UTC)

I am not sure if 2 users' opinion out of 5 users can win consensus to the 2 just because they were verbose and persistant. I guess you missed comments by User:Lanthanum-138 and User:Tamfang. Also, see below.

4th dimension and 4-dimensional space are not the same

Please see the complete discussion above. Since it is not possible to define dimensions without being a little technical (see Dimension, Hausdorff dimension), and since according to some contributors, the article cannot cross a threshold of technicality, I suggest the page be moved to "Four dimensional space". Then the first line be changed to "In mathematics, a four-dimensional ("4D") space...", which, at least, won't be mathematically incorrect while still respecting WP:BEGINNING.- Subh83 (talk | contribs) 17:59, 22 March 2011 (UTC)

There is no limit to how technical an article can be. However they should be written in the main to be accessible to people who would be able to understand them. Dimension in general isn't relevant to this article. Just because the meaning of dimension has been extended doesn't mean that all uses of dimension now need extra qualifications. This is just getting ridiculous. We don't need to start invoking Hausdorf dimensioons before talking about Euclidean geometry. Dmcq (talk) 18:12, 22 March 2011 (UTC)

There is no need to even go to Hausdorff. The basic informal definition from dimensions: "...the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it...". So dimensions is a number, which is distinct from the space itself. Iven if you don't invoke Hausdorff, 4th dimension and 4-dimensional space are still not the same. And as for ridiculousness, I guess invoking Hausdorff dimension is less of so than invoking "cats" and "big cats" in this discussion. - Subh83 (talk | contribs) 18:41, 22 March 2011 (UTC)

I have raised this matter at WT:WikiProject_Mathematics#Fourth_Dimension for more eyes as I really don't see a resolution with the current people discussing here. Dmcq (talk) 11:04, 31 March 2011 (UTC)

I think there are two points of discussion:

• What is the topic of the article?
• Is the title appropriate for the topic?

The topic, I believe, if four-dimensional Euclidean space, usually called four-dimensional space by mathematicians. I think that talking about Hausdorff dimension or topological dimension (of arbitrary topological spaces) would be a little off topic.

For the title, we cite a book by Hinton that uses the term "fourth dimension" to refer to the space, and four-dimensional space redirects here. So whichever name someone searches for, they will find this article. That makes me think the title is harmless. — Carl (CBM · talk) 11:23, 31 March 2011 (UTC)

Some observations: The term "fourth dimension" is not really used by mathematicians. A mathematician would be more precise in saying which four-dimensional space (vector space, Euclidean space, Minkowski space, real manifold, complex manifold, topological space, etc.) she wanted to talk about. The term "fourth dimension" may be used by physicists, especially when teaching students the fundamentals of relativity. Perhaps that is why some editors feel that four-dimensional space must be Minkowski space. The term "fourth dimension" also appears heavily in science fiction, and readers may come to this article for that reason.

In my opinion, the material in this article is almost all useful and interesting: history, hyperspheres and projections, diagrams, etc. The article could easily have sections about Euclidean 4-space and sections about Minkowski 4-space. Presenting them side-by-side could really help readers understand the difference! There could also be a "Generalizations" section that mentions that mathematics has many kinds of space and many kinds of dimension. Somewhere it should also be mentioned that our spacetime is not Minkowski space but rather (better modeled by) a Lorentzian manifold. Finally there could be an "In fiction" or "In popular culture" section that discusses the role of four-dimensional space in sci-fi, etc. Mgnbar (talk) 13:19, 31 March 2011 (UTC)

hi. Thanks for initiating the discussion. I guess the main issue is with the first sentence of the article that says "...fourth dimension, or a four-dimensional ("4D") space...", which precisely does not make much sense since the two terms "fourth dimension" and "four-dimensional space" mean two different things (although the former, as rightly pointed out by User:Mgnbar, may mean something quite vague). It would have been fine if the first sentence would have just mentioned something like "In mathematics, a four-dimensional ("4D") space, is an abstract concept...". However, as User:JohnBlackburne has rightly pointed out, that may violate WP:BEGINNING. So I suggested something like "In mathematics, fourth dimension is the codimension of any 3 dimensional subspace of a 4-dimensional space", which although conforms with WP:BEGINNING, precisely has 2 problems: i. it's becoming too technical for an introductory sentence, ii. it makes reference to 4-dimensional space without first defining it. So, the better solution would be to change the title to "4-dimensional space", and use a simple introductory sentence "In mathematics, a four-dimensional ("4D") space, is an abstract concept...", which will conform with WP:BEGINNING, won't be incorrect, and will also be simple enough for an introductory statement. - Subh83 (talk | contribs) 17:52, 31 March 2011 (UTC)

I didn't comment on the title in my previous post. I tend to agree with you: "4-dimensional space" or "Four-dimensional space" seems better than "Fourth dimension". I also agree with you that the "codimension" definition is too technical. However, please consider that replacing "Fourth dimension" with "Four-dimensional space" is not as simple as replacing "Fifth dimension" with "Five-dimensional space" would be. The term "Fourth dimension" encapsulates a powerful idea: We are all accustomed to three dimensions, and it is difficult but useful to imagine one dimension more than that. In other words, this article should not just be about 4D Euclidean and/or Minkowski space, but should help a novice reader bridge the 3D-4D gap. The projections, pictures, etc. are great for this reason. Mgnbar (talk) 19:11, 31 March 2011 (UTC)

In physics the four dimensional Minkowski space is referred to as a spacetime rather than a space, just calling it a space is more of maths thing. In Abbott's Flatland he deals with a Euclidean four dimensional space where we are confined to a three dimensional sheet and time is the fourth dimension and H G Well's The Time Machine sounds like it is based on the same idea. These are not spacetimes in the modern sense.

The only changes I think should be done to this article are to remove the 'a' in 'a four dimensional space' because it talks about the specific four dimensional space that people mean when they use the term. They don't normally refer to spacetime that way, it just is 'a' four dimensional space in maths terms. Also it should make it more clear that it is not dealing with the Minkowski one direct people there and that the Minkowski one is the straightforward one people mean in physics. Dmcq (talk) 21:37, 31 March 2011 (UTC)

You have a point that Minkowski space is called spacetime, not space. An article entitled "Four-dimensional space" could still be misconstrued to be about 4-dim vector spaces, but that's easy to clarify in the text. So you've convinced me that this article can just be about 4-dim Euclidean space. In that case, I feel more strongly that its title should be "Four-dimensional space", not "Fourth dimension". Mgnbar (talk) 22:11, 31 March 2011 (UTC)

The title of Hinton's book is The Fourth Dimension, but does the body of the book use fourth dimension as a synonym for four-dimensional space? The fourth dimension is the distinctive and exotic feature of 4-space, and thus a legitimate title for the book (likely to sell better than Four-Dimensional Space); there are any number of novels named for some attribute of the protagonist but not about that attribute. I said this above, but I guess I may have to say it again the next time someone asserts, relying on the title alone, that Hinton used the terms synonymously. —Tamfang (talk) 01:21, 1 April 2011 (UTC)

Well here's a quote from that book:

Take now the case of four dimensions. Instead of bringing before the mind a sheet of paper conceive a solid of three dimensions. If this solid were to become infinite it would fill up the whole of three-dimensional space. But it would not fill up the whole of four-dimensional space. It would be to four-dimensional space what an infinite plane is to three-dimensional space. There could be in four-dimensional space an infinite number of such solids, just as in three-dimensional space there could be an infinite number of infinite planes.

Dmcq (talk) 10:11, 1 April 2011 (UTC)

I guess that's a 'no' to my question, then. And do you reckon this supports preferring the less accurate term? —Tamfang (talk) 01:37, 5 April 2011 (UTC)

No he does not say 'Fourth dimension is a synonym for four dimensional space'. What he did was write 'What is the Fourth Dimension", that was the topic, and he described "four dimensional space" as a main item within that topic as seen by the quote I gave. Dmcq (talk) 09:13, 6 April 2011 (UTC)

What, does Hinton cover "items" other than 4space within the topic of The Fourth Dimension? Intriguing.

If you agree that fourth dimension is not a synonym for four-dimensional space, and that the latter is the subject of the article — I believe there is abundant precedent for using redirects from keywords which are not likely to be the titles of articles, to closely-related concepts which do have articles. (For a frivolous example, Hobbit-hole redirects to Hobbit.) —Tamfang (talk) 20:23, 6 April 2011 (UTC)