Talk:Five-dimensional space

This article seems to be written from the perspective that there's a preferred coordinate system (starting with "up/down, left/right and forwards/backwards"), with respect to which a fifth dimension might be identified. This is certainly not the favored approach of contemporary geometers. I suggest a move to Five-dimensional space and a corresponding overhaul of the article. --Trovatore 18:43, 16 November 2005 (UTC)

Reason for 'no': In addition to geometrical dimensions there are also physical dimensions, which are already height, width, depth and time (multiplied by velocity of light). Candidates for an additional 5th physical dimension are e.g. mass and probability.--Turul2 18:48, 04 December 2008 (UTC)

This article contains a formula to find out the volume of a 5 dimensional circle. Surely volume is not the correct word to describe the amount of space contained inside this 5d shape - just like area does not describe the amount of space inside a 3d object. Volume is measured in m³, which only describes 3 dimensions, the 5d objects amount of internal space would be measured in m⁵. —Preceding unsigned comment added by 86.134.9.179 (talk • contribs) 09:01, 17 June 2006

: The generic term is content. Will fix. —Tamfang 17:45, 17 June 2006 (UTC)

: Has been fixed to hypervolume. No appropriate article content. —Turul2 19:11, 04 December 2008 (UTC)

This article talks about the 5th dimension as an additional spatial dimension beyond the 4th, which it identifies as time. First of all, I don't think time can be regarded as a spatial dimension (at least, I doubt such an assumption is commonly accepted). Second of all, if the 5th dimension is merely another spatial one and non-temporal, then it is exactly like what the Fourth dimension article is talking about (4 spatial dimensions, potentially with one additional temporal one), except that the dimensions are labelled differently. In which case, I'm not sure I see the justification for a separate article. On the other hand, this article appears to be focusing on the geometry of 5 spatial dimensions, as it refers to 5D polytopes, and so it should not confuse the issue by labelling one of the dimensions as time. Besides, 4D space-time as defined by General Relativity is Minkowskian, not Euclidean, and as far as we know, the universe is by no means Euclidean with or without additional dimensions, so the discussion of polytopes doesn't really work in that context. I think this article should focus on 5D Euclidean space, and not try to rationalize it in terms of space-time.—Tetracube 00:36, 2 November 2006 (UTC)

:I whole heartedly agree! You have expressed this extremely well. Borg Master (talk) 09:41, 10 December 2023 (UTC)

both space and time are the 4th dimension, known as the spacetime dimension —Preceding unsigned comment added by 24.189.153.102 (talk) 13:10, 8 June 2008 (UTC)

As discussed in theory on the 4th dimension, time or speed is accepted as the 4th dimension. We have length, depth and height as the three other dimensions. So what is the 5th dimension? Is it a selector of the property of the 4th dimension? Lets just play with the thought and say that a value of 1 in the 5th dimension represents the property of time in a certain point in the 4th dimensional space, therefore the given value in the 4th dimension represents the time at which the 3rd dimensional image is given. This would mean that the image is moving according to the value in the 4th dimension.

The thing is that the value in the 5th dimension points to every possible version of the 4th dimension. The 5th dimensions give the 4th dimension all possible properties of the physical universe, probably also the abstract universe. In the 4th dimension we observe time when the value of the 5th dimension is 1. When the 5th dimension value is 2 it can represent atomic mass. Value 3 can represent taste. The value N represents all properties given in the relative universe of this specific case.

The full aspect of the 5th dimension is unthinkable to us, other than from a mathimatical point of view, as pain is for a rock.

• Daniel Holth, Norway 2nd November *

(Grammar edited by Dennis Standing)

One particular variant of Kaluza–Klein theory is induced matter theory and this theory

relates the 5th dimension to mass and momentum along it to charge (see discussion item 4 and space-time-matter consortium). --Turul2 (talk) 17:44, 18 August 2008 (UTC)

I think the link to "Imagining 10th dimension" should be removed, since it's mostly science fiction and nothing to do with current scientific concensus. —The preceding unsigned comment was added by 80.222.116.225 (talk) 12:39, 23 December 2006 (UTC).

• Seconded. \Mike(z) 17:40, 5 January 2007 (UTC)

I myself, studying dimensional theories, must disagree with the hypercube theory. Though it present an exceptional understanding of the dimensional pattern which my work is highly dependant on. It forgets that a line, one dimensional, extends infinitley. The square does NOT define two dimensional space. A parallelogram is the correct symbol. The parallelogram represents a plane once again extending infinitely. The cube, I will except as the symbol for three dimensional due to lack of excepted geometric symbol. The cube you must remember, as a representation of three dimensional, extends infinitely in all directions. Due to that fact the point, the line, the plane, and the cube are all co-cubular (lack of better word once again) so the hyper cube etc. are also co-cubular and not a shape exceeding the third dimension.

p.s. sorry for poor wording but mathmaticians have not yet had reason to define one where my words lack.

p.s.s. The person who orginated this theory was Intelligent and deserves credit. His theory is perfect in all ways except using a line segment in the stead of a line.

--Leon vautour 23:42, 6 January 2007 (UTC)Leon X Vautour

From the article: "Whether or not the real universe in which we live is somehow five-dimensional is a topic that is debated"

Isn't the 5th dimension sound?

Imagine:

1. a line
2. a plane
3. a shape
4. a shape rotating/moving through space
5. a shape rotating/moving through space and speaking.... ? (e.g., a holographic talkie, or a memory projection)

Which would make the 6th dimension "energy" transferred from one shape to another... ?

--Renice 13:34, 24 March 2007 (UTC)

Btw, this would also mean that Jedi knights are able to think in the 7th dimension -- the ability to force a 6th dimensional object to exert energy through suggestion. --Renice 14:20, 24 March 2007 (UTC)

This would also suggest that astroplaning may be a form of 7th dimensional thought. --Renice 14:42, 24 March 2007 (UTC)

The fifth dimesnion is described by some scientists as the 3rd dimension wrapping around itself. Look closely at the net of the hypercube and the resultant hypercube. The net is obviously 3-dimensional, unlike normal geometrical nets. In order to form the hypercube, the net appears to wrap around itself, joining the opposite ends of the net together, providing a visual representation of the 5th dimension. —The preceding unsigned comment was added by 64.149.188.82 (talk) 01:13, 10 May 2007 (UTC).

angelic conscioussness is not numeric —Preceding unsigned comment added by 77.165.5.122 (talk) 19:36, 28 February 2009 (UTC)

As far as spatial and temporal dimensions are concerned, the four are length, breadth, height, and time. The myth that there is a 5th seems to stem from the 60s pop group. I read here about some silly suggestion that the Star Wars 'Force' could be a dimension. That is not related to time (it does not interfere with it) or space (there are still 3 dimensions). These four dimensions can't be interfered with in any way in the real world, and they are all, by definition, infinite. Therefore, there is no 5th dimension. Gravity does not interfere with these dimensions, and it varies on location [the moon has less of a pull]. The physics view of a 5th dimension is complete myth.

In geometry, there are only 3 dimensions, except in the real world where the 4th may manipulate the other 3. The other 'dimensions' are just combinations of 2 of the 3 dimensions.

The article's neutral stance is unjustified because it has been acknowledged almost universally by the scientific community that there are only 4 dimensions.

4th dimension is spacetime, so it does exist and is its own dimension, the 5th dimension theory could be possible, as there may be infinite dimensions.

ex: 1st dimension a line 2nd dimension a line squared= a square 3rd dimension a square squared=cube 4th dimension cube squared= tesseract. possibility of a penatract. each dimension shows a side: the 1st is a line, so no sides, 2nd is a square showing 1 side, 3rd is a cube showing a full 6 sided view. 4th is spacetime and a tessaract showing many sides i havent counted yet. but the possibility of something past spacetime is low. there so far is a magnetic pole, a cosmic string, a domain wall, something about the 3rd dimension, and a spacetime tesseract. if there is a 5th dimension, it is impossible to find out what it is, except for the fact that it is a penatract24.189.153.102 (talk) 13:26, 8 June 2008 (UTC)

@24.189.153.102

You know what I think, the fifth dimension is probably a plane where objects travel far faster than light, which would mean that we are unable to see objects within it, even if it existed. I suspect if a type of "ghost" were real that it would live in that dimension. There are endless possibilities in this universe, you will never know all of what is out there. ~ali31 173.85.202.240 (talk) 18:43, 18 April 2011 (UTC)

Mathematicians and statisticians routinely work with Euclidean geometry in five or more dimensions. Michael Hardy (talk) 18:17, 18 May 2025 (UTC)

this section uses stuff like "we" and "lets" and "(you) think"

that makes it kind of personal and should be fixedSoyseñorsnibbles 01:52, 28 September 2007 (UTC)

Has a fifth dimension ever been actually mentioned in Smallville (TV series); can you name the episode? If not, I think it should not be listed here, even if some people speculate on the Mxyzptlk connection. 213.216.199.6 (talk) 13:15, 5 January 2008 (UTC)

:I find the whole popular culture section rather pointless really since only a few of the references really address the actual concept of a fifth dimension. Anything to do with comics books generally use the word "dimension" as a synonym to some alternate plane of existence and therefore irrelevant to the subject of the article. Takeshi357 (talk) 14:44, 1 February 2009 (UTC)

Kodjo and Togbey, (2000?), proposed that 4/5 is the exponent that represents allometric scaling of the brain. They reasoned that 5 is the natural extension of 1/2, 2/3, and 3/4 exponential (fractal) scaling laws of tissues. Since they found a close 4/5 scaling factor for neurons, and 5 is the denominator of that, they reasoned that the 5th dimension is that of thought. [Note: allometric scaling is non-linear variation of morphology based on mass or size, or other measurement of an organism or organ.] http://www.unomaha.edu/wwwmath/OurArchive/KerriganMinigrants/2006_2007/KodjoTogbeyReport.pdf for details. 74.195.25.78 (talk) 01:10, 8 January 2008 (UTC)

Any equation or set of equation whose exponents sum to 5 may be said to be 5-dimensional. Of course there are some sets of equations found in chemistry and mechanical engineering books that extend to 20 dimensions. Also, if E=mc^2 is correct, and c is not necessarily constant, then Energy (not the relationship between energy and what else) is five-dimensional. [c^2 would equal distance^2 / time^2; 2+2 = 4 (dimensions)] 74.195.25.78 (talk) 01:19, 8 January 2008 (UTC)

The word space has been relinked form vector space to Euclidean space.

This is a restriction, because there are also Hilbert-spaces and Riemannian manifolds.

As the following sentence says, the the 5th dimension will be the whole space,

but it should be a line of all locations, where the first 4 components remain constant and the fifth varies.

Suggestion: link space instead to euclidean space to Riemannian manifold etc. and reformulation of the definition, what a 5th dimension is.

Why has the reference to fourth dimension been removed ?

--17:25, 22 August 2008 (UTC)~ Turul2

:I know that Euclidean space is more restrictive, but this article mainly focuses on Euclidean space (e.g., 5-polytopes), so I thought it was more appropriate. The Euclidean space article is also easier for the general reader to grasp, whereas Vector space (the original link target) is much more abstract. From the POV of a casual reader, it would be quite difficult to understand how exactly the vector space article explains the word "space" in the intro.

:Also, I removed the reference to fourth dimension because it didn't seem to be directly pertinent to this article. Feel free to put it back if you feel otherwise.

:Anyway, I was just trying to make the article more readable to the casual reader. If you think that is detrimental, please feel free to rework it.—Tetracube (talk) 18:21, 22 August 2008 (UTC)

Out of curiosity: are there stellated shapes in 5 (and higher) dimensions - and what would they project as in our 3D world? Would they be 'chandelier-crystals-like' or 'coral-like'? Jackiespeel (talk) 15:39, 17 January 2013 (UTC)

:Yes, but none of them are regular. You can see cross-sections of some of them at [http://www.polytope.net/hedrondude/polytera.htm Jonathan Bowers' website]. Double sharp (talk) 15:45, 17 January 2013 (UTC)

::So would they be 5D versions of the images in crystal growth? How would 'polyhedra, simple or complex, stellated or otherwise, flexagons and other geometric objects' in general be seen in 'our' 3D world - and would there be 'seemingly disconnected objects that would move as a whole if you picked up one of them'? Jackiespeel (talk) 22:12, 20 January 2013 (UTC)

:::To understand how we see a 5D object psas through our 3D space, we need to do a little more dimensional analogy than we usually have to do for 4D objects. A 5D object forms a cross-section on a 4D plane if you stick the 4D plane through the 5D object. A 3D plane can be stuck through this 4D cross-section, creating a cross-section of a cross-section, otherwise known as a poke-section. (You could extend this down another dimension by sticking a 2D plane through this 3D poke-section, creating a jab-section, and so on until you hit the bottom.)

:::On your last question, yes there would – consider the outermost poke-sections of [http://www.polytope.net/hedrondude/polytera/QUITTIN.JPG the quasitruncated penteract]. (Poke-sections are cross-sections of cross-sections. A cross-section of a poke-section is called a jab-section.) You could pick up any one of the eight apparently disconnected shapes arranged like the vertices of a cube, and they would move together, for they are part of the same 5D object.

:::How each object will be seen in your 3D world depends a lot on the shape itself and at what angle you poke it into our 3D world. Double sharp (talk) 16:08, 10 April 2014 (UTC)

Putting it another way, as many of us cannot visualise 5D space:

The prongs of a 3D rake in Flatland would appear as a set of oblongs or circles (depending upon the intersection plane) which would move together 'for no apparent reason.' If it fell on its side it would 'somehow' become a single L shape.

How would the equivalent part of a 5D rake appear in our 3D space (one object or several, or 'depends upon the way it is positioned') - and what happens when 'it falls on its side' and several parts move out of our set of three dimensions? Jackiespeel (talk) 16:50, 21 January 2013 (UTC)

:It depends on what you think a 5D rake looks like. Since there is AFAIK no such thing, I do not really know what kind of object you are thinking of.

:What intrigues me is your use of Flatland as an analogy. Flatland is 2D, so you're going down one dimension, but your proposed 5D-to-3D goes down two dimensions. Perhaps it would be a better analogy to imagine a 3D rake passing through Lineland, or a 4D rake (tetraspace objects have been a lot more thought about because they're the simplest to visualize for all dimensions higher than 3!) passing through Flatland, if you want to understand how your proposed 5D rake would pass through our universe. Double sharp (talk) 16:08, 10 April 2014 (UTC)

::The 'issue' is the visualisations. We are all familiar with the 3D world and 'shadows, cross-sections and conic sections' - with rakes, doughnuts, stellations and other 'complex shapes' having different cross sections, possibly 'a group of linked but disconnected shapes' (rake, potato stamp, stellation of the stellation etc). That a 4D+ 'object' would have a 3D 'conic section' appears perfectly logical (even if it is difficult to visualize the object).

:::Indeed a 4D object would have a 3D cross section, but I don't see how a 5D object would have a 3D cross section. In the same way, a 2D object has a 1D cross section, but a 3D object doesn't have a 1D cross section. You can take the cross section of its 2D cross section, however, and that would be 1D. Similarly, an nD object has an (n − 1)D cross section. Double sharp (talk) 11:57, 11 April 2014 (UTC)

Another way - taking List of Wenninger polyhedron models - with 4D+ 'polyhedron of some form' or other objects would there be equivalents of the Fourth stellation of the cuboctahedron where the 3D 'facets/components' are likewise separate as we see them, but which would move as a single object? Jackiespeel (talk) 17:51, 10 April 2014 (UTC)

::Those facets are not disconnected as we see them, nor are they 2D. It is just that they intersect each other and pass through the inside of the polyhedron, so that you can only see the externally visible parts. As another example, in the small stellated dodecahedron, you can only see the spikes of each pentagrammic face, and the internal pentagonal areas of the stars are not visible from the outside.

::Nonetheless, if you mean a self-intersecting polytope (which would have areas of facets invisible to a viewer of the same dimensionality, because they are inside the polytope), quite a lot of the known nonconvex uniform polytopes (the gocco family seems a good candidate, starting og, gocco, gittith, ginnont, goxaxog, gososaz, gook, ganinov, godedak, gafefer, gizazac, etc.) Double sharp (talk) 12:03, 11 April 2014 (UTC)

:::I did ask the question a while back, and this is getting slighty outside my area of knowledge of the subject.

The question more generally is/was how would 'a five dimensional object appear in a particular 3D space as it moves through that space' (on a graph the x, y, z axes)? Going back to the 3D-2D analogue, the rake would present different shapes in Flatland as it rotates - would a 4D/5D/other D object do the same in our 3D universe?

Alternatively - consider the inhabitants of a '3D aquarium' in a 5D (v, w, x, y, z) world rolling 3D dice that can move in all 5 dimensions - would the dice at times appear to be only 'flat surfaces (eg lying along v, y, z axes) or lines (v, w, x) - and in a 6D universe would the dice sometimes 'disappear' entirely (being aligned u, v, w)? Jackiespeel (talk) 21:19, 11 April 2014 (UTC)

This section was removed, and I have no problem with the removal, except eventually someone will start adding things back. So any debate can go on talk, and this section below can be a references. Tom Ruen (talk) 22:09, 4 December 2014 (UTC)

{{hat|collapsed for readability}}

;In popular culture

In popular usage, the "fifth dimension" is often used to refer to unexplored or unknown aspects of the universe, and not necessarily to the mathematical concept of a 5-dimensional space.{{fact|date=November 2014}}

• For example, the opening narration of The Twilight Zone begins: "There is a fifth dimension, beyond that which is known to man."{{fact|date=November 2014}}
• In the fictional universe of DC Comics, the "fifth dimension" is said to be the place from which Mister Mxyzptlk, a Superman villain, comes.{{fact|date=November 2014}}
• The 1965 Lost in Space TV show episode "Invaders from the Fifth Dimension" features hostile aliens from the fifth dimension, and the Robot describes their spaceship by saying: "The craft is surrounded by a force field in the fifth dimension, which is... mathematically... impossible."{{fact|date=November 2014}}
• In 1966, The Byrds released an album titled Fifth Dimension, using the fifth dimension as a metaphor for unexplored and unknown aspects of the universe and oneself.{{fact|date=November 2014}}
• The 5th Dimension is the name of an American vocal music group popular in the late 1960s and early 1970s.{{fact|date=November 2014}}
• In Hindu philosophy, the fifth dimension of love of the Divine is termed by the Gaudiya Vaisnavas as turyatita, the dimension of the soul's Soul. The original Doctor Who episode hints at the 5th dimension being key to the abilities of the TARDIS.{{Citation needed|date=November 2009}}
• Other uses of the "fifth dimension" are closer to its mathematical meaning. For example, the novel The Boy Who Reversed Himself features 4-dimensional and 5-dimensional spaces, using the mathematical fact that a 3-dimensional object can be turned into its mirror image if additional spatial dimensions were available for it to rotate through. The characters in Madeleine L'Engle's novel A Wrinkle In Time use the fifth dimension{{citation needed|date=March 2010|reason=5th spatial dimension?}} as a "dimensional shortcut" to travel through space. A similar concept appears in the Powerpuff Girls episode Bring Back Jojo, where a creature able to see higher dimensions takes a dimensional shortcut through a fifth dimension to travel through time. The Red Dwarf episode "Parallel Universe" refers to the fifth dimension as the space in which multiple four-dimensional spaces exist.{{fact|date=November 2014}}
• Not all references to the "fifth dimension" in the mathematical sense involve time travel or space travel; Douglas Adams' book Mostly Harmless advances the idea of the fifth dimension being probability.{{fact|date=November 2014}}
• In Mikhail Bulgakov's The Master and Margarita, valet Koroviev in an explanation to Margarita attributes the expansion of a small apartment into the size of a large auditorium to the fifth dimension.{{fact|date=November 2014}}
• In the "original" or "boxed" version of the Dungeons & Dragons role-playing game rulebooks (discontinued in 2000), consisting of the Basic set, and then the Expert, Companion, Master, and Immortals expansion sets, a five-dimensional model was proposed in the Masters set where characters from the ordinary plane had the first, second, and third dimensions as their three-dimensional home, but other-dimensional beings called the third, fourth, and fifth dimensions home. Upon perceiving each other, each thought the other kind to be horribly deformed "demons".{{fact|date=November 2014}}
• In Family Guy, Mayor Adam West sends Alex Trebek, host of Jeopardy, to the fifth dimension by making him say his name backwards, commenting: "Only saying his name backwards can send him back to the fifth dimension where he belongs." This is a parody of Mr. Mxyzptlk's weakness in the Superman comics.{{fact|date=November 2014}}
• In a well-received 2010 CBS commercial for How I Met Your Mother, Accidentally on Purpose, Two and a Half Men, The Big Bang Theory, and the Late Show with David Letterman, an announcer states that "Everyone has 3-D, but only CBS has comedy in 5-D." This was a reference to the growing popularity of 4-D film in the later half of 2009, suggesting that certain programs on CBS feature 5-dimensional effects. By the end of the commercial, however, it is revealed that the letter D in 5-D does not stand for "Dimension," but stands for "Delightful, Delicious, Daring, Demented, and Dave."

{{hab}}

: Yes, masses of unsourced trivia and trivial uses. It probably could all be sourced to first party sources, to the relevant works of fiction etc. but it would still be trivia and excessive. If anyone can find any coverage in reliable secondary sources then it could be restored based on them but I don't hold out much hope, I certainly don't know of any sources or where to start.--JohnBlackburne^words_deeds 23:08, 4 December 2014 (UTC)

: See also #Smallville above; the reply I think makes another important point. Even when "the fifth dimension" comes up in popular culture it's very often nothing to do with this topic. It's often just shorthand for a mystical thing outside our space, and is utterly meaningless - it might as well be "hyperspace", "limbo". "the fifth dimension" sounds more formal and fits nicely after the four dimensions of space and time. There are exceptions in the above list but I suspect even they are largely using it as a buzz-word, not really considering a five-dimensional space. I doubt e.g. the the Powerpuff Girls episode Bring Back Jojo is a good example of a near mathematical use. But that's why we need reliable sources other than the works of fiction, TV programmes etc..--JohnBlackburne^words_deeds 02:32, 5 December 2014 (UTC)

::Agree that some of it is trivia or relates to a concept of the fifth dimension that is not accurate, but a label for a deus ex machina. However, some of the references seem to relate to the concept in physics. A Wrinkle in Time in particular stands out. Regardless, agree that it needs to be referenced by non primary sources. Plumpy Humperdinkle (talk) 00:42, 13 September 2015 (UTC)

The section on Hooft needs expansion with references. Searching peer reviewed literature has not resulted in much substantiation. Requesting help modifying/referencing this section. Plumpy Humperdinkle (talk) 17:16, 12 September 2015 (UTC)

Hello fellow Wikipedian. I am in the process of creating an article about the 4D shape, the “cubinder”. It was previously red linked on other articles (including the one you created) and I was surprised to see it was not already an item listed for creation by Wiki Projects Mathematics, as the duocylinder and spheriender are already articles. I require help to improve the draft, as I require more formulae, sources, and additional information to create this article. You can access this page at User:Darnburn98/Cubinder, please come on over and help improve this article to get into the main space! Darnburn98 (talk) 23:33, 20 January 2017 (UTC)

:{{tq|Fifth dimensional geometry is generally represented using 5 coordinate values (x,y,z,w,v), where moving along the v axis involves moving between different hyper-volumes.}} [ref omitted]

This strikes me as redundant and, in part, too narrow; moving along any line involves moving between different hyper-volumes. Your thoughts? —Tamfang (talk) 01:18, 30 April 2024 (UTC)

:In general the article focuses way too much on a particular coordinate system. It should be re-written from a more coordinate-free perspective. (Also the link redirects to Lebesgue measure, which has no immediate connection to the sentence.) --Trovatore (talk) 01:53, 30 April 2024 (UTC)

There is an obvious conflict in defining the "5th dimension" as "Five-dimensional space" - because 3D is space (spatial) the fourth dimension time to locate the fifth dimension as a word - in a simple consideration not 3D - quantum space is the only other possible spatial aspect (in the sense of ordinary perception). Since 5D would be based upon 4D i.e. spacetime - this would indicate 5 would have to have a time based aspect as spacetime is a fused reality. {{quote|for Kaluza, the introduction of the fifth dimension is legitimate as long as physical quantities do not depend appreciably on the fifth dimension. This, according to Kaluza, follows from the fact that neither the fifth dimension nor any of its effects whatsoever are physically perceivable.|Koray Karaca: 3 A Close Look at the Kaluza–Klein Theory www.journals.uchicago.edu/doi/10.1093/bjps/axr033}}

Searching "Five-dimensional space" scholar.google.com/scholar?start=10&q=Five-dimensional+space&hl=en&as_sdt=0,5 - has a var. of associated terms with 5D: "superspaces" (Kuzenko Linch III 2006), "gravity" (Wesson 2015), relativity theory (Leibowitz Rosen 1973), space-times (Halpern 1986). iask.ai/q/fifth-dimension-spatial-or-non-determined-5lp8u20: "the nature of the fifth dimension is subject to various theoretical interpretations in physics and mathematics. While some conceptualizations treat it as spatial, others propose different natures, indicating that its nature is not definitively determined to be solely spatial." "Recent research also suggests alternative interpretations, such as the space-time-matter approach and the spacekime representation, which uses complex-time instead of ordinary time" from which I found: socr.umich.edu/spacekime - so this indicates "space" is not the necessary / necessarily the determinant. (𒌋*𓆏)𓆭 10:53, 11 May 2025 (UTC)

:This article is about five-dimensional mathematical spaces, not about some particularly identified "fifth" dimension. I am going to revert your changes. The page move should also be reverted; I am going to tag the redirect for speedy so it can be done. Ah, no need to tag; turns out my move worked after all. I saw an error message which was why I thought it hadn't. --Trovatore (talk) 16:45, 11 May 2025 (UTC)

::The art. isn't about "five-dimensional mathematical spaces" since the subject physics is included. I indicated how 5D also inc. not space - "spacekime" for example. Since this example - and the others I included don't indicate space - why do you think 5D has to be + space. (𒌋*𓆏)𓆭 17:01, 11 May 2025 (UTC)

::I mean - I see "Physics" in the 1st sub. you see that? I'm sure you can see the same as me - except you state the art. is about "mathematical space". Let's just try and solve this problem: you see the words in the art. are they maths or words? As one example of how the art. isn't about maths i.e. the discussion is in words not numbers maths formula. 2nd any mention of physics indic. somthing which isn't maths (I'm reiterating here I know) - is this understandable? I thought it is obvs. but apparently I need to have some type of discussion. (𒌋*𓆏)𓆭 17:24, 11 May 2025 (UTC)

:::In my view this is primarily a mathematics article. We do also cover possible physical interpretations, but it's basically a geometry/topology article. --Trovatore (talk) 17:52, 11 May 2025 (UTC)

::::"primarily a mathematics article" is in contradiction to the existing source [1] "Engineers will marvel, no doubt, at its gleaming mechanisms, while mathematicians will be awestruck by the sheer quantity of its collected data and the powerful algorithms sifting through it. And physicists will wait eagerly for possibly the first evidence of a higher-dimensional realm beyond space and time." is 3 disciplines in addition - beyond space and time . Paul S Wesson (University of Waterloo, Canada & Stanford University, USA):[https://worldscientific.com/worldscibooks/10.1142/6029#t=aboutBook extra dimensions — beyond space and time] (𒌋*𓆏)𓆭 17:59, 11 May 2025 (UTC)

::::User:Onemillionthtree/sandbox#"Space"_determination - is a survey of source 1. (𒌋*𓆏)𓆭 18:05, 11 May 2025 (UTC)

::::Under the heading "Physics" which I showed above obvs. isn't maths. "Recent research suggests several alternative interpretations of the 5D extension of spacetime" is the beginning of the last para. that section (and this is obvs. c.f. "Physics") doesn't conclude that 5D is a mathematical construct (a pure mathematics / only a maths. subject). (𒌋*𓆏)𓆭 18:09, 11 May 2025 (UTC)

::::"In my view" has no value in this discussion - unless you are the named author of a paper to show your opinion could be used to prove/show/indicate your opinion is the same thing as the factual reality of the subject. (𒌋*𓆏)𓆭 18:11, 11 May 2025 (UTC) Do you have any sources which support your opinion? (𒌋*𓆏)𓆭 18:14, 11 May 2025 (UTC)

::::I'll do some research survey on sources in my sandbox so we can look at the evidence. (𒌋*𓆏)𓆭 18:15, 11 May 2025 (UTC)

::::: There is no "reality of the subject"; what we're arguing about is how to organize and name the material. As I see it the stuff on Kaluza–Klein and extensions thereof should be covered primarily at Kaluza–Klein theory. That is not about five-dimensional space in general, but about a particular physical theory that involves a five-dimensional space. --Trovatore (talk) 19:04, 11 May 2025 (UTC)

I have notified both the math and physics WikiProjects about this discussion; let's try to talk it out calmly. One possibility, I suppose, is that there really ought to be two articles, one on the mathematical notion and one on the physical notion. I am dubious of this mainly because I am not convinced there is any such thing as "the" physical notion, unless it's specifically Kaluza–Klein, in which case I think that should be treated at Kaluza–Klein theory. --19:34, 11 May 2025 (UTC)

: Oof. The section {{slink|Five-dimensional space#Physics}} comes across as an essay, mainly tangentially about Kalusa–Klein theory, in the style of science journalism. It does not particularly seem to be about the title topic, just on themes related to it. —Quondum 20:35, 11 May 2025 (UTC)

:The question that is unclear to me, though, is whether there is much to say about the mathematical 5-dimensional spaces (I guess $\backslash mathbb\{R\}^5$ although $\backslash mathbb\{C\}^5$ is also a possibility) that is not merely a repetition of the things one would say about higher-dimensional spaces more generally. If so, it isn't represented well in this article. If not, maybe we don't need an article on this topic. —David Eppstein (talk) 20:39, 11 May 2025 (UTC)

::That is sort of a fair point, though I would call $\backslash mathbb\{C\}^5$ ten-dimensional rather than five-dimensional, and more generally would take the article to be about spaces of topological dimension 5. I don't know much that is particularly special about that exact collection of spaces. --Trovatore (talk) 20:54, 11 May 2025 (UTC)

::: Technically, one cannot meaningfully assign a dimension to a space without saying over which field (or whatever) it has that number of dimensions. Although I suppose this point is not of any significance in the debate about whether we need an article on spaces of five dimensions specifically. I agree entirely with David Eppstein on this. —Quondum 21:24, 11 May 2025 (UTC)

:::: To the first sentence, no, not true. See Lebesgue covering dimension for example. --Trovatore (talk) 21:36, 11 May 2025 (UTC)

::::: I'm somehow not wrapping my brain around that. You did qualify this as for topological dimension, but I hardly see why that should be assumed. In geometry (and manifolds generally), there should be no assumption that the space has a topology. —Quondum 23:25, 11 May 2025 (UTC)

:::::: Hmm? Geometry is extra structure on top of the topology. "Geometry" usually means you have a metric, so the metric topology is understood. --Trovatore (talk) 23:30, 11 May 2025 (UTC) Given the context of this discussion I suppose I should clarify that I'm not talking about metrics in the sense of pseudo-Riemannian geometry, but about genuine distance functions satisfying the triangle inequality and distinguishing distinct points. For a pseudo-Riemannian manifold, I'm not sure whether you always have a distinguished metric in the sense I mean, but in any case you have the atlas witnessing that it is a manifold. Usually we define atlases on top of a topological space to begin with, but if you didn't, you could still recover the topology from the atlas, so we still have a clear topology. --Trovatore (talk) 23:38, 11 May 2025 (UTC)

::::::: Finite geometries are over a finite fields. It would not seem to me that topologies or metrics apply there. But you still have well-defined dimension, points, lines, planes, etc. —Quondum 23:46, 11 May 2025 (UTC)

:::::::: Oh, yeah, there's always some corner case, isn't there? When I think of geometry I really don't consider finite geometry. But I suppose you would use the discrete topology, which is zero-dimensional. --Trovatore (talk) 23:50, 11 May 2025 (UTC)

::::::::: Yeah. Anyhow, I would think that it is evident that one should specify the type of dimension, rather than assuming that the reader will infer this from the title (or even understand that there are different kinds). —Quondum 00:01, 12 May 2025 (UTC)

:::::::::: I guess I feel like, in most areas of mathematical analysis, a topology is pretty much the minimal structure you can have in something called a "space", so dimension means topological dimension by default. It might be different in algebra. --Trovatore (talk) 00:44, 12 May 2025 (UTC)

::::::::::: Or in geometry, which seems relevant here, so I'm not persuaded. In some areas of geometry (primarily projective geometry, I imagine), the concept of "geometric dimension" is very much of interest (in theorems), and the topological dimension has no real use. (Disclaimer: I'm not mathematically trained and in principle I could be speaking out of turn.) —Quondum 01:25, 12 May 2025 (UTC)

:::::::::::: I think in any ordinary geometric context (not counting finite geometry), geometric dimension and topological dimension are going to coincide. But I'm not up-to-date on some of the things they're calling geometry these days. --Trovatore (talk) 01:37, 12 May 2025 (UTC)

::::::::::::: Which, I think, is a good note to end this on. I've learned about topological dimension, at least. —Quondum 12:58, 12 May 2025 (UTC)

::Agreed. My first intuition here is that this could basically be merged into something about higher-dimensional spaces (including probably most of the higher-dimensional physics). Sesquilinear (talk) 02:47, 12 May 2025 (UTC)

:The entire "Physics" section is a confused mess. It's what you'd get by throwing together half-understood remarks about the original Kaluza--Klein theory along with secondhand knowledge of other theories (which are ten- or eleven- or twenty-six dimensional), and obscure proposals on the boundaries of the physics literature that nobody cares about. Kaluza--Klein theory isn't even about five-dimensional space! It uses a spacetime with four space dimensions and one time dimension. Minkowskian geometry is not Euclidean!

:I picked one sentence at random: "'T Hooft has speculated that the fifth dimension is really the "spacetime fabric"." It has two citations [https://web.archive.org/web/20220118151328/https://www.scientificamerican.com/article/what-is-spacetime-really-made-of/][https://www.wired.com/2015/05/spooky-quantum-action-might-hold-universe-together/]. Neither of them say that 't Hooft said that. The first one doesn't even use the word "fabric". The second mentions 't Hooft only in regard to a different topic. In short, delete it all. 64.112.179.236 (talk) 00:02, 12 May 2025 (UTC)

:: I think you're likely justified in removing the section. I would point out though that four space dimensions and one time dimension would be a five-dimensional space. --Trovatore (talk) 00:15, 12 May 2025 (UTC)

:::: I just noticed that my second sentence uses the word "space" in two completely different senses. The first instance is using "space" in the physical sense (similar to spacelike) whereas the second is in the mathematical sense (which for me almost always means "topological space possibly with extra structure"). {{u|64.112.179.236}}, is that what you were getting at when you said KK theory wasn't about five-dimensional space? I thought it was clear enough that that's not the way "space" is being used here. --Trovatore (talk) 03:16, 12 May 2025 (UTC)

::: I concur. I think it would help if we had a rule that reporting (news, popsci, and suchlike) is disqualified entirely as a source for technical topics, and may only be used for colour/explanation. —Quondum 00:19, 12 May 2025 (UTC)

::::OK, I cut the "Physics" section and then added a brief paragraph setting Kaluza--Klein in context. 64.112.179.236 (talk) 00:43, 12 May 2025 (UTC)

:

:The popular term {{qi|The 4th dimension}} is essentially meaningless. In a Euclidean space you can pick whatever set of independent coordinate axes you please. Prior to Einstein's theories of relativity, it made sense to refer to time as the fourth dimension because the direction was unambiguous, but after special relativity there were infinitely many equally valid choices. Adding another dimension doesn't change that; you have a free choice of coordinate systems.

:Further, five dimensional spaces in mathematics need not be tied to physics, and even for physical applications may be used for something other than space or space-time. For example, neither the Lie group $\backslash mathrm\{SU\}(5)$, nor the Lie algebra $\backslash mathfrak\{su\}(5)$, represents space-time in $\backslash mathrm\{SU\}(5)$ field theories. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:42, 12 May 2025 (UTC)

"Fifth dimensional geometry is generally represented using 5 coordinate values (x,y,z,w,v), where moving along the v axis involves moving between different hyper-volumes." e.math.cornell.edu/people/Morgan_Weiler/4540/SB.pdf (𒌋*𓆏)𓆭 12:37, 11 May 2025 (UTC)

In mathematics, a sequence of N numbers can represent a location in an N-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. (𒌋*𓆏)𓆭 17:44, 11 May 2025 (UTC)

:Not quite. In both special and general relativity there are spacelike and timelike directions but there is not a unique time dimension or unique space axes. Worse, there are theories with nonstaandard signatures, e.g., {{brace|+,+,+,-,-}}. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:22, 12 May 2025 (UTC)

::

:: I don't follow what point is being made in either of the two posts here. The quoted sentence is overly simplistic to characterize geometries of five dimensions ("generally represented" is off the mark), but it does not claim any uniqueness of coordinates either. —Quondum 14:00, 12 May 2025 (UTC)

:::The basic point I was making is that the concepts don't depend on a special coordinate system, so the number of dimensions is fixed but the choice of what directions to call dimensions is not. A second point made elsewhere is that even in physics a 5-space may be something totally different from space-time, e.g., $\backslash mathfrak\{su\}\backslash mathrm(5)$. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:32, 12 May 2025 (UTC)

I removed the subsection on the 4-sphere because it's not a five-dimensional space, similar to the way a circle is 1-dimensional, not 2-dimensional, and a sphere is 2-dimensional, not 3-dimensional. We could treat the 5-sphere if we wanted. --Trovatore (talk) 04:47, 12 May 2025 (UTC)

:For consistency, you'd also have to remove Four-dimensional_space#Hypersphere, which discusses the 3-sphere. Instead, you might consider restoring the content removed here and rewrite it slightly to emphasize a five-dimensional space contains a 4-sphere, which is perfectly fine. This is normally done do in lower dimensions, with the circle discussed in the context of planes and spheres in 3-dimensional spaces. PS: I've just edited n-sphere and sphere for clarity. fgnievinski (talk) 13:18, 12 May 2025 (UTC)

::I don't think we should focus on the dimensionality of the embedding space. Spheres, for example, can be defined completely intrinsically as 2-dimensional manifolds, with no need to embed them in a 3-dimensional space. --Trovatore (talk)

:I don't understand this. Surely the point was to talk about the 4-sphere because it's an object that exists within $\backslash mathbb\{R\}^5$, and thus explain something about what kind of a place $\backslash mathbb\{R\}^5$ is. Otherwise, we'd have to remove the stuff about polytopes, too, because their surfaces are lower-dimensional. 64.112.179.236 (talk) 15:46, 12 May 2025 (UTC)

::The article is about five-dimensional spaces, and $S^4$ is not a five-dimensional space. The article is not specifically about $\backslash mathbb\{R\}^5$. --Trovatore (talk) 16:43, 12 May 2025 (UTC)

:::But E5 is the majority of the article, with a minor section on Five-dimensional_space#Other_five-dimensional_geometries. And can you address the comment above about 5-polytopes being bounded by 4-polytope facets -- are they going to be removed here, too? fgnievinski (talk) 13:31, 13 May 2025 (UTC)

:::Furthermore, the 5-ball bounded by a 4-sphere is a perfectly legitimate topic for discussion in the context of 5-space. So by just renaming the section, from Hypersphere to "Hyperball", it'd allow retaining a mention of the 4-sphere. fgnievinski (talk) 14:41, 13 May 2025 (UTC)

:::: I higher-level question to be resolved before such considerations is the topic of the article. If the topic is geometric objects in 5-dimensional Euclidean space, the title is wildly off (and it should be merged with the many other existing articles on that topic). It is about general is general 5-dimensional spaces, then it discussing specific shapes in Euclidean 5-space is just weird (and it should probably be merged with other articles on that topic. Either way, I do not see much reason for this article to exist spearately. —Quondum 15:28, 13 May 2025 (UTC)

:: I think we have a general problem of organization of content in Wikipedia. If we think that an article such as this should mention every conceivable object that exists in five dimensions, this article (and articles generally) would be unmanageable. Material on 5-polytopes belongs in that article, with not more than a link to that article in the see-also section, and should not be copied here (i.e., I agree with removal). —Quondum 16:47, 12 May 2025 (UTC)

:::Alternatively we might merge both 5-polytope and 5-manifold here — I think the article would still be of manageable size. --Trovatore (talk) 16:52, 12 May 2025 (UTC)

:::: That would be madness. What about Regular 5-polytope, Uniform 5-polytope, and who knows how many others (which, incidentally, might be better merge targets). This would also, as I was saying, be a major problem of how information is managed in WP. We already have inclusion of information by editors that is far too specific for a general article. Heck, polytopes are a specialized part of Euclidian geometry, not of interest at the level of general 5-dimensional spaces, which need not even be Euclidean. 5-manifold does warrant mention here, but it, too, falls under manifolds rather than immediately under 5-dimensional space. —Quondum 17:06, 12 May 2025 (UTC)

::::: I mean, in my view, an article called "five-dimensional space" should be about five-dimensional spaces, which are basically the same thing as 5-manifolds, with minor quibbles. --Trovatore (talk) 17:07, 12 May 2025 (UTC)

:::::: Interpreted broadly, any set that may be parameterized by 5 parameters is a 5-manifold. More narrowly, we might mean something more specifically geometric (a topological manifold). But here, we are already running into a bifurcation of definition (which might use different definitions of dimension, e.g.Hamel vs. Lebesgue). I would consider a merge to 5-manifold more appropriate, if we were to choose that topic as the one being covered. —Quondum 17:24, 12 May 2025 (UTC)

::::::: Any set that may be locally parameterized by five parameters. The "locally" is important. A five-dimensional space doesn't have to involve any particular global way of parameterizing by five parameters. That's a key point to keep in mind in the intro of this article. --Trovatore (talk) 17:44, 12 May 2025 (UTC)

::::::::Typically there are no global coordinate systems to choose from. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:01, 13 May 2025 (UTC)

::::::: "Hamel dimension" seems to apply to vector spaces, not geometric spaces. Along those lines, the finite-geometry stuff doesn't really look like what I would call geometry; it seems more like linear algebra. But it is true that it's called geometry, and I suppose it's conceivable that someone could get here looking for five-dimensional vector spaces, though I don't know why they would be of particular interest. --Trovatore (talk) 19:14, 12 May 2025 (UTC)

:::::::: You really have a topology-centric view, don't you? Felix Klein might have felt insulted by your characterization. Are you familiar with the idea of defining geometry through axioms? I expect that this played a far larger role in the history of geometry than topology did. —Quondum 19:25, 12 May 2025 (UTC)

::::::::: Hmm. When I think of geometry (in the modern sense) I think of Riemann more than Euclid, I suppose. Is that topology-centric? Riemannian geometry is not just topology, whereas Klein apparently did a lot in topology. It would be fair to say I have a structure-centric (rather than axiom-centric) view. --Trovatore (talk) 23:51, 12 May 2025 (UTC)

:::::::::: There is the whole area of groups actions on spaces, pioneered by Klein. This is not axiomatic per se, but is built on structure (the axiomatic approach does not appeal to me), and covers an incredible array of homogeneous geometries; it does have a strong correspondence to the axiomatic incidence-based geometries, though. There are many theorems in this class of geometry that work very naturally. In this setting, it makes little difference whether a geometry is finite or continuous: the theorems of dualism, incidence, etc. don't care particularly. These theorems typically refer to what is effectively the Hamel dimension. It includes classical geometric theorems, and covers affine, finite, projective, Euclidean, elliptic, hyperbolic, Minkowski, de Sitter, anti-de Sitter, ... . It is often concerned with the transformations of the space that preserve properties (e.g. the Euclidean group, Poincaré group, etc.). I imagine that much of interest here would be deliberately ignored in a topological approach and would more difficult to derive. My point is simply that geometry, as a discipline or field of study, should not be presumed to be subsumed in the "modern" topological approach. The topologies tend to be simple. —Quondum 02:16, 13 May 2025 (UTC)

::::::::::: Well, but per that article, the Erlangen program is primarily concerned with homogeneous spaces. That doesn't have much to do with this article. There is nothing in the phrase "five-dimensional space" that suggests even remotely that the space should be homogeneous, which is a pretty severe restriction. --Trovatore (talk) 19:47, 13 May 2025 (UTC)

:::::::::::: I feel that I am unable to communicate the logic of my point. So maybe I should leave it be. —Quondum 20:01, 13 May 2025 (UTC)

:::::::::Surely J. Random Topologist is familiar with the Erlangen program . I would say that the boundaries between disciplines have been fuzzy for a century, and that anybody serious about topology will be using, e.g., homological algebra. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:01, 13 May 2025 (UTC)

::::::::How would you classify Euler's analysis of the Seven Bridges of Königsberg? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:01, 13 May 2025 (UTC)

::::::::: Graph theory. --Trovatore (talk) 19:51, 13 May 2025 (UTC)

:::::::Only if the image of each chart is open. A 5-manifold with boundary, or, more generally, a 5-orbifold, may be locally parametrized by five parameters. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:01, 13 May 2025 (UTC)