Talk:Four-dimensional space/Archive 1

Which dimension is time ? (open)
There should be a mention that the fourth dimension is also commonly used to refer to time. 86.135.161.60 16:57, 15 December 2005 (UTC)

The disambiguation page has a link to Spacetime, which is the correct name for what you are referring to. Saying that the fourth dimension is time is misleading. This is because we know that there are at least three spacial dimensions and one temporal. If there was actually four spacial dimensions then that would make time the fifth dimension. Having said that, people do call it the fourth, so if someone puts a comment about that can they include that it is a misleading way of naming it? Thanks. --Aceizace 18:27, 19 February 2006 (UTC)

Time is the way we percive distance in the fourth dimension. Time is not a dimension of space, so time by itself is not a dimension. Time multiplied by the speed of light is the distance through which the observer has moved in the fourth dimension. There is no mention of a fifth dimension in relativity, so references to it ought to be removed.FVP 07:43, 10 September 2006 (UTC)


 * This article is not talking about 4-dimensional space-time as defined in relativity. It is talking about 4-dimensional Euclidean space. Time, if one wishes to consider it in this context, could be regarded as a "fifth dimension". This has nothing to do with whether such a thing exists in Einstein's theory.&mdash;Tetracube 15:05, 19 October 2006 (UTC)

There is no common definition, which physical dimension time is. It could be zeroth, fourth or fifth, or the last of ten or eleven. The author of this article insists not to define it as the fourth. &mdash;Turul2 20:34, 04 December 2008 (UTC)

SPOILER! (fixed, marked for delete, expiration 30 June 2009)
I feel The Star Ocean reference should be removed as it is a spoiler and the basis for a BIG plot twist in the game. &mdash;The preceding unsigned comment was added by 80.213.106.141 (talk &bull; contribs) 13:06, 16 January 2006 UTC.

Reality and the fourth dimension
I think this section needs cleaning up. There are style issues such as the need to enter the mathematical forumlae with the wikipedia provided function, but also the way it is written is very hard to follow. I am unsure whether the author is trying to prove the existance of the fourth dimension, or say there can only be four (ie no fifth or higher spacial dimensions). --Aceizace 18:27, 19 February 2006 (UTC)

Although I am not good enough at mathematics/physics to say for certain whether the following is right or not, it sounds to me very suspicous. It seems to be either original research, or nonsense, or if its neither its not explained very well. In particular, the idea that you can prove a physical property of our universe based on mathematical theorems? Talk about a leap from the a priori to the a posteriori!
 * The fact that our universe can only be described by four dimensions is clear from the Pythagoras theorem and vector multiplication. The proof is as follows: The rule for multiplying two vectors is to multiply their magnitudes and add their angles. [1] Pythagoras' theorem for a triangle with one 90 degree angle is h^2=x^2+y^2 where h is the magnitude of the hypotenuse (opposite the right angle) and x and y are the magnetudes of the other two sides.
 * But y can be expressed as a length in the x direction multiplied by 'j' where 'j' is unit length in the x direction with the 90 degree angle that rotates the x direction into the y direction. But h^2=(x1)^2+(jx2)^2 is not correct as j^2=-1 so we must consider that both the x and the y directions have been rotated from the z direction. Now h^2=(iz1)^2+(jz2)^2 =-(z1^2+z2^2) so the squares of the two vectors add. But the Pythagoras theorem works in three dimensions so a fourth dimension must be present which is at right angles to the x and y and z directions.

I will remove this. If someone with a maths/physics background confirms there is truth in this, they can re-add it (and please clarify it at the same time.) --SJK 12:03, 23 February 2006 (UTC)

This article is the result of original research carried out by the author. It forms the basis for understanding the how the Universe works. What mass is and how mass inevitably generates gravitational, inertial and electromagnetic interactions and many more. As these insight have not been recognised what is said must be different and therefor strange. Please bear with me, am doing my best to make it clear, many of the concepts are strange and have taken me many years to appreciate.

I have just found the Editing Talk pages and am greatful for the feed back that helps me to understand what problems that have been encountered with what I have written. I did not know about wikipedia mathematical functions. As a result of your comment I have found that they exist and where they are. Thank you for telling me..

This artical proves the nature of the fourth dimension as follows. All four orthogonal dimensions are identical. The Universe exists as a mathematical surface made of nothing expanding at the speed of light away from its point of origin. The fourth dimension is the direction or dimension that the observer is moving. The Pythagoras theorem proof resolves the logical contradiction between vector multiplication and the Pythagoras theorem and proves that there must be an invisible fourth dimension of space. The Lorentz equation shows that the fourth dimension is a surface moving at the speed of light, which explains why we can not see the fourth dimension with light. Relative velocity means that the observer and the observed object are moving at the speed of light in different directions.

Higher dimensions do not form part of this artical on the fourth dimension. The possibility of higher dimensions is not ruled out but I have not found any reason to postulate their existence as every physical observation that I have considered is explicable to me with the four dimensions that are considered here.

Regarding a leap from a priori to a posteriori I am not sure what is meant here.

A priori means proceeding from from causes to effects, or logically independent of experience or not submitted to critical investigation. Well I agree that the fourth dimension must be inferred from the experience and logical deduction from the other three dimensions we are therefor proceding from the effects to infer the cause so that is not a priori. The fourth dimension is not logically independant of experience, so that is not 'a priori'. The fact that I am presenting this to the world is submitting it to your critical investigation so that is not 'a priori'. However I will grant you that it has not been subject to critical acceptance by the world as, apart from giving some lectures and discussing it with friends, I have not found a way to publish it.

A posteriori means inductive, empirical, moving from effects to causes, prior knowledge being used to deduce what comes after. This seems to me to be what I am doing here. FVP 00:29, 27 February 2006 (UTC)

I see that despite my editing efforts this entry has been deleted and rewritten by someone introducing the dreaded word space-time. Interestingly enough the artical does not mention time in connection with space time. However the connection between Pythagoras and the Lorentz equation has been preserved even if its derivation has been removed. Great chaps lets just say it works no need to have all that tedious derivation logic who bothers with derivation anyway.

I'm also very skeptical about proving existence of fourth dimension through maths. For example some real-life engineering problems can only be solved using i (the square root of minus 1) yet nothing measuring i cm will ever be found to exist in reality. I didn't have enough maths to understand the vector proof but it seems possible that all that is being proved is the need for a 'mathematical' fourth dimension as opposed to a real one. Also, don't really understand dimensional analogy. For example if you shine a light on a 3D object you get a 2D image. If you shone a light at a 4D object surely you wouldn't get a 3D object? MikeyMike


 * If you shone (4D) light at a 4D object, you will get a 3D shadow, just like how shining (3D) light on a 3D object makes a 2D shadow. That's what the dimensional analogy section is driving at. Of course, this will take a while to grasp; understanding 4D is not simple.&mdash;Tetracube 17:56, 31 August 2006 (UTC)

I hope this new entry is acceptable. I need to show that the fourth dimension is a reality rather than a mathematical abstraction. 'Spacetime' is of course only space. The fourth dimension of space is measured in metres defined by seconds multiplied by the speed of light.

Shadows of Ourselves
This might have been mentioned before, but I read that 3d objects cast 2d shadows. So, what if our mind where we think and compute our thoughts is indeed 4d so the bodies that are cast are our 3d forms. Even more so, the prospect of ghosts could be defined as a 5d figure with a 4d shadow, which shadow is 3d. Only something to wonder about.--Dige 00:43, 22 June 2006 (UTC)

Directions
The directions up and down are based on gravity; north, south, east, and west are based on the orientation of the Earth; and foward, backward, left, and right are based on our own bodies. What are these other directions like ana and kata based on? —Keenan Pepper 05:52, 4 September 2006 (UTC)


 * A (hypothetical) 4-dimensional being's body. Or the orientation of a 4-dimensional planet, as the case may be. The idea is that ana and kata are simultaneously perpendicular to all of north, south, east, west, up, and down. Of course, this is only possible in 4-dimensional space. Note that there is no established consensus whether ana and kata are absolute terms (based on 4-dimensional planetary orientation) or relative terms (based on the orientation of a 4-dimensional being's body). Some authors use it one way, others use it another way. Also, some authors use up and down as horizontal directions when in 4D (i.e., perpendicular to 4D gravity), and ana and kata as the 4D equivalents of up and down (colinear with 4D gravity). The common factor is simply that ana and kata refers to the extra pair of directions available when in 4D space.&mdash;Tetracube 16:40, 4 September 2006 (UTC)

Writing style
I'm sorry, but this reads like a textbook, especially when words like "we" are used. Is there any way of cleaning it up (or simplifying it to make it more readable) without losing important information? SKS2K6 09:21, 19 October 2006 (UTC)
 * Go ahead and fix the "we" references. That should give us a start. As for the accessibility of the article itself, I think it is in need of a major reorg, but I'm not sure how to go about it.&mdash;Tetracube 15:07, 19 October 2006 (UTC)

My head hurts!
Is a 2D representation of a 4D object even possible? It seems too far removed in dimensions. Then, can a 4D object be efficently depicted using 3D paper? Also, how come the 4D cubes are only cubes joined on their sides, not their top and bottom faces? Is it just a coincidence that the net of a tesseract is composed of 6 cubes just as the net of a cube is composed of 6 square? The net of a square is composed of only 4 lines, and the net of a line is an infinite number of points, no? Aaadddaaammm 09:09, 20 October 2006 (UTC) PS. No idea what I'm talking about.


 * The net of a tesseract has 8 cubes. --WikiSlasher 09:32, 23 October 2006 (UTC)


 * Well, a 2D representation of a 4D object is of course possible, and so is a 2D representation of an object of any larger dimension. A better question would be, is a 2D representation of a 4D object sufficient to convey the geometry of the 4D object? As you note, too much information is lost in going from 4D to 2D, which is why many Java applets you see out there trying to draw 4D objects with lines end up showing an incomprehensible tangle of lines which is very hard to understand.
 * As for whether a 4D object can be adequately represented on "3D paper", the answer is yes!, because our own eyes only see in 2D, but we have no problem perceiving 3D depth from the images. A hypothetical 4D being with eyes similar to ours would have a 3D retina, and thereby perceive 4D objects from 3D images. Therefore, projecting 4D objects to 3D is a good way to visualize them.
 * However, to fully appreciate such projections, we'd need to be able to see every point of a 3D volume simultaneously, which is impossible for us because our eyes only see in 2D. Therefore, some simplification is needed. Usually, this is done by drawing only the faces or edges of the 4D object, leaving plenty of blank spaces for our 2D eyes to be able to see the internal structure of the 3D projection image (we are really projecting from 4D to 3D, and then from 3D to 2D). The animation of the 4D hypercube recently added to this article is a good example of this: it "fattens" the vertices and edges of the projected image so that when rendered on the 2D screen, we properly perceive the 3D depth of the image. (Otherwise, our eyes will get very confused by optical illusions caused by ambiguity in 3D depth.)
 * Of course, to understand what we're looking at is another matter altogether. For this we need to use dimensional analogy, which is briefly discussed in the section with that heading.
 * And yes, don't be surprised that your head hurts trying to grasp this. :-) Nobody said visualizing 4D was easy. In fact, many mathematicians still have trouble with the concept, even if they can mathematically manipulate these objects.&mdash;Tetracube 00:21, 2 November 2006 (UTC)


 * It helps a lot if you draw out a few different perspectives of a hypercube by manipulating the perspectives of the individual cubes forming it's sides. Thats how I got my first glipse of a hypercube.--Scorpion451 04:17, 15 July 2007 (UTC)

Simpler?
Can this article be made to be understandable by us less-smart people who are just curious? Or is the concept of the fourth dimension just an incredibly complex topic? Dylanga 01:52, 15 December 2006 (UTC)

I also want a simpler explanation :S--71.62.178.53 05:43, 7 January 2007 (UTC)


 * I think it's difficult to avoid the complexity. It's a simple concept, just a natural extension of the sequence (1, 2, 3... ) of dimensions. But it is very difficult for us three dimensional creatures to visualise. You can use mathematics but it is hardly easy, and there are some additional complexities in four (and higher) dimensional mathematics that don't help.
 * If anything this article is less technical than it could be, with no formulae or mathematical discussions, except for the volume of the hypersphere.

JohnBlackburne 16:18, 20 January 2007 (UTC)
 * Hmmm... while I agree this article could be a lot more complex with the formulas and what not, perhaps we should have an article which serves as an intro to higher dimensions. Actually, that gives me an idea. I will add a section to the talk page about it. Jaimeastorga2000 13:56, 9 August 2007 (UTC)

Incorrect caption for the animation
The animation in the article is actually a two dimensional projection of a three dimensional projection of a fourth dimensional object (http://upload.wikimedia.org/wikipedia/en/5/55/Tesseract.gif), not a three dimensional projection as the caption (3D projection of a rotating tesseract) states. A computer screen cannot create a three dimensional projection. —The preceding unsigned comment was added by 24.187.17.94 (talk) 00:04, 2 January 2007 (UTC).


 * Every image is 2D. So an image of a 3D object, real or virtual, is a 2D projection. Because it is always true it does not need to be stated and hardly ever is. JohnBlackburne 22:43, 23 January 2007 (UTC)


 * I think the caption is correct as it stands, as the person above said, there is no need to call a picture of a three dimentional apple a 2-dimentional projection of an apple--Scorpion451 04:20, 15 July 2007 (UTC)

Do we live in four-spatial dimensions space?
General relativity says that Universe is a curved space.Scientists proved the that massive objects such as  SUN  bend the light coming from the stars.

The question is

Does that also mean that this Universe a four spatial dimensions universe since it is a close/ curved space universe?.

86.147.252.83 12:59, 15 January 2007 (UTC)

General Relativity says that four dimensional spacetime is curved, i.e. non-Euclidean. Some people like to imagine it as embedded in a fifth dimensional space, as this space can be Euclidian. This space is also used by some science fiction writers for faster than light travel. There is no mathematical justification for it though.

JohnBlackburne 23:29, 16 January 2007 (UTC)

Scientists???
Can somebody make a list with all the scientists involved with the study? Izaak 08:44, 15 April 2007 (UTC)

Image mistake
The uppermost image on the page seems to be missing an arrow (the ones that point perpendicularly into the 4th dimension). There are only 7 arrows, but 8 vertices on the cube. The front upper right vertex is missing an arrow. Leon math 21:32, 20 April 2007 (UTC)

Physics???
This article is about 4-dimensional Euclidean space, not about 4-dimensional Minskowskian space-time. I. e., it is an article about geometry and not about physics, even though there is a tenuous mathematical connection. I don't think this article is relevenat to WikiProject Physics.&mdash;Tetracube

My brain, it BURNS! Seriously, that animated gif blows my mind.--Daniel Berwick 23:50, 9 April 2007 (UTC)


 * Physics and mathematics are basically the same in reguards to geometry. One mathematicians deirivative is another physicists velocity.--Scorpion451 04:23, 15 July 2007 (UTC)

I think it is important to cover all the aspects of the topic.Southafrica6 (talk) 00:15, 29 April 2008 (UTC)

Pictures
What about a real picture: http://www.physorg.com/news7409.html might be of some help. —Preceding unsigned comment added by 130.127.3.249 (talk) 23:45, 8 December 2008 (UTC)

I know the rotating 3D tesseracts are pretty and all, but is it really necessary to have 2 of them? If nobody raises an objection I'm going to delete the second one because it doesn't look as good as the first one. RageGarden 23:34, 28 April 2007 (UTC)


 * The first is of a rotating (4D) tesseract, the second is of a rotating 24-cell. Not only are they different but the second one is the only 24-cell pictured. I think a rotating version of it works well, as it gives it some depth - a 2D picture of a 24-Cell is a lot less interesting. JohnBlackburne 12:10, 17 May 2007 (UTC)


 * There used to be another 4D tesseract, but I deleted it already. The one I was talking about was [[Image:Changingcube.gif]] RageGarden 04:22, 18 May 2007 (UTC)


 * I think rotating tesseracts are entirely unnecessary. They make me want to throw up, and they aren't any better at explaining the concept than a stationary 2D tesseract. PyroGamer 20:18, 24 June 2007 (UTC)


 * I agree, or at least I think they're pretty, but not helpful here. Tom Ruen 21:07, 24 June 2007 (UTC)


 * Personally, in my experience it's the only way to get a good idea of what they look like, even if it is only a zoetrope view. And of course it will make some people sick to your stomach, Its something the brain isn't hardwired to handle.--Scorpion451 04:30, 15 July 2007 (UTC)

Should this clarification be added?
I propose we add the following clarification to the header:


 * This article refers to a fourth proposed spatial dimension. For the einsteinian concept of time playing the role of a fourth dimension, see spacetime.

PyroGamer 20:16, 24 June 2007 (UTC)


 * This would be helpful, yes. I second this.&mdash;Tetracube 01:30, 5 October 2007 (UTC)


 * You can add this yourself if you think it would be helpful. Be bold!   Pi is 3.14159  |  Talk  01:40, 5 October 2007 (UTC)

Good idea!Italic textSouthafrica6 (talk) 00:24, 29 April 2008 (UTC)

New Introductory Level Article?
Well, after reading http://en.wikipedia.org/wiki/Talk:Fourth_dimension#My_head_hurts.21, I thought that maybe an introductory level article to higher dimensions (or indeed, the 4th dimension only, as I doubt anybody who is trying to understand higher dimensions wishes to go any higher) would be appropriate. After all, we have seen introductory level articles to complex subjects before, such as http://en.wikipedia.org/wiki/Introduction_to_general_relativity or http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics. Therefore, by http://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_accessible, an article which would make this one more accessible is desirable. I suggest we use the already written section of dimensional analogies as a base or as a guide, because of two reasons:


 * -I fully believe dimensional analogies are the best method of introduction and the beginning of true understanding of the 4th dimension.


 * -As http://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_accessible states:
 * Use analogies to describe a subject in everyday terms. The best analogies can make all the difference between incomprehension and full understanding.  As an example, the Brownian motion article contains a singularly useful entry entitled Intuitive Metaphor:


 * Consider a large balloon of 10 meters in diameter. Imagine this large balloon in a football stadium or any widely crowded area. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted from the left side and the right side are imbalanced in favor of the left side; the balloon will move slightly to the left. This imbalance exists at all times, and it causes random motion. If we look at this situation from above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.


 * Now return to Brown’s pollen particle swimming randomly in water. A water molecule is about 1 nm, where the pollen particle is roughly 1 µm in diameter, 1000 times larger than a water molecule. So, the pollen particle can be considered as a very large balloon constantly being pushed by water molecules. The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted by the small liquid molecules on the particle.

What do you think? Jaimeastorga2000 14:31, 9 August 2007 (UTC)


 * I fully second this idea. I also believe that dimensional analogy is the best way to develop an intuitive feel of 4D space. This is not to discount most mathematicians' preferred approach, which is with algebra and hard equations. Rather, an intuitive grasp of 4D space can complement and enhance one's facility with algebraic manipulation. Dimensional analogy is best because it relates unfamiliar things in 4D with similar things in the 3D space we're familiar with. I would say that dimensional analogy of perspective/parallel projections is probably one of the best tools for visualizing 4D objects. Many regular and Archimedean polytopes may be re-discovered by simple application of this method, once one is familiar with how 4D works. (Of course, such results have to be verified mathematically, but the fact that one can arrive at them using just dimensional analogy speaks of its usefulness as a visualization method.)&mdash;Tetracube 01:30, 5 October 2007 (UTC)

It would be good to create a jumping of point that deals with our everyday objects instead of the more advanced stuff.Southafrica6 (talk) 00:23, 29 April 2008 (UTC)

The weight of the Tardis
Rephrasing my question to this context: if the Tardis can be treated as a tesseract of a given weight, how would one calculate the weight in each variant visible in three dimensions (ie how much would the police box in the series, rather than the Tardis-complete) weigh?

A suitable website redirect would suffice. Jackiespeel 15:30, 5 October 2007 (UTC)

Time OR Spacetime
This article, and a lot of the rhetoric I've found on the fourth dimension, does not seem to discuss a difference between Einstein's concept of spacetime and the generally understood construction of "time." Is this article saying that the fourth dimension is the union of space and time that Einstein called spacetime OR is it simply time by itself? Most people, I would think, would agree that time can be defined by change -- in video, for example, the change between frame 1 and frame 2 being a measure of time -- and that this traditional flavor of "time" can exist on all dimensional levels. Whether I'm a dot, a square, or a cube, if I'm late to a meeting, the meeting is no longer at the same x,y, and/or z coordinate at a particular time t and I will have to get the meeting's minutes at a later time t + n. This situation seems to change when I'm a hypercube.

What I believe the article is arguing is that the fourth dimension is spacetime, and whenever the article mentions the word "time" it is really talking about "spacetime." Since I am not a physicist, I do not know if this is accurate, but if it is, we should by no means make the assumption that everyone who reads this article will understand that "time" and "spacetime" are the same, even if they actually are.

If my assumptions are correct, we should clarify a couple things. The solutions I can think of off the top of my head would be: either make a note in the introduction about time and spacetime being the same, OR replace all instances of "time" with "spacetime" where appropriate. // Montag 05:20, 10 October 2007 (UTC)

I think making a distinction between "fourth spatial dimension" and "spacetime/temporal dimension" (or similar term) would resolve the unclarity - being the way most people would divide them. Jackiespeel 13:38, 11 October 2007 (UTC)


 * I believe the article was originally intended to discuss 4-dimensional Euclidean space, rather than Einsteinian space-time. Contrary to popular understanding, these are two very different things. Einstein's concept of space-time is not Euclidean, but Minkowskian. The fourth dimension in Euclidean 4-space has absolutely nothing to do with time, just as the 3rd dimension in 3-space has nothing to do with 2D time! Einstein's concept of space time may be regarded as 3+1 dimensions, where the extra dimension is different from the other 3 in the sense that it is temporal, and behaves differently from the spatial dimensions. Even mathematically, the time dimension in relativity does not behave like the other dimensions: it has a negative sign whereas the other dimensions have positive sign. Another major difference is that you cannot arbitrarily interchange the time dimension with the other spatial dimensions, the way you can interchange the spatial dimensions via rotation.
 * The 4th dimension this article is discussing is Euclidean rather than Minkowskian, which means that all four dimensions behave exactly the same way, and are freely interchangeable with each other via rotations. They are spatial and not temporal. Only in this context, geometry makes any sense: geometry by definition deals with shapes and angles, etc., in an absolute sense. Geometry doesn't apply to temporal dimensions (although this is often attempted, esp. when discussing relativity, because it appeals to people's imagination), at least, not in a sensible way like it does to spatial dimensions. For example, 4-dimensional polytopes, which are geometric objects, do not make very much sense in Einstein's space-time, where they represent mutating shapes over time. (And I'm not even sure if this is a valid interpretation, as they probably violate fundamental laws of Einsteinian space-time, such as the fact that the rate of shape change cannot possibly exceed light-speed, which severely limits the allowed angles between facets lying across the time dimension.) Most of their beauty and symmetry is lost because space-time does not allow you to rotate these polytopes the way you can rotate them in Euclidean 4-space. The common attempt to understand them as "snapshots" of shapes in space-time is misguided, and does not really explain their true geometry.
 * Anyway, Einstein's theory is already discussed adequately in spacetime. The focus of this article is really to discuss Euclidean 4-space. (Unfortunately, that last paragraph in the intro about considering a 4th spatial dimension as the "5th dimension" only serves to confuse this issue more.)&mdash;Tetracube 01:56, 16 October 2007 (UTC)

The methaphysics of the fourth dimensional space
According to my knowledge, whatever corresponds to the fourth-dimensional space in physics and maths is a spiritual image of a higher world, the world of perfect ideas in Plato, the Kingdom in Christian thought, etc. Anyone? —Preceding unsigned comment added by 202.80.43.11 (talk) 02:00, 13 October 2007 (UTC)

Variants of the "hypercube"
''"During studies, dimensions relate by multiplying itself by itself (squaring it), then multiply it by it's next dimension. This will give you the amount of "drawn lines" within the next dimension.

''I.E.(1) 1st Dimension, multiplied by itself, Gives you 1. Multiply 1 by the next dimension (2), and you receive the answer of 2, which is how many lines represent the 2nd dimension. (Right Angle)

''I.E (2) 2nd Dimension, multiplied by itself, gives you 4. Multiply 4 by the next dimension (3), and you receive the answer of 12, which is how many lines represent the 3rd dimension. (Cube)

''I.E (3) 3rd Dimension, multiplied by itself, gives you 9. Multiply 9 by the next dimension (4), and you receive the answer of 36, which is how many lines represent the 4th dimension."''

Would it be more practical make a study of the required vertices (corners) involved in these shapes? Such a equation would be far easier to explain with:

$$2^D=V$$

Where d is the dimension count (0, 1st, 2nd, 3rd etc) and V is the required amount of vertices.

For example a dot in the 0 dimension has one point. A line has 2, a square has four, a cube with 8 and the cube within cube design has 16. This would require consensus that a cube-within-cube design is a valid tesseract of course. Lagginwagon (talk) 08:16, 23 November 2007 (UTC)

Something to ponder
Thinking on fromwhat was touched upon in the section above 'Shadows of ourselves'. Doesn't the 4d concept ever strike you as an explantion for paranormal activity? I'm kinda thinking of ghosts or Mothmen - the idea of just having random apparitions appearing (as a 3d cross-section of a 4d being passing through the plain of our deminsion) and being able to comunicate without being see and also know the contents of a sealed container (like us seeing the contents of a 2d container). Anyways I quite liked the idea and thought it might be worth sharing, if not the plot to a cheap movie(!). ArdClose (talk) 18:38, 20 December 2007 (UTC)
 * Coming to a theater near you... GhostCube! J- ſtan ContribsUser page 18:44, 20 December 2007 (UTC)

Where to find out about the eccentricities in higher dimensions?
I know there are many properties of higher dimensions that can't be discovered by simple Dimensional analogy, as described in this article, because the third dimension is, in some sense, a special case. An example I've heard (just heard, from someone who has about 90% reliability) is that a knot can't exist in higher dimensions. Another one I found from a tesseract form of the Rubik's cube (computer simulation). In the tutorial, if I remember correctly, was something about a 'turn' being different in the 4th dimension; it said that our version of a turn, a rotation around an axis, is a quirk of the third dimension (or something like that).

Where can I find out about that and other strange properties of higher dimensions (within the realms of a high-school education)? --70.124.85.24 (talk) 15:19, 7 January 2008 (UTC)


 * Thanks for your help. Much appreciated. &lt;/sarcasm&gt; --70.124.85.24 (talk) 22:28, 19 January 2008 (UTC)


 * Knots do exist in 4D. Just that 1D objects (ropes, strings, vines, etc.) can't knot; you need a 2D sheet to knot in 4D. This gives rise to such objects as the Klein bottle, the real projective plane, and other interesting objects. Also, rotations in general should be thought of as motion in a 2D plane, rather than around some axis. You're quite right that rotational axes only exist in 3D: even in 2D space, there is no such thing as an axis of rotation (because the axis we like to imagine lies outside of the 2D space!). There is only a central point everything else revolves around. Similarly, in 4D, you rotate around a plane, not an axis. 4D also has compound rotations (sometimes called "Clifford rotations"), in which there are two independent rates of rotations. Such rotations happen about a point. Hope this helps. (P.S., no need to be sarcastic, sometimes it just takes a while for people to notice your question.)&mdash;Tetracube (talk) 13:48, 5 April 2008 (UTC)


 * Here's another tidbit for you: In 2D there is an infinite family of regular polygons; in 3D there are nine regular polyhedra (five convex and four stars); in 4D there are sixteen regular polychora (six convex and ten stars); and in each higher space there are only three regular figures (all convex). —Tamfang (talk) 07:29, 21 May 2008 (UTC)

I thought that objects such as ropes and vines were 3 dimensional, as they have height length and width. If this is right, shouldn't 4th dimensional knots have to be created by 4th dimensional objects?Southafrica6 (talk) 21:37, 12 May 2008 (UTC)


 * Physical ropes have three dimensions, but the objects considered in knot theory are made of abstract strings with zero thickness. —Tamfang (talk) 07:25, 21 May 2008 (UTC)

Assessment comment
Substituted at 20:34, 2 May 2016 (UTC)

Charles Hinton was the first to consider the possibility of a fourth dimension?
A recent edit claimed that Charles Hinton was the first person to consider the possibility of a fourth dimension. Is this really true? Do we have a citation for this claim?&mdash;Tetracube (talk) 22:16, 13 November 2008 (UTC)


 * In fact, a quick Googling turned up the following:


 * 


 * This shows that as early as 1854, Riemann had challenged Euclid's proposition that space can only be of 3 dimensions. So I dispute the claim that Hinton was the first to consider such a possibility; more likely he got the idea from Riemann and developed his own theories based on that.&mdash;Tetracube (talk) 22:24, 13 November 2008 (UTC)

OK, I've gone ahead and reworded the statement; Higher dimension does explain that Riemann was the first to consider the possibility of geometry in more than 3 dimensions.&mdash;Tetracube (talk) 23:18, 13 November 2008 (UTC)

Terminology, please
Just look at this chart here:

Looking at that, we can see that there is a lack of terminology here. Does anyone know what the proper terms are? And if you do, could you put them here so that they may be put in the article. It would help. —Preceding unsigned comment added by Wikool (talk • contribs) 23:37, 8 December 2008 (UTC)


 * Some of the analogies in the table are dubious; longitude and latitude are more specialized than this would imply; and spissitude is not exactly standard. There's no standard successor to over/under, though ana/kata are sometimes used (see, which my bookmark randomizer drew from a hat as I was writing this!).  A hyper-pyramid is commonly (i.e. in my experience) called a pyramid; in 3space you have your polygonal pyramids, and in 4space your polyhedral pyramids.  —Tamfang (talk) 04:17, 12 December 2008 (UTC)

We're gonna need eyes, lots of eyes ?
Would having 3 or more 2d eyes work for being able to have depth AND spissitude perception in 4d ? ps:Yeah, I know, talk page is meant for the article and not what the article is about, but it seems people here are more accepting of these kind of off-topicness, I apologize if I interpreted the reactions (or lack of them) here incorrectly --TiagoTiago (talk) 09:08, 16 January 2009 (UTC)


 * The Tetraspace forum is the best place to discuss such topics, where you don't have to worry about whether it's encyclopedic or not. :-)&mdash;Tetracube (talk) 18:35, 16 January 2009 (UTC)


 * Assuming that the 4D animal has a 3D retina (as ours is 2D), you only need two to measure parallax. —Tamfang (talk) 01:28, 13 March 2009 (UTC)

How can one say that eye is 2d instrument,certainly not!An eye is cetainly a 3d object built of transparant matters and the 2d curved surface (the retina)expandes itself in three cngruent euclidians,hence is in fact a three d surface,and the picture we draw upon our retina is also not 2d but is 3d real picture.Thanks for reading me. — Preceding unsigned comment added by 27.97.174.110 (talk) 09:17, 13 January 2012 (UTC)

an open-source 4d maze game,should this be included?
should http://www.urticator.net/maze/ be included in external links or somthing? --TiagoTiago (talk) 06:04, 14 January 2009 (UTC)


 * I'm not sure if we should be linking to every 4D-related game out there, as there are quite a number of them, and their relevance to this article is a bit doubtful.&mdash;Tetracube (talk) 18:25, 14 January 2009 (UTC)


 * Perhaps a new page titled "List of 4d games" or something. Pastastraw   ( talk ) 15:21, 17 February 2022 (UTC)


 * Seems to severely lack notability. - DVdm (talk) 15:32, 17 February 2022 (UTC)


 * This is quite an old part of discussion (a bit above 13 years!) so it's likely the old participants are not interested in the topic at all now. How about starting a new thread at the bottom of the page...? --CiaPan (talk) 21:46, 17 February 2022 (UTC)
 * Ha yes, 13 years it is . No problem, but I don't think this is sufficiently notable to start a new discussion. - DVdm (talk) 23:34, 17 February 2022 (UTC)
 * I agree. --CiaPan (talk) 02:56, 18 February 2022 (UTC)

Ambiguous naming
In mathematical terms, a fourth dimension could mean any number of things. I think this article (presumably about the popular concept of the fourth physical dimension) should be renamed more specifically. --Fusionshrimp (talk) 05:24, 22 January 2009 (UTC)


 * Any suggestions?&mdash;Tetracube (talk) 14:42, 22 January 2009 (UTC)


 * Well the first thing that comes to mind is Euclidean 4-Space, but this might put off some less mathematical readers. Also there is mention of Minkowski spacetime and a lot of linear algebra sections in the page that don't specifically involve the fourth dimension at all. So what I'm thinking is a disambiguation page for "Fourth Dimension" where users can be directed to separate pages for "Euclidean 4-Space" (or something similar, which would contain the geometrical aspect and methods of visualization), "Minkowski Space", and a page on the vector space interpretation of dimension. I would look and see what pages exist but I have to go to class in like 5 minutes --Fusionshrimp (talk) 20:10, 22 January 2009 (UTC)


 * The linear algebra sections are really not that essential if we simply link to vector space. Or parts of it could be salvaged and moved to vector space or an introductory-level article if one exists (I'm thinking mainly the parts that explain the concept of dimension to a lay audience: it's not really specific to 4-space alone).&mdash;Tetracube (talk) 20:42, 22 January 2009 (UTC)


 * Did you mean 'spatial' instead of 'physical' in the brackets of the second sentence ? Turul2 (talk) 17:54, 27 January 2009 (UTC)

Okay, I removed the linear algebra material; it was all unsourced anyway so a rewrite would be necessary if one wished to reproduce them on such an introductory page. --Fusionshrimp (talk) 05:04, 23 January 2009 (UTC)


 * Stop editing ! The article is ment to be an article in the sense of Charles Howard Hinton - as a terminus technicus - and not as an enumeration. Please read first chapter 3 of Michio Kaku's book "Hyperspace" (see references ^6):
 * The man, how saw the Fourth Dimension
 * A dinner party in the Fourth Dimension
 * Class struggle in the Fourth Dimension
 * Fourth Dimension as art
 * Bolsheviks and the Fourth Dimension
 * Bigamists and the Fourth Dimension
 * Hinton's cubes
 * The contest on the Fourth Dimension
 * Monsters in the Fourth Dimension
 * Building a four-dimensional House
 * The useless Fourth Dimension
 * Subsequently follows chapter 4: The secret of light: Ripples in the fifth dimension. Turul2 (talk) 17:54, 23 January 2009 (UTC)


 * Stop editing? I believe you misconstrue the ideals and purpose of Wikipedia. This article is NOT meant to be an article in the sense of anybody. And you're correct, it is not supposed to be an enumeration. This is an 'encyclopedic' article that covers 'encyclopedic' content concerning the fourth dimension. We here at Wikipedia edit articles to make them better 'encyclopedia' entries, and if you have an issue with how I'm editing, we can discuss it here and hopefully resolve it. This is a Start-class article that needs to be improved. --Fusionshrimp (talk) 19:25, 23 January 2009 (UTC)


 * Sorry, just wanted to astound you to get you to reconsider the direction of editing or splitting this article. My hope is that you consider Hinton and the time from 1870 till 1900, which is far before Einstein's work and Minkowski's work but short after the invention of Riemann's geometry by Riemann and dealt with 4D Euclidean space and the 4D cube for which the term tesseract was found. Turul2 (talk) 07:05, 24 January 2009 (UTC)


 * If I have time, I will certainly read more of Hinton's work. We just have to remember NPOV; that is, we cannot be biased toward his work for the purpose of this article. And if you must have the linear algebra parts, I moved them to my user page. Someone just needs to cite those bad boys. --Fusionshrimp (talk) 05:20, 26 January 2009 (UTC)

For a possible next step, a suggestion for the "Fourth Dimension" disambiguation page (Science):
 * Fourth dimension, the concept of an additional spatial dimension (after Hinton)
 * Spacetime, the concept of time as an additional dimension (after Minkowski)
 * Kaluza-Klein-theory, the concept of charge/mass as an additional dimension to spacetime (after Kaluza, when starting counting dimensions with zero = time)
 * 4-D vector space ................................  Turul2 (talk) 17:54, 27 January 2009 (UTC)


 * Note that Hinton is not the first person to consider the concept of a fourth spatial dimension. Mathematicians who study polytopes have considered geometrical objects in such a space long before Hinton came on the scene. As early as 1852, some 30 years before Hinton came on the scene, the 6 convex regular 4-polytopes have been enumerated by Ludwig Schläfli, and Bernhard Riemann in 1854 put the concept of higher-dimensional space on firm grounding by considering a "point" to be any finite sequence of numbers.&mdash;Tetracube (talk) 14:51, 21 February 2009 (UTC)

the case of the asymmetric abstractions
In case anyone wants to know why I'm changing this sentence:


 * This fourth spatial dimension is a distinct concept from that of the time dimension in spacetime.

it's because the phrase "that of" implies a distinction between the time dimension and the concept of the time dimension, with the latter (but not the former) on the same level of abstraction as the fourth spatial dimension. —Tamfang (talk) 08:18, 20 February 2009 (UTC)

Repeated reverts to an old version of the article
The editor at IP address 219.78.119.49 seems bent on reintroducing a large amount of text removed some time ago in the process of cleaning up the article, in spite of repeated reverts from another editor. Such large scale controversial needs to be discussed in a civil manner, instead of turning into an edit war. So I ask the editor concerned to explain his/her reasons for reintroducing this material here.&mdash;Tetracube (talk) 14:26, 21 February 2009 (UTC)

Please stop repeating the change until we reach a consensus!! There is a reason the text was removed. It contains a lot of duplicate information available in other articles. The correct way to reference this information is to link to the relevant articles, NOT copy-and-paste it. Can we please discuss what exactly should be put back first, before copy-and-pasting the old stuff back again?&mdash;Tetracube (talk) 04:37, 22 February 2009 (UTC)

Note
Someone should add information about the fourth dimension being time. - Thanks. --219.78.126.250 (talk) 23:21, 11 March 2009 (UTC)


 * There is already an article that talks about this: spacetime.&mdash;Tetracube (talk) 23:50, 11 March 2009 (UTC)


 * Oh yeah. Sorry. --219.78.126.250 (talk) 08:40, 12 March 2009 (UTC)

This article seems to make a lot of assumptions about 4th Dimension existence, which is not proven, just required by modern science. Perhaps there should be some indication of this? - 72.220.125.54


 * A fourth "geometric" dimension exists as a mathematical concept. I agree it would be good to have a sourced section that explains the difference between imagined dimensions, and any evidence of its reality. Modern physics talks about more dimensions for instance, but that includes time, and in string theory, other dimensions that loop back on themselves at a scale too small to experience. Tom Ruen (talk) 02:58, 28 September 2009 (UTC)

Dimensional analogue
There are properties of geometry which are added for each dimension and don't have dimensional analogy for lower-dimensional space. For an example:


 * 1st dimension: Ability for objects to move.
 * 2nd dimension: Infinite variety of shapes, ability for objects to go around each other.
 * 3rd Dimension: Ability to form knots.

But what new properties do objects in four spatial dimensions have? --Artman40 (talk) 22:30, 17 March 2009 (UTC)


 * Many. Here are a few:
 * The inability for 1D strings to form knots.
 * The ability to knot 2D closed surfaces (a Klein bottle is essentially a "knotted sphere").
 * The ability to rotate in two independent planes with two different rates of rotations.
 * Hair on a 3-sphere (4D sphere) can be combed in such a way that no cowlick forms. This is impossible on a 2-sphere (3D sphere).
 * (Assuming we generalize geographical features to a 4D world) You don't need a bridge to cross a river; you can simply walk around it. Similarly, a linear road in 4D does not divide the city into blocks; there is no need for crosswalks, since you simply walk around the road. (In 2D, rivers completely cover the surface; you can't even cross a river, you can only go upstream or downstream.)
 * There are many other such properties, too many to list here.&mdash;Tetracube (talk) 16:48, 18 March 2009 (UTC)
 * These can still be compared with dimensional analogy. --Artman40 (talk) 20:50, 17 June 2009 (UTC)

can someone add some of these examples to the dimensional analogy section? right now it simply explains what a dimensional analogy is, using 3d examples in a 2d world. it doesn't seem to relate to the article. i don't feel confident that i can do it, since i don't understand any of the analogies here.- Uncleosbert (talk) 21:48, 29 April 2010 (UTC)


 * Tetracube, a Klein bottle isn't a sphere, nor is it (on its own) knotted. So it's certainly not a knotted sphere in any sense.  I suggest just deleting (or clarifying) what's in brackets -- after "The ability to knot 2D closed surfaces". Rybu (talk) 17:11, 16 May 2010 (UTC)


 * Yes, the statement about knots in 4D is misleading, as stated. As Rybu says, a sphere, torus, and Klein bottle are intrinsically different topological manifolds. In contrast, a knot in 3D has the same intrinsic topology as a circle in 3D; they differ in how they lie in 3D.  The question is, can a surface in 4D be non-homeomorphic to a sphere in the topology of 4D, but homeomorphic to a sphere in the induced topology of the surface?  (same question for torus or Klein bottle) (I would like to know!) Stevan White (talk) 08:37, 10 December 2016 (UTC)


 * It seems pretty clear to me that it should be easy to knot the 2-sphere in 4-D, but how one actually proves such a knot can't be turned back into a spherical surface without going through itself is something I've never looked up. Dmcq (talk) 15:12, 10 December 2016 (UTC)

4D objects in the real world?
A shirt, shorts or pants can be folded inside out, indefinitely, like that tesseract image at the top of the article. What's going on there? 99.246.243.175 (talk) 14:21, 15 May 2009 (UTC)(Tetra Vega)


 * But a sock can't,much less a ball, and if you just rotate a shirt, or a short or whatever, as a whole, it doesn't turn inside out, the tesseract is solid, the cube inside isn't being distorted nor turned inside out when the tesseract is rotated that way, it just appears to be because of how the projection works, some of the rotations in 4+d look quite weird for people used to just 3d and 2d rotations. Btw, don't forget to sign. --TiagoTiago (talk) 16:45, 12 May 2009 (UTC)


 * And put new topics at the bottom, please. (Use the "new section" button.) —Tamfang (talk) 20:43, 12 May 2009 (UTC)

The tesseract would be made of parts though, molecules/atoms, and such, as the shirt/shorts/pants are made of particles, thread, or sew it would seam... Thus in theory, it's a physical interperetation of a "4-dimensional" object. Or is the tesseract like a ring shaped drop of water, flowing like a conveyor belt? 99.246.243.175 (talk) 14:21, 15 May 2009 (UTC)(Tetra Vega)
 * If a solid object rotates, its shadow might seem to stretch and deform. The deforming-elastic looking projection of the tesseract is in that similar to a shadow: the actual tesseract being represented isn't deforming at all, only its "shadow". Dan ☺ 14:53, 15 May 2009 (UTC)

Euclid vs Minkowski

 * This fourth spatial dimension is a concept distinct from the time dimension in spacetime, since movement in time generally is considered to be unidirectional, or at least not so free as movement in space. Hence spacetime is not an Euclidean space but a Minkowski space.

Um, thanks for trying, but the 'Hence' is invalid. We could imagine a Euclidean space in which movement on one (or more) of the dimensions is similarly constrained, but that would not make the geometry Minkowskian. —Tamfang (talk) 00:48, 15 June 2009 (UTC)


 * Greg Egan's forthcoming novel Orthogonal is set in such a universe. —Tamfang (talk) 20:23, 4 December 2010 (UTC)

arithmetic
In the History section I suggested that Hamilton started arithmetic in four dimensions with his quaternions. Tamfang changed it to defined, saying this is a better word. Actually several arithmetics in four dimensions were used soon after Hamilton, such as coquaternions and hyperbolic quaternions. What Hamilton started was the use of a single variable to represent four real variables, extending the complex number concept where a single variable represents two real. This innovation was practice later championed by Peter Guthrie Tait and his school at Edinburgh. The word "started" conveys the impression of getting the ball rolling, while "defined" seems to refer to a static and settled situation. Therefore I prefer the word "started".Rgdboer (talk) 21:14, 6 September 2009 (UTC)


 * You wrote "An arithmetic of four dimensions called quaternions was started by William Rowan Hamilton", which suggests only that Hamilton did the preliminary work toward quaternions, rather than what you say you intended. How about "The first of several four-dimensional number systems was ..." ? —Tamfang (talk) 05:12, 7 September 2009 (UTC)

Perhaps that is the right phrase for now. One of the weaknesses of the article is that it does not mention linear algebra, the topic relevant to the lead as now stated. This article is one of the 500 most viewed math articles, so it should be a high priority. Having worked on the linear algebra article, and noting little enthusiasm to improve it, has shown me some of the challenge in this corner of mathematical exposition. The impression is often given that the science is settled and little interesting material is likely to turn up. Reviewing the parallel evolution of linear algebra and hypercomplex numbers has in fact been very interesting to me. So far this article seems to be keeping the introductory level necessary to serve the thousands that are reading it; technicalities can be avoided by offering hyperlinks, hopefully to other articles within the range of readers.Rgdboer (talk) 23:31, 9 September 2009 (UTC)

An adjective
The noun "space" has been appropriated frequently, such as the cases of topological space and metric space. The meanings are quite distant from three-dimensional physical space. However, the adjective "spatial" is almost always associated with the mundane space of physical reality. Currently there is a section in the article titled "Fourth spatial dimension". The intended meaning is a fourth Euclidean dimension. The use of the adjective "spatial" here may suggest that some believe there is a hidden parameter of reality. Edits should be made to avoid such a suggestion. Just because the noun has been extended in meaning, it does not follow that the cognate adjective is also extended in meaning.Rgdboer (talk) 02:06, 25 October 2009 (UTC)