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The 4th Dimension





_AVG_
I'm really really confused right now ... thinking of the fourth dimension does that to me ... so, I think I should get my questions answered once and for all:

- First of all, what is the 4th Dimension? Is there a 4th Dimension?
- Secondly, can it be represented theoretically in a diagram or something? If it can, how can it?
- Thirdly, most popularly 'time' is referred to as the 4th Dimension sometimes. Why's that and if it is, how exactly is it the 4th Dimension?
- Lastly, I've heard these terms somewhere: hyper-sphere, hyper-cube, hyper-cuboid. What are they?

I've never been so befuddled ... Confused Confused
Bikerman
OK, let's try to deconfuse the issue.
According to relativity there are 4 dimensions - 3 spatial and 1 temporal. We can therefore specify a point in the universe using 3 spatial dimensions (x,y,z) and one temporal dimension (t). Thus we can say that this 'brick' (or whatever) exists at a particular place and at a particular time - giving a spacetime co-ordinate (x,y,z,t).
OK so far?
Now, before Einstein it was assumed that the time dimension was the same for everyone - ie you can move in space anyway you want, but everyone moves the same in time.
Einstein showed that this is wrong with Special Relativity.
Movement in space is also movement in time. If someone is moving faster than you in space, then they are moving slower than you in time (and vica-versa). The total movement is c (the speed of light). Therefore something moving in space at c does not move in time at all.
Thus, in the 'real' universe (as opposed to the special bit of it that we live in) space and time are combined into a 4-dimensional 'whole' called spacetime.
Now, you can represent spacetime in diagrams - they are called spacetime diagrams (strangely enough).
http://www.astro.ucla.edu/~wright/st_diags.htm
http://www.phy.syr.edu/courses/modules/LIGHTCONE/events.html
http://en.wikipedia.org/wiki/Minkowski_diagram

Various theories (particularly string/m theory) say that there are more than 3 spatial dimensions. M theory, for example, specifies 6 extra ones curled-up at very small scales - well beyond our ability to detect at present. This is, however, another story Smile
Indi
_AVG_ wrote:
I'm really really confused right now ... thinking of the fourth dimension does that to me ... so, I think I should get my questions answered once and for all:

- First of all, what is the 4th Dimension? Is there a 4th Dimension?
- Secondly, can it be represented theoretically in a diagram or something? If it can, how can it?
- Thirdly, most popularly 'time' is referred to as the 4th Dimension sometimes. Why's that and if it is, how exactly is it the 4th Dimension?
- Lastly, I've heard these terms somewhere: hyper-sphere, hyper-cube, hyper-cuboid. What are they?

I've never been so befuddled ... Confused Confused

Bikerman covered the time-as-4th-dimension idea, but didn't go into much detail on the idea of a 4th spatial dimension, so i'll fill in the gap.

So, to answer your questions:

  • (Puts on Bill Clinton hat) That depends on what you mean by "is". ^_^;

    Is there physically in existence a 4th spatial dimension? We don't know. Bikerman mentioned that String theory and M-theory posit higher spatial dimensions (because if you solve Maxwell's electromagnetic equations in higher dimensions, for example, neat things happen that relate the fundamental forces together that don't happen in 3+1 dimensions). But they're just hypotheses at this point.

    But mathematically you can have as many spatial dimensions as you want. It turns out that mathematically speaking, dimensions spaces of 3 or lower are privileged in various ways, but while that may explain why our multi-dimensional universe collapsed into 3 dimensions, it doesn't rule out higher dimensions existing. And certainly on paper you can calculate in as many dimensions as you want.

    So does a 4th spatial dimension exist? i dunno. Is there a 4th spatial dimension - at least conceptually/theoretically? Yes, definitely.

  • Well, technically yes, but...

    Think of it this way, how do you represent 3 dimensions on a 2 dimensional piece of paper (or screen, i suppose, since this is the 21st century)? It's really not easy, is it? You can draw your x, y and z axes, and put a point somewhere... but there's no way you can visually determine where that point is supposed to be in 3-space.

    You'd have the same problem depicting 4-space in a 3 dimensional image, with one extra catch... we are biologically wired to see in 3 dimensions, so we are inclined to see depth even in a flat image... but we are not inclined to see hyperdepth in a 3D image.

    And as for depicting 4-space in a 2 dimensional image... wow.

    There are various animations that exist that show a 4D hypercube projected into 3D space, rotating around (by analogy think about a cube rotating on a flat screen). It's kinda trippy. If you've got 3D glasses - and who doesn't right? - try this one.

    But in my experience, 4D problems are usually reduced to 3D or 2D problems if they are going to visualized. Otherwise, they're just left as matrices, not actually visualized.

    Still, if you want to try and blow your mind... ^_^; Start with one dimension, and draw a 4-part line: ---- You can fold this line one of two ways out of the one dimension (up or down) and get a square - a 2 dimensional shape. Now go to 2 dimensions, and draw 6 squares connected in a cross shape. You can fold these squares in one of two ways out of the two dimensions (in or out) and get a cube - a 3 dimensional shape. Now go to 3 dimensions, and draw 8 cubes connected in a cross with two extra arms. You can fold these cubes in one of two ways out of the 3 dimensions ("ana" and "kata") to create a hypercube - a 4 dimensional shape.

  • This was what Bikerman covered.

  • These are all 4D analogies of the common 3D and 2D shapes.

    First, consider a shape formed in n-dimensions where every point in that shape is the same distance from the centre. In 1 dimension, that's a pair of points. In 2 dimensions, that's a circle (every point on a circle is the same distance from its centre). In 3 dimensions, that's a sphere (every point on a sphere is the same distance from its centre). In 4 dimensions... a hypersphere. Mathematically, a hypersphere is defined by all points that satisfy the equation R = p + q + r + s (which, if you start dropping dimensions, gives you the other shapes: R = p + q is a circle).

    You can also go backwards: start with a hypersphere in 4 dimensions, and project it into 3-space... you get a sphere. Take a sphere in 3-space and project it into 2-space, you get a circle. Project a circle into 1-space you get a pair of points.

    The same notion goes for a hypercube. Start in 1 dimension - the shape that satisfies (p1 ≤ p ≤ p2) is a line from p1 to p2. In 2 dimensions, the shape that satisfies (p1 ≤ p ≤ p2) and (q1 ≤ q ≤ q2) is a square with corners at (p1, q1), (p1, q2), (p2, q2), (p2, q1). In 3 dimensions, the shape that satisfies (p1 ≤ p ≤ p2), (q1 ≤ q ≤ q2) and (r1 ≤ r ≤ r2) is a cube. In 4 dimensions, the shape that satisfies (p1 ≤ p ≤ p2), (q1 ≤ q ≤ q2), (r1 ≤ r ≤ r2) and (s1 ≤ s ≤ s2) is a hypercube.

    Again, you can go backwards: if you look at one of the "faces" of a hypercube, you will see a cube. If you look at one of the faces of a cube, you will see a square. If you look at one of the faces of a square, you will see a line.

    As for a hypercuboid - a cuboid is just a stretched cube, the same way a rectangle is a stretched square. So if you had a hypercube and stretched it along one of the dimensions, you'd get a hypercuboid.
_AVG_
Thanks a lot for your explanations and the links really helped as well ... but there's one drawback about the 4th dimension ... it seems to be only mathematical / theoretical ... at least to our eyes.
Bikerman
_AVG_ wrote:
Thanks a lot for your explanations and the links really helped as well ... but there's one drawback about the 4th dimension ... it seems to be only mathematical / theoretical ... at least to our eyes.

You will have to specify what you mean (now that Indi and I have explained, you should be able to do so). Do you mean the temporal 4th dimension (time) ? If so then it seems pretty real to me...
If you mean a 4th spatial dimension then, yes, there is currently no specific evidence to support the notion. There are lots of hints, however, that there might be more dimensions. One big one is gravity. Why the heck is gravity so weak compared with the other forces? String theory offers a possible solution by positing that gravity is actually about the same strength as the other forces, but at a very small scale it is 'divided' by 6 extra dimensions - thus making it appear very weak to us.

So, yes, I agree that the notion of extra spatial dimensions is, at present, only a mathematical one, unsupported by empirical evidence. It is, if you like, a beautiful mathematical model which may be complete hokum. Personally I'd say it is even money......
ocalhoun
To add my 2 cents in, the best way I know of to 'visualize' a 4th (or more) spatial dimension is this:

Place a stick somewhere (an imaginary one will do): 1D
Place another one at a right angle to it: 2D
Place another at a right angle to all the others (vertical): 3D
Place yet another one at a right angle to all the others: 4D (you (probably) won't be able to do that... if you do manage it, you've made a huge breakthrough.)
Repeat the 4th step for more and more dimensions.
Bikerman
That probably counts as the most useless contributions to the science forums yet seen.....
ocalhoun
Bikerman wrote:
That probably counts as the most useless contributions to the science forums yet seen.....

...

I've seen worse
richard270384
Cmon guys. Useless??? Maybe.

But this is such a fascinating topic. Let's let everyone make contributions. Even if they are very simple...

For us non-phicisists the more angles we get things explained to us from, the more likely we will understand.
Indi
i recommend reading through Flatland, by E. A. Abbott. Admittedly it's a little tough to get through because of the really old language used, and it is a parody of Victorian principles which are eons out of date to us today... but it's short it it has pictures. ^_^;

Now, it won't help you grok the fourth dimension directly. But what it does cleverly is show how beings in lower dimensions would see beings that exist in higher dimensions. For example, a four-dimensional being (four spatial dimensions), would be able to see your insides - and your outsides at the same time. They would be able to appear and disappear at will.
_AVG_
Is the following assumption correct:
- A 3D object will appear to a person with 4D vision (spatial) like a 2D object appears to us.
metalfreek
The 4th dimension is time. Einstein in his theory of relativity discussed the concept of space time fabric to explain the gravity.
Bikerman
Indi wrote:
i recommend reading through Flatland, by E. A. Abbott. Admittedly it's a little tough to get through because of the really old language used, and it is a parody of Victorian principles which are eons out of date to us today... but it's short it it has pictures. ^_^;

Just to add to this....Flatland is available on-line HERE
ocalhoun
_AVG_ wrote:
Is the following assumption correct:
- A 3D object will appear to a person with 4D vision (spatial) like a 2D object appears to us.

Sort of, they would still see it in three dimensions, but still be able to see all parts of it at once, as we can a 2D object.
_AVG_
So, now, talking about a Hypercube ... how many sides would one have? [How do you calculate it?]
Bikerman
http://www.traipse.com/hypercube/
http://en.wikipedia.org/wiki/Hypercube
Indi
ocalhoun wrote:
_AVG_ wrote:
Is the following assumption correct:
- A 3D object will appear to a person with 4D vision (spatial) like a 2D object appears to us.

Sort of, they would still see it in three dimensions, but still be able to see all parts of it at once, as we can a 2D object.

Technically they would only be able to see its exterior and interior from one side. This is a quibble point in some cases, and a critical distinction in others.

To understand why this is true, think about looking at a line (a one-dimensional "cube") in two dimensions. From any vantage point in two dimensions, you cannot see both sides of the line - you can always see the start-point, the end-point, and one side of the line segment (except when you look at it edge on, where you can only see the start-point - or end-point).

To understand why this is important, think about looking a more complex 2-D shape in 3 dimensions. For example, think about the letter "N". If you look at it from above the 2-D plane, you will see "N"... but if you look at it from below the 3-D plane, you will see "И" (which looks like the Cyrillic capital letter I). No matter how you change your orientation (by turning the plane, getting closer or farther, or leaning it one way or the other), there is no way you can go from "N" to "И" without crossing the plane.

In the case of a 3-D shape, you can look at it in 4-D from the "ana" or "kata" direction (the 4-D version of left/right, forward/back and up/down). Although both ways you will see "all" of the 3-D shape, you will only see half of the possible orientations of that "all" (same way as you can only see half of the possible two shapes the 2-D "N" shape makes in 3-D)... so is it really "all"? Well, that depends on the context.

Of course, for shapes with even just one line of symmetry - which includes lines, squares, cubes and hypercubes - the two different orientations make no difference at all.

_AVG_ wrote:
So, now, talking about a Hypercube ... how many sides would one have? [How do you calculate it?]

Although i could give you the equations, you could very easily logic it out.

First, think of the number of vertices. The number of vertices in a point, line, square and cube are 1, 2, 4 and 8 respectively (you can just count them). But why?

Well, start with 0-D - the point. To make a line out of the point, you take that point and echo it into the first dimension... then connect the original to the higher-dimensional echo. Voil, a line. To make a line into a square, you echo both points of the line into the second dimension, then connect each original point to its echo. To make a square into a cube, you echo all four points of the square into the third dimension, then connect each source and echo points. And, logically, to create a hypercube, you take a cube and echo it deeper into the fourth dimension, then connect the original points and connect them to their echoes.

So every time i move up a dimension, i make an echo of the current dimension's characteristic shape (and then connect the points). What happens when you make an echo, or a copy, or something? You create a duplicate... you double it. If there are n points in dimension N, then dimension N+1 will have n*2 points.

That's not bad right there, because we can already write a recurrence equation for the number of sides. If the number of sides in dimension N is S(N), then S(N) = 2 * S(N-1). Which means you can already solve for the number of points in a hypercube: S(4) = 2 * S(3) = 2 * 8 = 16. There are 16 vertices in a hypercube.

But you can take it to the next level by noting that S(0) = 1. That means that S(1) = 2 * 1, S(2) = 2 * 2 * 1, S(3) = 2 * 2 * 2 * 1... and in fact, S(N) is N 2's all multiplied (then multiplied by 1). N 2's all multiplied is simply another way of saying 2^N.

Which means the number of sides S in an N-dimensional cube is: S(N) = 2^N.

If you follow similar logic, it is trivial to find recurrence relations for other things (like the number of sides, the number of faces, etc.). It can get a bit tougher to find non-recurrence relationships, but it is not hard. This is good practise for the mathematically inclined in high-school.
dupnitzabg
hmmm.....here's this theory i came up with (pretty sure its wrong though)....


basically, with each dimention, we're adding a unit of depth to an object:
1) adding depth to a point would be a line
2) to a line would make it a plane
3) to a plane would make it a 3-dimentional object...
4) ....not there yet....

ok, so now that we got that down, started wondering what it was getting at......a perfect object maybe?
as far as i can tell, a sphere is the "perfect" figure....everything in the universe attempts to to reach this shape after all....so shouldn't that be the final dimention?
....the way i can picture this best is by imagining a cube and picking one of it's edges as a center, then projecting that same cube in all possible directions...wouldn't that make it a perfect sphere?

i can't say that's the 4th dimention (that would be just stupid.....cus it would mean that every single dimention after that will simply be a sphere of a bigger size), but it certainly seems to me to be the shape of all objects within the final dimention....what do you guys think?


...and yes....i do realize that the "final" dimention will simply be an infinite dimention meaning it can never be reached..................so does that mean we can never have a perfect sphere...ever???



oh, and i think it's just stupid to consider time as a dimention btw.....it's just our perception on reality after all, nothing to do with any of this....
Gagnar The Unruly
Conceptually, it may be impossible for humans to imagine an object in four dimensions. Mathematically, however, it's really easy.

Picture a square. It is made up of two sets of two parallel lines that intersect one another at 90 degree angles.

Now, a cube. Three sets of two parallel planes that intersect one another at 90 degree angles. Also, each side of the cube is a square.

So a four-dimensional cube, if I'm not mistaken, will have another set of parallel surfaces added on. But for the 4D cube, each side is actually a 3D cube, and the 3D cubes are parallel to one another, in sets of two, and orthogonal to the rest. Can you picture it? Probably not.

Mathematically, it's easy to define.

The volume of a 2D cube (a square) is x^2 (where x is the length of a side)
For a 3D cube, its x^3 and for 4D its x^4.

The surface area of a 2D cube: 4x
For a 3D cube: 6x^2
For a 4D cube: 8x^3

In space, the coordinates of the vertices of a 2D cube are:
(0,0)
(0,x)
(x,0)
(x,x)

For 3D cube:
(0,0,0) (0,0,x)
(0,x,0) (x,0,0)
(x,x,x) (x,x,0)
(0,x,x) (x,0,x)

For a 4D cube:
(0,0,0,0) (x,0,0,0) (0,x,0,0)
(0,0,x,0) (0,0,0,x) (x,x,0,0)
(x,0,x,0) (x,0,0,x) (0,x,x,0)
(0,x,0,x) (0,0,x,x) (x,x,x,0)
(x,0,x,x) (x,x,0,x) (0,x,x,x)
(x,x,x,x)

Conceptually, the math makes no sense, but mathematically it is clear and simple. If you try to figure it out conceptually, you'll just lead yourself to the wrong conclusion.
ParsaAkbari
Quote:
Conceptually, the math makes no sense, but mathematically it is clear and simple. If you try to figure it out conceptually, you'll just lead yourself to the wrong conclusion.


And what do you mean by conceptually? I understand that each conrner is in one place. xyz but where does the forth one come from?

And im pretty sure time is a constant right? Obviously if you are moving faster you are covering more ground in one second. But the image im getting is that if you are moving faster you are using less seconds to cover ground Question
Parkour_Jarrod
ParsaAkbari wrote:
Quote:
Conceptually, the math makes no sense, but mathematically it is clear and simple. If you try to figure it out conceptually, you'll just lead yourself to the wrong conclusion.


And what do you mean by conceptually? I understand that each conrner is in one place. xyz but where does the forth one come from?

And im pretty sure time is a constant right? Obviously if you are moving faster you are covering more ground in one second. But the image im getting is that if you are moving faster you are using less seconds to cover ground Question


okay lets say that your going in a general direction X=<direction> Y=<Direction Z=<Direction> well your not exactly following any particular direction but your making direction W so the 4th dimension is the combination of the 3 dimensions

That's what i understand and believe, but my Understanding and Beliefs will be changed when there is a universal agreement on what the 4th dimension is
Indi
dupnitzabg wrote:
basically, with each dimention, we're adding a unit of depth to an object:
1) adding depth to a point would be a line
2) to a line would make it a plane
3) to a plane would make it a 3-dimentional object...
4) ....not there yet....

3) would be a volume, or a space.
4) would be a hypervolume, or hyperspace.

dupnitzabg wrote:
ok, so now that we got that down, started wondering what it was getting at......a perfect object maybe?
as far as i can tell, a sphere is the "perfect" figure....everything in the universe attempts to to reach this shape after all....so shouldn't that be the final dimention?

A sphere is a shape, not a dimension.

The reason things tend to spherical in our 3-space is because a sphere has the property that every point on the sphere is the same distance from its centre. That means it is the minimum energy configuration of a closed surface (just like the minimum energy configuration of a closed line is a circle - take something stiff and bend it until the ends touch and you get a circle). i don't know if that makes it "perfect" - that's a matter of opinion.

In 4-space, the minimum energy configuration of a volume is a hypersphere. In 5-space it is a 5-sphere... and so on. There is no "final dimension", and if there were it would have nothing to do with a sphere (or hypersphere for that matter).

dupnitzabg wrote:
....the way i can picture this best is by imagining a cube and picking one of it's edges as a center, then projecting that same cube in all possible directions...wouldn't that make it a perfect sphere?

i'm not sure, because i don't understand what you mean.

First of all: what is "all possible directions"? In 2 dimensions, "all possible directions" does not mean the same thing it does in 3-space, and so on.

Second: project on to what? You can project a cube onto a line, a plane, another volume, an elliptical space, a hyperspace... pretty much anything. What you get when you project it depends on what you project it onto, and from which orientation.

dupnitzabg wrote:
i can't say that's the 4th dimention (that would be just stupid.....cus it would mean that every single dimention after that will simply be a sphere of a bigger size), but it certainly seems to me to be the shape of all objects within the final dimention....what do you guys think?

Dimensions are not spheres of any size. Dimensions have no shape. They are simply "directions". You can think of them as straight lines (usually) stretching out front and back into infinity... but they are not lines either.

dupnitzabg wrote:
...and yes....i do realize that the "final" dimention will simply be an infinite dimention meaning it can never be reached..................so does that mean we can never have a perfect sphere...ever???

You can have a perfect sphere in anything higher than 2 dimensions. A perfect sphere is simply a 3 dimensional shape.

dupnitzabg wrote:
oh, and i think it's just stupid to consider time as a dimention btw.....it's just our perception on reality after all, nothing to do with any of this....

That is wrong. Time is not just our perception of reality, and it is indeed a legitimate dimension. A dimension is a way that two points can differ. In two dimensions, two points can only differ in two ways: a difference in up/down and a difference in left/right. In three dimensions, two points can differ in three ways: up/down, left/right, forward/back. In our world, two points can differ in four ways: up/down, left/right, forward/back... and when. So if i wanted to go from right here, right now to another point in spacetime, you need to give me 4 directions: up/down, left/right, forward/back, and how far to travel in time.

Time may not be a spatial dimension... but it definitely a dimension as meaningful as any of the spatial ones.

Parkour_Jarrod wrote:
okay lets say that your going in a general direction X=<direction> Y=<Direction Z=<Direction> well your not exactly following any particular direction but your making direction W so the 4th dimension is the combination of the 3 dimensions

That's what i understand and believe, but my Understanding and Beliefs will be changed when there is a universal agreement on what the 4th dimension is

No, that makes no sense. Any combination of 3 dimensions is still 3 dimensions.

To understand that, go back to 2 dimensions - a flat piece of paper. You can move in any combination of those two dimensions - up, down, left and right - and you will never leave that sheet of paper. You can't unless you move in a third dimension that is independent of the first two.

That is what makes a dimension a dimension - it must be independent of other dimensions. Not made up of a combination of them.

If you want a mathematical example, then take any two 2D lines that you want:
Ax + By = C
Dx + Ey = F
Then add them, subtract them... do whatever you want to them without introducing a third dimension... and see if you can do anything to get a 3D line:
Gx + Hy + Iz = J

You will find you cannot.
tukun2009manit
_AVG_ wrote:
I'm really really confused right now ... thinking of the fourth dimension does that to me ... so, I think I should get my questions answered once and for all:

- First of all, what is the 4th Dimension? Is there a 4th Dimension?
- Secondly, can it be represented theoretically in a diagram or something? If it can, how can it?
- Thirdly, most popularly 'time' is referred to as the 4th Dimension sometimes. Why's that and if it is, how exactly is it the 4th Dimension?
- Lastly, I've heard these terms somewhere: hyper-sphere, hyper-cube, hyper-cuboid. What are they?

I've never been so befuddled ... Confused Confused



i don't know much but the 4th dimension should be time because while measuring anything in coordinate system time can also give the estimate of point if 4th dimension is considered
myviny
The universe that we live in has only three spatial dimensions. We are limited to length, width, and height, and we can only travel along three perpendicular paths. While people generally call time the 4th dimension in the universe we live in, time will be the 5th dimension in my hypothetical universe.

Many fascinating possibilities exist when a spatial fourth dimension is present. Several types of wheels are possible, very complex machines can be built, and many more shapes are possible. Objects can pass by each other more easily, but they are harder to break into multiple pieces. Energy reduces much faster with distance than in the 3rd dimension, so both light and sound are weaker. Much more things can be compacted into a small space, but its much easier to get lost.

Many people have explained a lot above. I will post more later to give an even better idea of the fourth dimension. Till then the best way to get an idea would be to watch the explanation by Carl Sagan, the great astrophysician. Here is the link :
www.youtube.com/watch?v=Y9KT4M7kiSw
orkutthemes
what we see is not always what there is.....and what there is not always as we see it.
there must be another form of energy in universe to provide existance to presented energy.
might be in other dimension.
and other dimension can have other medium then EM as it's all we got here.
EM creats light,energy,mass.....everything ....(here only)
Bikerman
orkutthemes wrote:
what we see is not always what there is.....and what there is not always as we see it.
there must be another form of energy in universe to provide existance to presented energy.
might be in other dimension.
and other dimension can have other medium then EM as it's all we got here.
EM creats light,energy,mass.....everything ....(here only)

Err...
a) EM is NOT all we have in this universe. It is one of the fundamental forces which also include the strong and weak nuclear force and gravity.
b) EM does not 'create' everything. We don't actually know, from the standard model, what causes objects to have mass. One current theory is that particles (photons, electrons etc) move through a 'field' called the Higgs Field. This field interacts with charged particles to produce a 'drag' and it is this resistance to movement (inertia) that we perceive as mass.
Raidation
The 4th dimension is time.

I know this is really confusing. I was confused the first time I heard it.

Assume an object with vertex C is at position (5,6) at time 1.
It moves to position (7,8 ) at time 2.

We would measure the vector of vertex C.

In 2 or 3 dimensions, nothing "moves". Everything is placed in peaceful coexistence. In a 4th dimension, objects can move. Two objects can be in the same place in different times without coexisting.

This was based off of Einstein's idea of relativity.
myviny
Oh yes, Einstein was really brainy
RanVijay
Well, first lets remember that we live in 3 dimensional space. How do we know? - We have three directions of free movement:

1.) Left/Right
2.) Back/Forth
3.) Up/Down

Because we have three possible directions, it will take three values to describe the position of any point (x,y,z). In two dimensions, we only need two points: (x,y). In three dimensions, I could tell you to go forward 8 feet, right 12 feet, then you reach a rope. Climb up the rope 6 feet. Notice that each of these three directions are perpendicular to each other... That means that any of the directions is its own and not combined of any of the other directions. That means I can go forward or backward all I want but I won't be going left or right, nor up or down. So if we were to consider a fourth dimension (x,y,z,t), it would have to be a new perpendicular direction in addition to the three directions we have free movement in.

Since it is hard to try to directly picture the fourth dimension in our minds, perhaps using analogy can help us. In 1884, Edwin Abbot wrote a book called "Flatland". The book writes about A. Square and his world, Flatland. You may have already guessed, but Flatland is a 2 dimensional, flat plane and A. Square is a square shaped guy who lives there. He has two dimensions of free movement. He can go left/right and back/forth, however because he is restricted to his 2 dimensional Flatland plane, he cannot go up/down off the plane. By analogy, we humans are restricted to our "plane" of existence... and it would be impossible for us to freely move in the fourth dimension. Let's go back to A. Square again. Note that A. Square can only see what lies in his plane of existence, which means if a 3 dimensional sphere were to pass through Flatland, A. Square would not see the sphere, but just 2 dimensional "slices". Taking this further, imagine if a sphere passed halfway through Flatland but stopped in the middle. the sphere would intersect. Flatland as just one circle and A. Square could see it! Furthermore, imagine if as the sphere approaches Flatland, A. Square watches as the sphere slowly moves through his plane. What would A. Square see? Recall that A. Square can only see 2-d slices of the sphere (or circles) so what A. Square would perceive is a circle suddenly appearing, then growing... then reaching a maximum size as the sphere was halfway through and as it exited, the circle would grow smaller until it disappeared. This means that 3d objects could be explained to a 2d being as a bunch of "slices stacked" on top of each other. Try to imagine taking a bunch of circles and stacking them. They would begin to form a skeleton framework of the actual 3d image. Similarly if a 4d hypersphere would intersect our plane of existence, we would see a 3d sphere appear out of no where. It would grow until the hypersphere was halfway through, then it would shrink back to nothing. Theoretically, we could stack these spheres to form a hypersphere, but we can't stack them in the usual sense, but rather it would have to extend in the fourth dimension which takes us back to the original dilemma of trying to visualize it.
Jinx
I've heard of the theory of tightly curled extra dimensions, but I have yet to read an explanation of what is meant by "tightly curled"? What does that mean? Does it mean that anything that (hypotheticaly) exists in a 4th spacial dimension is all tightly packed together like sardines in a can? Does it mean that if we can figure out how to move kata or ana (instead of left/right, up/down, back/forth) we could travel anywhere very quickly because things are close together in the 4th dimension?

Why are these extra dimensions so compact? If a human being were to make a wrong turn into the fourth dimension that they would be compressed to fit? (what i mean is: picture a shape drawn on a 2d piece of paper laying on a desk, pick that paper up so that it's standing on it's end relative to the desk, now that figure, wile still being 2d, has moved in the third dimension. Like a flatlander standing up on one of it's sides to peer out into the space above it's plane. The third dimension isn't compressed, so it wouldn't affect size, but if a three dimensional being suddenly "stood up" into the fourth spacial dimension, what would happen?)

And would time flow at the same rate in a higher dimension, or would it flow faster or slower from the viewpoint of someone with a third dimension pov?
Bikerman
Let me try to give you my understanding of this.
Superstring theory (M-theory) posits that as you go down in scale, beyond the macro scale that we see, beyond even the atomic scale (much smaller than atoms and nucleii) you get closer and closer to a fundamental limit called the Planck Length. Current physics theory suggests that this is the smallest distance about which we can know anything.
Now, conventional physics says that as we approach the planck length then everything is fairly normal - 3 spatial dimensions and one time dimension.
Superstring theory, on the other hand, posits that as we approach the planck length things get very complicated - there are an extra 6 spatial dimensions curled up into something called a Calabi-Yau manifold. This means that at the smallest scale of existence each 'point' in spacetime is actually a 6 dimensional space. Now, because this is at such a small scale, we don't experience these extra-dimensions (remember we are talking at a scale WAY smaller than atomic).
The question now becomes - is this science? If it IS science then there has to be a way to test the hypothesis - otherwise it is what I would call mathematical philosophy.
So, how can we test this hypothesis? Well, one interesting possibility is that as we get closer to the planck length then gravity changes. At the scales we are used to, gravity obeys the inverse-square law (ie it increases/decreases in proportion to the square of the distance). If there are an extra 6 dimensions at the smallest scales then one would expect gravity to stop obeying this relationship at very small scales - in fact one would expect gravity to suddenly become much stronger. This is actually the reason that some physcists have speculated that the LHC might produce microscopic black holes. Conventional physics would say that the gravitation would be insufficient to form black holes at the sort of energies (distances) that the LHC can produce but, if superstring theory is correct, then gravity is much larger at these energies (distances) and that might be enough to allow the formation of black holes (allbeit for a tiny instant of time).

Hope that helps....
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