Freethought & Rationalism Archive

Aldehyde · 07-31-2002, 03:41 PM

A supid question nonetheless. What exactly is the 4th dimension? Some people say it's time and that we live in the 4th dimension instead of the third. Yet others say it's in a parellel universe beyond many's (especially my) understanding (eg objects such as the tesseract.)

Secular Elation · 07-31-2002, 03:44 PM

I'm having difficulty visualizing 4 dimensions. It is clear what 3 dimension is: for example, a cube: length, width, height, as represented by, say, (x,y,z) if we were to graph a cube. I can't imagine a four-variable sistem.

tronvillain · 07-31-2002, 04:35 PM

I suspect that it is impossible to visualize four dimensions, though it can be represented other ways. A three dimensional figure can change over time as a fourth variable changes, or you can show many three dimensional figures, each of which corresponds to a different fourth variable, and so on. Time is often treated as a dimension, so it might be accurately labelled "the fourth dimension."

Undercurrent · 07-31-2002, 05:01 PM

It's really not that complex. A four dimensional space is one that requires four numbers to describe it. Even before Einstein, people knew that it took four numbers to specify a point in space and time. Be they x,y,z (rectagular) coordinates around some origin and time, or r,theta,phi (polar) coordinates and time, or some more exoitc corrdinates and time, you always need three numbers to specify the position and one to specify the time.

What was tacitly assumed before Einstein, however, is that you could always change coordinate systems in space without affecting the coordinate system in time. That is, any sensible transformations of coordinates you made in space would only involve the space coordinates, and not the time coordinate.

What Einstein posulated is that transformations of coordinates could also include the time coordinate as well. Thus t was connected to x, y, and z in a way not thought of before.

m.

Abacus · 07-31-2002, 06:13 PM

If String Theory is correct, then it may turn out that we live in a 10 or even 11 dimensional universe.

Don't blow a fuse trying to visualize that.

echidna · 07-31-2002, 06:14 PM

Quote:

Originally posted by tronvillain:
<strong>I suspect that it is impossible to visualize four dimensions, though it can be represented other ways. A three dimensional figure can change over time as a fourth variable changes, or you can show many three dimensional figures, each of which corresponds to a different fourth variable, and so on. Time is often treated as a dimension, so it might be accurately labelled "the fourth dimension."</strong>

Actually it�s easier than one might guess. Try this as a projection of a 4-D hypercube. :

<a href="http://dogfeathers.com/java/hyprcube.html" target="_blank">http://dogfeathers.com/java/hyprcube.html</a>

Similarly, the way one forms a crucifix by flattening a 3-D cube onto a 2-D surface, one can unwrap a 4-D hypercube into 3-D space. It becomes a block form of the crucifix (6 blocks) with 2 extra cubes perpendicular to the 2 individual cubes (if that makes any sense).

echidna · 07-31-2002, 06:19 PM

Quote:

Originally posted by Random Number Generator:
<strong>If String Theory is correct, then it may turn out that we live in a 10 or even 11 dimensional universe.

Don't blow a fuse trying to visualize that.</strong>

Consider ourselves lucky. Back in the 80�s the N=23 Supergravity Theory was quite popular where N was the number of dimensions required, of which 2 were chronological !!!

Aldehyde · 07-31-2002, 06:26 PM

I find visualizing the hypercube (or tesseract) very difficult. I think it's because I lack the mathematical mind.

Quote:

From dictionary.com:

hypercube

A cube of more than three dimensions. A single (2^0 = 1) point (or "node") can be considered as a zero dimensional cube, two (2^1) nodes joined by a line (or "edge") are a one dimensional cube, four (2^2) nodes arranged in a square are a two dimensional cube and eight (2^3) nodes are an ordinary three dimensional cube. Continuing this geometric progression, the first hypercube has 2^4 = 16 nodes and is a four dimensional shape (a "four-cube") and an N dimensional cube has 2^N nodes (an "N-cube"). To make an N+1 dimensional cube, take two N dimensional cubes and join each node on one cube to the corresponding node on the other. A four-cube can be visualised as a three-cube with a smaller three-cube centred inside it with edges radiating diagonally out (in the fourth dimension) from each node on the inner cube to the corresponding node on the outer cube.

Each node in an N dimensional cube is directly connected to N other nodes. We can identify each node by a set of N Cartesian coordinates where each coordinate is either zero or one. Two node will be directly connected if they differ in only one coordinate.

The simple, regular geometrical structure and the close relationship between the coordinate system and binary numbers make the hypercube an appropriate topology for a parallel computer interconnection network. The fact that the number of directly connected, "nearest neighbour", nodes increases with the total size of the network is also highly desirable for a parallel computer.

I do not understand the above without thinking about it, and I do not wish to. The time concept as the another dimension seems more comprehendable. But if time = another dimension, then do we live in at least a 4th dimension? I really think I should read up on the String Theory.

Secular Elation · 07-31-2002, 06:54 PM

I did not mean to say that 4 dimensions was bunk. I only stated it was difficult to visualize, but that is of course no reason to dismiss it.

DNAunion · 07-31-2002, 07:02 PM

DNAunion: Michael gave a good answer. I will just add some. (Please note that this is off the top of my head people - I don't mean this to be the ultimate disseration on the matter!).

Yes, time is the fourth dimension. Originally, it was believe there was three-dimensional space and there was time: two distinct things. Then Einstein found that (or was it Minkowski, who was Einstein's mathematics professor, who found this based on Einstein's special theory of relativity) it is not space and time, separately, but rather space and time fused together into four dimensional space-time. (Minkowski said something like, "From this point forward, neither space nor time will persist, only a union of the two shall". Now I just know people will look up the quote and hammer me because I didn't get it exact....oh well).

Now, keeping with motion in three dimensional space for a moment... First, note that the term vector just means a quantity with both magnitude (such as 50 mph) and direction (such as 30 degrees north of east).

Okay, if an object has a velocity of 50 mph and is traveling due East, then the full 50 mph will belong to West-East direction (dimension). But if that object instead moved at an angle of 30 degrees to the North of due East, then the total velocity vector would be broken up into two components - an East-West component and a North-South component. The magnitude of the East-West velocity vector would be 50 mph * the cosine of 30 degrees = 43.3 mph, while the magnitude of the North-South velocity vector would be 50 mph * the sin of 30 degrees = 25 mph. Note that the magnitude of both component vectors is less than the total 50 mph vector. This only makes sense, since the total velocity is being split into two directions (dimensions), the velocity in each direction (dimension) is less than the total.

So what's the point? Let's take this to four dimensions.

All objects move through four dimensional space-time at the speed of light (this according to "The Elegant Universe"). If an object is at rest, that is, it is not moving through three-dimensional space (ignore what is called peculiar motion), then its full velocity belongs to the time dimension. On the other hand, an object that is moving through space is sharing its motion through time with at least one spatial dimension. Thus, the total velocity, c, is shared among multiple dimensions and the magnitude of each vector cannot be as large as the original total velocity. Remember, time is one of the dimensions sharing that motion: yes, time no longer has a monopoly. Thus, the "velocity" through the time dimension is less for an object in motion than for an object at rest. This is relativistic time dilation.

07-31-2002, 03:41 PM	#1
Aldehyde Junior Member Join Date: Jul 2002 Location: Vancouver, Canada Posts: 24	The 4th Dimension A supid question nonetheless. What exactly is the 4th dimension? Some people say it's time and that we live in the 4th dimension instead of the third. Yet others say it's in a parellel universe beyond many's (especially my) understanding (eg objects such as the tesseract.)

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07-31-2002, 03:44 PM	#2
Secular Elation Veteran Member Join Date: Apr 2001 Location: California Posts: 6,196	I'm having difficulty visualizing 4 dimensions. It is clear what 3 dimension is: for example, a cube: length, width, height, as represented by, say, (x,y,z) if we were to graph a cube. I can't imagine a four-variable sistem.

07-31-2002, 04:35 PM	#3
tronvillain Veteran Member Join Date: Oct 2000 Location: Alberta, Canada Posts: 5,658	I suspect that it is impossible to visualize four dimensions, though it can be represented other ways. A three dimensional figure can change over time as a fourth variable changes, or you can show many three dimensional figures, each of which corresponds to a different fourth variable, and so on. Time is often treated as a dimension, so it might be accurately labelled "the fourth dimension."

07-31-2002, 05:01 PM	#4
Undercurrent Veteran Member Join Date: Jan 2001 Location: Santa Fe, NM Posts: 2,362	It's really not that complex. A four dimensional space is one that requires four numbers to describe it. Even before Einstein, people knew that it took four numbers to specify a point in space and time. Be they x,y,z (rectagular) coordinates around some origin and time, or r,theta,phi (polar) coordinates and time, or some more exoitc corrdinates and time, you always need three numbers to specify the position and one to specify the time. What was tacitly assumed before Einstein, however, is that you could always change coordinate systems in space without affecting the coordinate system in time. That is, any sensible transformations of coordinates you made in space would only involve the space coordinates, and not the time coordinate. What Einstein posulated is that transformations of coordinates could also include the time coordinate as well. Thus t was connected to x, y, and z in a way not thought of before. m.

07-31-2002, 06:13 PM	#5
Abacus Veteran Member Join Date: Mar 2002 Location: South Dakota Posts: 2,214	If String Theory is correct, then it may turn out that we live in a 10 or even 11 dimensional universe. Don't blow a fuse trying to visualize that.

07-31-2002, 06:54 PM	#9
Secular Elation Veteran Member Join Date: Apr 2001 Location: California Posts: 6,196	I did not mean to say that 4 dimensions was bunk. I only stated it was difficult to visualize, but that is of course no reason to dismiss it.

07-31-2002, 07:02 PM	#10
DNAunion Veteran Member Join Date: Jan 2001 Location: USA Posts: 1,072	DNAunion: Michael gave a good answer. I will just add some. (Please note that this is off the top of my head people - I don't mean this to be the ultimate disseration on the matter!). Yes, time is the fourth dimension. Originally, it was believe there was three-dimensional space and there was time: two distinct things. Then Einstein found that (or was it Minkowski, who was Einstein's mathematics professor, who found this based on Einstein's special theory of relativity) it is not space and time, separately, but rather space and time fused together into four dimensional space-time. (Minkowski said something like, "From this point forward, neither space nor time will persist, only a union of the two shall". Now I just know people will look up the quote and hammer me because I didn't get it exact....oh well). Now, keeping with motion in three dimensional space for a moment... First, note that the term vector just means a quantity with both magnitude (such as 50 mph) and direction (such as 30 degrees north of east). Okay, if an object has a velocity of 50 mph and is traveling due East, then the full 50 mph will belong to West-East direction (dimension). But if that object instead moved at an angle of 30 degrees to the North of due East, then the total velocity vector would be broken up into two components - an East-West component and a North-South component. The magnitude of the East-West velocity vector would be 50 mph * the cosine of 30 degrees = 43.3 mph, while the magnitude of the North-South velocity vector would be 50 mph * the sin of 30 degrees = 25 mph. Note that the magnitude of both component vectors is less than the total 50 mph vector. This only makes sense, since the total velocity is being split into two directions (dimensions), the velocity in each direction (dimension) is less than the total. So what's the point? Let's take this to four dimensions. All objects move through four dimensional space-time at the speed of light (this according to "The Elegant Universe"). If an object is at rest, that is, it is not moving through three-dimensional space (ignore what is called peculiar motion), then its full velocity belongs to the time dimension. On the other hand, an object that is moving through space is sharing its motion through time with at least one spatial dimension. Thus, the total velocity, c, is shared among multiple dimensions and the magnitude of each vector cannot be as large as the original total velocity. Remember, time is one of the dimensions sharing that motion: yes, time no longer has a monopoly. Thus, the "velocity" through the time dimension is less for an object in motion than for an object at rest. This is relativistic time dilation.

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