Besides being a very popular R&B/soul/pop/psychedelia group in the late 60's, the fifth dimension is the one that, for me, starts the mindfuck. I can get my head around the orthogonality of time to the 3 dimensions we can perceive in our universe. So this cube moves through time, and as it does so, it travels in another dimension. Great. We don't know why time exists, or what it really is, but we all perceive it. In order for things to move, there must be time. OK. Once you tack on another dimension though, as physicists are want to do, I just don't understand where the dimension is supposed to go! What does it do? How do we move on it, and what does that mean? I know that thinking about these things really doesn't lead anywhere other than math problems I don't like to do, but I have to think these things.

The fifth dimension had me stumped a looong time. But then along came Hitchhiker's Guide to the Galaxy to clear it all up for me, specifically the last book in the series, Mostly Harmless. Douglas Adams described the fifth dimension as probability or in other words alternate realities. For instance, if somebody had killed my grandfather 60 years ago my father and myself would never have been born, and in another universe that just may be the case. Well, anyway, once I read Mostly Harmless it was all too clear on where this theory for the fifth dimension came from (Douglas Adams didn't really go into exactly what the fifth dimension is, I had to sit and think about it for quite a while). Basically the fifth dimension would just be several fourth dimensions stacked on top of each other (kinda like the 2nd dimension(y) is a bunch of 1st dimensions(x) stacked on top of each other, and the 3rd dimension(z) is just the whole (x,y) plane stuck together side by side). So, you take the timeline of the fourth dimension where every point is an entire (x,y,z) cartesian graph which is the universe and stack it on top of each other, kinda like a second (x,y) plane, except now its a (time, probability) plane. Since the 5th dimension is made up of an infinite number of timelines, it'd be logical to assume that each one of these lines is a whole other spin on what *could have* been and what *can* be.

I do not claim to be a great mathematician by any means. Nor do I claim to have discovered a grand insight into dimensional analysis or spatial theory or whatnot. I simply find that this is an interesting way for people who wish to imagine worlds with four of five dimensions to do so.

Here is a way to visualize objects with one dimension up to five dimensions.

A one dimensional world would be a line (x). An object populating a one dimensional world would be a point or line segment moving along a line in two degrees of freedom.

X <---------o---------->

A two dimensional world would be a plane (x,y). An object populating a two dimensional world would be a line segment with height in the form of a shape (i.e., square) sliding around the plane in four degrees of freedom.

  ^
  |
  |       ___
  |      |   |
  |      |___|
  |      
  |
  |
  |
 Xo-------------------->  
  Y

A three dimensional world would be a space (x,y,z). An object populating a three dimensional world would be a planar shape with depth (i.e., cube) moving around the space in six degrees of freedom. This could be a balloon.

  ^
  |             ______             
  |     _      /____ /|
  |     /|    |     | |
  |    /      |     | /
  |   /       |_____|/  
  |  /
  | /
  |/
 Xo-------------------->  
 ZY
 

A four dimensional world would be a hyperspace (d1,d2,d3,d4). An object populating a four dimensional world would be a cubic shape with depth2 (i.e., hypercube) moving around the space in eight degrees of freedom. Ok, this is kind of tricky. Because we don't live in a world with freedom to move through four dimensions, we have to express a four dimensional object in terms of three dimensions.  Image that the object below is a cube that can exist in three dimensions. Imagine that the cube is at Position 1. Then, the cube is moved, through time, to Position 2. The hypercube would be every progressive state of the cube during it's deepening2 from P1 to P2. So, imagine a line connecting AP1-AP2, BP1-BP2, and so on. A hypercube, therefore, is a four dimensional object represented in a three dimensional world. This could be a balloon between 12:00am and 12:10am existing all at once in three dimensions and at one time.

  ^                    <-"d4"->
  |               (P1)         (P2)
  |
  |            A______B     A______B
  |     _     D/____C/|    D/____C/|
  |     /|    | |   | |    | |   | |
  |    /      |F|___|_|E   |F|___|_|E
  |   /       |/____|/     |/____|/
  |  /        G     H      G     H
  | /
  |/
d1o-------------------------------------->  
d3d2  

Using the techniques to visualize a four dimensional object, another way to view a three dimensional object would be to envision a "hyper-square." This isn't really an appropriate name because hyper suggests four dimensions. Perhaps a more appropriate name would be a "trans-square." Basically, imagine a square that would have two positions, or sizes. The progression of this square from P1to P2 would be a three dimensional object (or in this case, a simple cube) represented in two dimensional space. Such a technique could be used to explain three dimensional objects to someone living in a two dimensional world.

  ^
  |     _________P2    _
  |    |\       /|     /|
  |    | \ ___ / |   "Z"
  |    |  | P1|  |  _/
  |    |  |___|  |   |
  |    | /     \ |  
  |    |/_______\|
  |
  |
 Xo-------------------->  
  Y
 

A five dimensional world would be a series of hyperspaces (d1,d2,d3,d4,d5). An object populating a five dimensional world would be a closed hypercubic shape with depth3 (i.e., a hypercube with an alternative P2 and/or P1) moving around the series of hyperspaces in ten degrees of freedom. Basically, imagine a cube "moving" or "growing" through three dimensional space with two or more possible ending positions, or states. Imagine the points AP1-AP2a-AP2b,  BP1-BP2a-BP2b, etc. are connected with planar triangles (as opposed to the lines that connect the hyper-points in a four dimensions object). Now, imagine that this object exists all at once and in three dimensions. This could be a balloon between 12:00am and 12:10am and that same balloon between 11:50am and 12:10am having been popped at 12:10am; the entire thing existing all at once in three dimensions (P2a, not popped; P2b, popped).

 

  ^                    <-"d4"->
  |               (P1)         (P2a)   <-"d5"->  (P2b)
  |
  |            A______B     A______B          A______B
  |     _     D/____C/|    D/____C/|         D/____C/|
  |     /|    | |   | |    | |   | |         | |   | |
  |    /      |F|___|_|E   |F|___|_|E        |F|___|_|E
  |   /       |/____|/     |/____|/          |/____|/
  |  /        G     H      G     H           G     H
  | /
  |/
d1o-------------------------------------------------->    
d3d2  
Another Dimension
Whether this is the "5th" dimension or some other dimension, I couldn't really say. Nor could I say what this dimension would exactly relate to, but it's there in the math. It seems that way to me anyways, according to Science's current take on Special Relativity.

The Lorenz Transformation is a matrix transformation by which we translate one frame of reference to another, in Special Relativity. If you spend enough time playing with this transformation in 2D using one dimension to represent space and one to represent time, you start to realize that as one accelerates (ie. changes from one reference frame to another) the 2D graph looks like it is being rotated in 3D. So, 3D is nice and simple, we see 3D every day of our lives, it is easy to comprehend. This makes the whole nature of acceleration in special relativity much more intuitive.

It only gets really weird when you realize that that 3rd dimension isn't time, and it isn't a spatial dimension. It's something else. Why? The 2D graph we are drawing already accounts for time and space albeit in a dumbed-down 2D instead of 4D sense.


I took a course at NYU where among lots of other really well formatted material we were exposed to a number of lectures on Special Relativity. It was during some of the lessons about the Lorenz Transform that I noticed it looked like a 3D rotation and did a nice simple mock-up in Maya to show it. I asked the Proffessor after class and he told me he remembered something about a paper that claimed something similar. I could find only semi-relevant info about this on the web at the specialrelativity Yahoo Group.

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