The 4th Dimension


In everyday life, we are all limited to 3 axes of motion. What if we could move one axis furthur? To be able to enter the 4th dimension? This section discusses this thought, and some of my views on it.

What is the 4th dimension?

No, we are not talking about time here. Most of us have come to associate the 4th dimension with time from reading too much science fiction. That may be true, but the 4th dimension has a much more direct interpretation. Our world as many people know is 3 dimensional. That means that we have 3 perpendicular 'axes' of motion, in other words, we can move up and down, forwards and backwards, left and right. If we were to be only 2 dimensional beings, we would only be able to move forwards and backwards, left and right, or any combination of 2 of the three pairs of ranges of motion in 3 dimensions. If we were to be furthur limited to one dimension, then we would only have one range of motion.

Now, lets take a step forward. What about the 4th dimension? Well, for a start, we would have not 3 but 4 axes of motion. Imagine how much freedom we would have! But how is this possible? For us 3 dimensional beings, it is impossible to picture the 4th dimension because we have never been there, and even if we did our 3 dimensional minds would not be able to comprehend the 4 dimensional objects and 'see' them. In order to fully understand the 4th dimension in this section, analogies from the 2nd dimension to the 3rd dimension will be used. Using analogy, we will first try to get a 'feel' of 4-space, or 4 dimensional space.

Lower dimensions

Zero-dimensional space(Fig 1.) would be a point as it would have 0 axes of motion. Imagine a creature living in such a space. Its life would be rather dull, wouldn't it, being unable to move, having nothing to see. 1 dimensional space(Fig 2.) or a line starts to become more interesting. Motion is now possible among 1 axis. However, the beings living in such a world would not be able to pass each other, as they cannot go around pr over each other in 1-space. Going one step furthur, we come across two-space(Fig 3.) or a plane(surface). With 2 axes of motion, life becomes more bearable. However, they lack many things which are possible in three space. One of these things are knots, for 'lines' cannoth pass over each other. Now, we come to our familiar three-space(Fig 4.), or simply space. I'm sure I don't have to elaborate on it, having lived in such a space our entire lives.

.

Fig. 1 A point.


________________________________

Fig. 2 A line.


___________________________
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|                          |
|__________________________|

Fig. 3 A plane.


     __________________
      /                 /|
     /                 / |
    /                 /  |
   /                 /   |
  /                 /    |
 /                 /     |
/_________________/      |
|                 |      |
|                 |      |
|                 |      /
|                 |     / 
|                 |    /  
|                 |   /   
|                 |  /    
|                 | /     
|_________________|/

Fig. 4 3 dimensional space(a cube).


A geometrical view of the 4th dimension

Diagramming 4 space is a bit more interesting. Since we are 3 spatial, we cannot 'draw' out a 4 dimensional figure. However, we can draw 3 dimensional representations of the 4th dimension. One such example is the 4 dimensional cube, or 4-cube, or hypercube, or tesseract. Just as we can draw a 2 dimensional figure to represent the edges of a 3 dimensional cube(Fig 5.), we can also draw a 3 dimensional representation of the 4-cube(Fig 6.). Another example is the plan for a cube(Fig 7.) which is a T-like arrangement of squares. A similar plan(Fig 8.) made up of a T-like arrangement of cubes can be 'folded' to form the hypercube. Remember, a 3 dimensional object has no 'thickness' in 4 dimensions just as a 2 dimensional figure has no thickness in 3 dimensions which it does have in 2 dimensions. Thus a 3 dimensional figure can be folded in 4-space just like a piece of paper being folded through 3-space. 

____________________
|\                 /|
| \               / |
|  \_____________/  |
|  |             |  |
|  |             |  |
|  |             |  |
|  |             |  |
|  |             |  |
|  |             |  |
|  |_____________|  |
|  /             \  |
| /               \ |
|/_________________\|

Fig. 5 A 'flatenned' cube.


       ____________________
        /|\                _/|
       / | |             _// |
      /  | |            / /  |
     /   | |          _/ /   |
    /    | |        _/  /    |
   /     | |      _/   /     |
  /      | |_____/    /      |
 /       |/     /|   /       |
/________/_____/_|__/        |
|\______/_____/__|_/|_______ |
|       |     |  /__|_______\|
|       |     | /   |       / 
|      /|_____|/    |      /  
|     / /      \    |     /   
|    / /        \   |    /    
|   / /          \  |   /     
|  / /           |  |  /      
| / /             \ | /       
|/_/_______________\|/

Fig. 6 A 'flatenned' hypercube. A smaller cube in a larger cube with their corners connected.


       _____       
      |     |      
      |     |      
 _____|_____|_____ 
|     |     |     |
|     |     |     |
|_____|_____|_____|
      |     |      
      |     |      
      |_____|      
      |     |      
      |     |      
      |_____|

Fig. 7 A cube plan.


             ______        
            /     /|__     
           /_____/ | /|    
       ____|     | |/_|___ 
      /    |     | /     /|
     /_____|_____|/_____/ |
     |     /     /|     | |
     |    /_____/ |     | /
     |____|     | |_____|/ 
          |     | / |      
          |_____|/| /      
            |_____|/|      
            |     | |      
            |     | /      
            |_____|/

Fig. 8 A hypercube plan. Folded through 4-space.

Folding the hypercube


It may prove hard to understand how the hypercube(Fig 6.) can be folded from the plan in Fig. 8, so this sections purpose is to clarify that. First, well take a look at how Fig. 5 is folded from the cube plan in Fig 7. 

       _____                                                                 
      |     |                                                                
      |     |                                                                
 _____|_____|_____     _____ _____ _____     _____ _____     _____     _____ 
|     |     |     |   |     |\___/|     |   |\___/|     |   |\___/|   |\___/|
|     |     |     | ->|     |     |     | ->||    |     | ->||   || ->||___||
|_____|_____|_____|   |_____|_____|_____|   |/____|_____|   |/___\|   |/___\|
      |     |               |     |         |     |         |     |          
      |     |               |     |         |     |         |     |          
      |_____|               |_____|         |_____|         |_____|          
      |     |               |     |         |     |         |     |          
      |     |               |     |         |     |         |     |          
      |_____|               |_____|         |_____|         |_____|


Each of the outermost squares is 'turned" into the innermost square over the sides of the innermost squares. This is the motion as seen through 3 dimensions. In 2 dimensions, where the squares cannot rotate about the sides of the innermost square, they are seen as having 'flipped' into the innermost square. This 'flipping' will allow us to understand how the hypercube is folded from its 3-dimensional plan which follows:

        ______                                                      
       /     /|__                  ______                           
      /_____/ | /|                /     /|                          
  ____|     | |/_|___     _______/_____/_|___     ______     ______ 
 /    |     | /     /|   /      /\___//     /|   /\___//|   /\___//|
/_____|_____|/_____/ |  /______/_/__//_____/ |  /_/__// |  /_/__// |
|     /     /|     | |->|     /     /|     | |->|\___/|||->|\___/|||
|    /_____/ |     | /  |    /_____/ |     | /  | |__||||  | |__||||
|____|     | |_____|/   |____|     | |_____|/   |/___\|/|  |/___\|/ 
     |     | / |             |     | / |        |     | |           
     |_____|/| |             |_____|/| |        |     | |           
       |_____|/|               |_____|/|        |_____|/|           
       |     | |               |     | |        |     | |           
       |     | /               |     | /        |     | /           
       |_____|/                |_____|/         |_____|/
Each of the outer cubes in 'inverted' or 'flipped' to the other side of the innermost cube to form the hypercube. This is similar to folding the cubes into the innermost cube through the 4th dumension, as we can see by analogy from our previous floding of the cube.

Movement in the 4th dimension

Let's take a closer look at the diagrams. Notice that the 3 dimensional representations are all similar to the 2 dimensional representations. It shows the analogous realtionship between different dimensions which we will frequently use here. Just as the plan of the cube's 6 squares make up the 6 faces of the cube, the hypercube has 8 'faces' or rather spaces, as these faces are one less dimension, or 3 dimensional in nature. These spaces are shown as the 8 cubes in the plan(one hidden) but not the hypercube itself. The corners and edges of the hypercube can also be determined similarly from looking at those in Figure 6. We say that the diagram in figure 6 is 'topologically equivalent' to the hypercube, as they both essentially have the same connections between their spaces, faces, edges and corners. Looking at Figure 6 again, we see that moving along its edges from corner to corner is 'equivalent' to moving in 4 dimensions. From each corner, you can move horizontally forward or to the side, or vertically to another corner in the same cube. However, moving to the inside or outside cube constitues the motion through 4 dimensions, namely moving from one 3-space to another. Just as 3-space is made up of a series of 2-spaces 'stacked' together(Fig 9.) and 3-spatial movement constituting of all 2 spatial movement AND moving to the previous or next 2-space, the 4th dimension can also be imagined of being made up of a series of 3-spaces 'stacked' together(Fig 10.), with 4 spatial movement being all the 3-spatial movements and the abilty to move to the previous or next 3-space. This provides us with an even clearer idea of movement through 4-space. 

        __________
       /         /
      /         /_
     /    +    / /
    /_____|___/ /_
     /  <-0->  / /
    /____/|___/ /_
     /    +    / /
    /_________/ / 
     /         /  
    /_________/

Fig. 9 3 dimensions as 2 dimensional planes 'stacked' together. Motion includes forward, backward, left and right motion, as well as moving to the previous or next plane, marked with a +(up and down).


      ________       ________       ________       ________ 
     /       /|     /       /|     /       /|     /       /|
    /       / |    /       / |    /       / |    /       / |
   /_______/  |   /_______/  |   /_______/  |   /_______/  |
   |       |  |   |    |/ |  |   |       |  |   |       |  |
   |    +<-|--/---|   -0- |--/---|--->+  |  /   |       |  /
   |       | /    |   /|  | /    |       | /    |       | / 
   |_______|/     |_______|/     |_______|/     |_______|/

Fig. 10 4 dimensions as 3 dimensional spaces 'stacked' together. Motion includes up, down, left right, forward and backward motion, as well as to the previous or next 3-space(4th dimensional movement).


The possibilities of the 4th dimension

Now that we have a firm grasp of this concept of a 4th dimension and what motion in such a space is like, what are the possibilities that the 4th dimension holds? For a start, one could disappear from 3-space by simply entering the 4th dimension through a direction that we cannot see as 3-dimensional beings. Then, we could choose to reappear anywhere in our 3 dimensional space without having to pass through 3-space. You can imagine this by picturing the 2 dimensional being who suddenly disappears from the 2nd dimension by entering 3-space, then entering back into 2 space wherever he wants without having to go through 2-space. To his 2 spatial friends, he would seem to disappear suddenly and then appear again from out of nowhere. Explains how David Copperfield does it, don't you think? However, it's probably very dangerous to do this. Suppose you were a cuboid. In 3-space your volume would be your length multiplied by your breadth multiplied by your breadth. However, in 4 dimensions, your 'hypervolume' is essentially zero, for your 'depth', or your 4th axis would be zero, just as a 2 dimensional object has a volume(which we call area) in 2 space but none whatsoever in 3-space because its thickness is 0. Thus we would be nothing whatsoever in 4-space. The reason why we take up any space at all is because we are bounded by the limits of 3-space. An alternative to entering 4-space for unobstructed travel would be to 'fold' space' or instead of moving the person, one brings the destination to the person. Imagine a 2-dimensional being on a flat piece of paper. To get to one end of the paper to the other, it would have to move across the sheet of paper. An alternative to this would be to fold the paper such that the corners touch, bringing the corner to the being instead of having to move him there. We could similarly 'fold' 3-space through 4 space, bringing our destination to us instead of moving there. The energy cost of doing this should be practically zero, as our 3-space is nothing in 4-space, having no 'depth'. This method also has its dangers, as such folding may contort and damage the stability of 3-space and excessive use may even prevent normal 3-spatial movement.

Four dimensions may not always be advantageous to our 3-space though. Life would be extremely complicated with that one other axis of movement. Knots using strings(lines) would be impossible, as moving them through 4-space would untangle them immediately. I suspect however, that knots with surfaces(planes) are possible in the 4th dimension. People would have to see and understand things 3 dimensionally instead of 2 dimensionally like us. Machines would be so much more complicated, having a 4th axis of motion to work on. Mathematics would also become a harder to learn subject, with geometry, trigonometry, topology and many other subjects brought to greater heights. Their world would be so many times larger than ours. Imagine what a map would be like. One could so easily get lost in 4 space, with all the freedom of motion. In fact, having a 4th dimension means that there are far more possibilties in which molecules can be formed, giving rise to many new materials. The world would be such a complicated place. Now aren't you glad that you can stick to these simple 3-dimensions?

The 4th dimension as time

The idea of time being the 4th dimension is believed by many. Although what I have discussed so far may seem to contradict this notion, I can demonstrate otherwise. This is only a theory of mine and has not been demonstrated to be true, so take it with an open mind. Imagine all the 3-spaces at every moment in time. You would get an infinite number of sich spaces, each of which depicts the situation of the world at exactly one moment in time. You could call each of them a '3 dimensional snapshot' of the world. Now, stack all these 'photographs of time' together in the 4th dimension. We would have created a 4 space made up of an infinity of 3 spaces, each of which is a moment in time. 'Time' would then be the movement of a 3 space through this 4-space, at each moment passing through one of the 'stacked' 3-spaces and thereby taking its form. This may be hard to understand, so let's use a dimensional analogy. Imagine a string of 2-spaces capturing every moment of time. Stack them chronologically one on top of the other to form a 3-space. Now, by moving another plane(2-space) through this 3-space from bottom up, the images captured on this 2-space as it moves would be time, as it passes from one image of time to the next. A similar photographing, stacking and moving procedure could account for time as we know of in our 3 dimensional world. So time may be related to the 4th dimension discussed here after all. Both these concepts could be correct, and linked to each other.

Higher dimensions

Now that we have discussed the 4th dimension, I'm sure that you are hungry for more. What about the 5th dimension? The 6th dimension? Even higher dimensions? To start you off, here is a means of picturing such higher dimensions. Using the 'stacking' of 3-spaces to form a 4-space, we will again use this idea to understand the 5th dimension. The 5th dimension can be split up into 4th-dimensions stacked onto each other(Fig 11.), the 6th dimension being 5th dimensions stacked on each other and so on and so forth. In general, the n th dimension can be depicted as being made up of n-1 th dimensions stacked on each other. We can then furthur break down these n-1 th dimensions into n-2 th dimensions and so on and so forth until we arrive at our all familiar 3rd dimension. Using this method, we can thus reduce higher dimensions to being made up of numerous 3rd dimensions, stacked together to form higher and higher dimensions. Again, the same rules of movement apply: for an n th dimension, one can move in all directions available in the n-1 th dimension, and to the next or previous n-1 th dimension. The movement in the n-1 th dimension can be broken down again, and so on and so forth. Well, I have talked about all the dimensions now, ranging from the lowliest of them all, 0 dimensions, all the way to infinity. These higher dimensions are now for you to think about.

      ________       ________       ________       ________ 
     /       /|     /       /|     /       /|     /       /|
    /       / |    /       / |    /       / |    /       / |
   /_______/  |   /_______/  |   /_______/  |   /_______/  |
   |       |  |   |       |  |   |       |  |   |       |  |
   |       |  /   |   +   |  /   |       |  /   |       |  /
   |       | /    |   |   | /    |       | /    |       | / 
   |_______|/     |___|___|/     |_______|/     |_______|/  
----------------------|-------------------------------------
      ________       _|______       ________       ________ 
     /       /|     /       /|     /       /|     /       /|
    /       / |    /       / |    /       / |    /       / |
   /_______/  |   /_______/  |   /_______/  |   /_______/  |
   |       |  |   |   |/  |  |   |       |  |   |       |  |
   |   +<--|--/---|  -0-  |--/---|-->+   |  /   |       |  /
   |       | /    |  /|   | /    |       | /    |       | / 
   |_______|/     |_______|/     |_______|/     |_______|/  
----------------------|-------------------------------------
      ________       _|______       ________       ________ 
     /       /|     / |     /|     /       /|     /       /|
    /       / |    /  |    / |    /       / |    /       / |
   /_______/  |   /___|___/  |   /_______/  |   /_______/  |
   |       |  |   |   |   |  |   |       |  |   |       |  |
   |       |  /   |   +   |  /   |       |  /   |       |  /
   |       | /    |       | /    |       | /    |       | / 
   |_______|/     |_______|/     |_______|/     |_______|/  
------------------------------------------------------------
      ________       ________       ________       ________ 
     /       /|     /       /|     /       /|     /       /|
    /       / |    /       / |    /       / |    /       / |
   /_______/  |   /_______/  |   /_______/  |   /_______/  |
   |       |  |   |       |  |   |       |  |   |       |  |
   |       |  /   |       |  /   |       |  /   |       |  /
   |       | /    |       | /    |       | /    |       | / 
   |_______|/     |_______|/     |_______|/     |_______|/

Fig. 11 5 dimensions split up into 'double stacked' 3 dimensions. The dotted lines show its division into stacked 4-dimensions which are furthur divided into stacked 3-dimensions. In 5 dimensions, one can move in the usual 3 axes, as well as to the next or previous 3-dimension(far extending left and right arrows) or to the next or previous 4th dimension(far extending up and down lines).


Furthur topics:

4th dimensional geometry(yet to come)
A more technical view of the 4th dimension(yet to come)



This page is the first of many mathematics pages that I will create and upload, to bring such wonderful and fascinating ideas to the world. However, I'm not sure how it will go down with the general public. Should you feel that my page is too technical, too difficult to read or the text graphics messy and incomprehensible, please tell me about it and if possible suggest changes. GIF graphics for the diagrams will be available soon. Remember, this page talks about the 4th dimension as only a 'theory'. It may not be true that higher dimensions do exist, in fact we may never even know. Please do not take what has been said here to be the truth, rather it should be considered as an idea for one to think about.

Anything you don't understand in this page or feel should be included? E-mail me. If you have any opinions that you would like to put forth, don't hesitate to tell me about them.

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