Fifth dimension (idea)

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 A_P1-A_P2, B_P1-B_P2, 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 A_P1-A_P2a-A_P2b,  B_P1-B_P2a-B_P2b, 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