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
(d

_{1},d

_{2},d

_{3},d

_{4}). An object populating a four dimensional world would be a cubic shape with depth

_{2}(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 deepening

_{2}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 (d

_{1},d

_{2},d

_{3},d

_{4},d

_{5}). An object populating a five dimensional world would be a closed hypercubic shape with depth

_{3}(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