Dimensions... again

In a previous post, I was wondering if it was possible to describe a dimension solely in terms understandable by the previous dimension(s) (i.e. describe (n)D in terms of (n-1)D). While thinking about this, I stumble on something interesting. But first lets start with some basic setup:

The Point
It all starts with the point. We say that the point is in 0D (no dimension).

The Line
To make a line starting from that point we can go in any directions around it. We than stretch that point into that direction and draw a line between the two points. We can then say that we had total freedom of movement to create that line. We would also be drawn to conclude that this freedom of movement was in a 3D volume. (Spherical coordinate system.)

The Square
Making the square is a similar process but this time we have stay perpendicular to the previous line. We then stretch the line anywhere around the original line and draw two extra lines liking the copied points. We had some freedom of movement to create that square and conclude that this freedom of movement was in a 2D plane. (Circular coordinate system)

The Cube
Repeating the procedure again will generate a cube. This time our movement have to stay perpendicular to the squares plane. We then stretch the square anywhere along a perpendicular line from the plane and draw four lines liking the copied points. Again, we had some freedom of movement to create that square. We would conclude that this freedom of movement was on a 1D line. (Linear coordinate system)


Looking back at what happen, we can see that as the dimensions are increasing from 0D to 3D the freedoms of movement decreases from 3D to 1D, leaving no dimension to continue stretching the cube into a possible next dimension. In a way, I was somewhat happy. It looked like a perfect circle, from 1D to 3D and back. I showed those results to my wife and the first thing she thought was “Of course you start describing the points freedom of movement as 3D because that's the max for you, they would not say that in line world”. She was right.

I ask myself: how would 1D and 2D describe the same transformation?

The Line
The line would say that the point as no dimension (0D). It would say that to make a line, you have to stretch the point into the first dimension. It then stretch the point anywhere along the existing Line World (1D) and draw a line between the two points. It can say that It had some freedom of movement to create that line. It would conclude that this freedom of movement was on a 1D line.

The Square
The square would also say that the point as no dimension (0D). It would also agree that to make a line, you have to stretch the point into 1D. It then stretch the point anywhere around the original point, while staying in the squares plane, and draw a line between the two points. It can say that It had some freedom of movement creating that line. It would conclude that this freedom of movement was in a 2D plane.
The square would then be able to go one step further than the 1D Line, by stretching the newly created line into it's 2D world with a freedom on a 1D line.


Since the maximum freedom of movement is bounded by our knowledge we cannot picture stretching beyond that limit. But, we as 3D beings, know that there is a second dimension even though the Line World can't see it.
A 4D being would say that the freedom of movement of the point, when creating the line, is in four dimensions. It would also be able to stretch the cube into the fourth dimension and understand why we can't see it.

Can that scale of dimensions be infinite?