Perspective of 3-dimensional objects from the 4th dimension

[wolfpax50]

This question will probably make the most sense to those who have read Edwin Abbott Abbott's novel Flatland. But I'm sure many others know the answer.

To explain, I'll have to use some dimensional analogy.

Let's say you're a 2-dimensional being. You live in a two dimensional world and thus you are limited to 1-dimensional sight. If you were, for example, to look at the flat side of a square, and I was to draw your field of vision, it would be a line. Just as, if a 3-dimensional being (like a human) being were to look at the flat side of a cube, and I was to draw its field of vision, it would be a square.

Now if you, as a lowly 2-dimensional being, were to be gifted the power of 2-dimensional sight (like a human), and were lifted upward and rotated, you could see the square you had been looking at from above. Now, rather than just seeing a side, you would be seeing the entirety of the square at once. If I was to draw your field of vision, it would be square.

Now let's say you're a 3-dimensional being (like a human). As I stated prior, if you were to look at the flat side of a cube, and I was to draw your field of vision, it would be a square. Now if you, the 3-dimensional being, were to be gifted 3-dimensional sight (as the 2-dimensional being was gifted 2-dimensional sight) and were lifted into the fourth dimension and rotated, you could see the cube you had been looking at from a new direction. Now, rather than just seeing a square, you would be seeing the entirety of the cube at once. If I was to draw your field of vision, it would be cube.

Now that that's over with, here is my question. If I were to draw your field of view as a 3-dimensional being, staring at the corner (not the side) of a solid cube, and were to draw successive views as your view is lifted and rotated across the fourth dimension. What would these views look like? At the end would the cube still be solid? What about halfway through the transformation?

If the first view is of a 2-dimensional representation of a 3-dimensional cube and the last view is of a 3-dimensional representation a 3-dimensional cube, and I were to draw all of this on 2-dimensional paper, did anything change? Is it possible to represent this change?

My head hurts.

Any input is appreciated. Please include pictures if possible.

 

[Bill Simpson]

This book
http://www.bestwebbuys.com/Experiments-in-Four-Dimensions-ISBN-9780830615414?isrc=b-search
is a series of exercises that the author really wants you to actually do, step by step. You can do them on paper, with the aid of a calculator or with a computer. He walks you carefully through 1D, 2D, 3D and finally, with the understanding and practice you have gained if you really did all the exercises, 4D. This sounds very much like what you are thinking of. But it has been many years since I last opened that and I don't remember the details.

There are other authors, particular in topology, who have described methods of visualizing 4D in various ways. I believe Steenrod has a chapter or section in one of his books on a method to do this.

A century or more ago, in the victorian tradition Charles Howard Hinton created and sold a little pamphlet with a set of colored cubes. He claimed that by carefully following his directions and doing the exercises you could learn to visualize 4D. Some who saw him wrote that he must either really be able to do this or he had just become extremely good at solving questions in 4D. But it is said that he had an incredible memory. He supposedly produced an even larger set of colored cubes later, but I am not sure if those were ever sold. I read somewhere that Rudy Rucker was able to reproduce the smaller set of cubes, but I have never found the details or been able to see the original directions from Hinton. A very very long time ago I did read reprints of several of Hinton's textbooks on 4D and you might be able to find a university library that has those buried in storage somewhere.