I am one and a half years late in joining this forum, and I might not have read the entire page tonight, so please excuse me if the same thing has already been posted. And please excuse me, if any of you have changed your opinions on the matter since this was posted, since my reply is based on your posts one and a half years ago. I have commented on a few quotes of 4Dspace and wendy.

4Dspace wrote:And so, returning to the "problem", i.e. a well-defined cube made of 6 square planes (the exact definition came about not from the start but in the process of discussion in that thread), the task was to see this cube, colored red outside and green inside, in 4D. Of course, in 3D, we cannot see that it is colored green inside. But, having moved our cube into 4D, we can easily see inside it and thus can tell that its 6 planes are colored green inside and red outside.

Outside to what, one may ask. Why, the answer is the same as in 3D: outside to the inner 3d-space this cube occupies. It is still there. It is not 0 as in 3D analogy.

Given the fact that all 6 faces of the cube are visible in their entirety from most positions in 4D, what color they appear to a fixed POV in 4D?

And here, I believe from what you posted above, you would say that some faces will be seen green (the counterclockwise aspect of a writing on its face), and some, red. Agree?

First of all, we don't "move the cube into 4D" (This might be a misunderstanding of terms). We could see the interior as well as the exterior of the cube only with a 3D array of retina cells at the outermost layer. And from such a POV, we would not see some faces as green and the rest as red. First, I would like to emphasize that, in the hollow cube you mention, the fact that the outer side of a face can be coloured differently from the inner side shows that there is a small distance between the two sides, i.e. they are not on the same plane and are not the same square. And I

have to use analogy to present my opinion on this. I have given an image, which is of a square, under similar conditions as that of your cube. (My apologies if the image is not successfully posted.)

[img]ikariasquare.bmp[/img]

Suppose there was a Flatlander on the same plane as this square. The Flatlander would see only the outer red side of the square's perimeter. And the faces of the cube as seen from the 4D POV would be similar to the way we see the edges of this square. We cannot say that some sides are red and the rest are green. Similarly, we cannot say that some faces of your cube are red while the others are green.

4Dspace wrote:And so, how do you see the 6 faces of such a cube in 4D? I say, half are seen red and half, green. That's 3 and 3. The 3 'near' ones are red and 3 'far' ones are green.

Now, what's near and far? A hyperplane, which is a 3D object defined by 3 bounding 2d planes orthogonal to each other, separates 4D into 2 halves. It's a wall you cannot penetrate. If the wall is not infinite (as the case is with our cube == it's a bounded 3d subspace in 4D), then you can walk around it. But it's still a "wall" in the sense that you cannot walk through it, and, unless it is transparent, you can't see its other side. You POV determines which side is seen. One at a time.

So, a cube in 4D is bound by "2 parallel hyperplanes" (both parallel to each other and perpendicular to the POV). These two hyperplanes carve out a 3D subspace out of the 4D space they are in. If here we are to use an analogy with 3D, this would be equivalent to two 2d-planes sandwiching a section of 2D space between. Seems redundant, since the 2D space has neither direction nor length in the 3rd dimension (==from 3D POV). Yet, in this analogy that's how it is, a 0. And so, going out of the analogy into what is, we realize that 0 (which is the total volume a 2D plane occupies in 3D) is not equal to x cubed in 4D. Here we make a leap from 2D seen from 3D, to 3D as seen from 4D. There is no correspondence in this step. Here the analogy is misleading: A gazillion of 2d-planes will never amount to a cube in 3D. Simply cause 0 times a gazillion is still 0. But, funny enough (!) eight 3d cubes in 4D do in fact make up a tesseract, a bona fide, real, 4d-object.

I agree that "A gazillion of 2d-planes will never amount to a cube in 3D. Simply cause 0 times a gazillion is still 0." But if it is zero times infinite, then it can amount to any finite value and not zero. This concept should be familiar to anybody who knows calculus. This means that a cube is made of infinite squares stacked together. And as for your statement, "eight 3d cubes in 4D do in fact make up a tesseract, a bona fide, real, 4d-object", it is just as sensible as saying that six 2d squares in 3d do in fact make up a cube, a bonafide, real, 3d object. In other words, the six squares just make up a boundary of the cube, not the entire thing. Similarly, eight cubes just make up the boundary of a tesseract and it has another interior which is 4D and each point inside it can be represented by 4 Cartesian coordinates. You might feel that I am going to the very basic concepts, but I felt that it was necessary from what I understood from your post. It is obvious that you cannot visualise the fourth dimension (neither can I, though the concepts are familiar to me and I find no fault with any of the statements that you have contradicted).

wendy wrote:This view comes from looking at a squashed polytope too. No, it makes a hollow box or the surface. You need to add bulk or substance to it to make a real 4d object, just as you have to fill the six squares to make a cube. Otherwise, it's a drawing on a peice of paper.

I agree with this statement by wendy. Eight cubes connected by their surfaces make a hollow tesseract, and a solid tesseract is made when infinite cubes are stacked together in the "Upsilon"/"Delta" directions.

4Dspace wrote:wendy, this is the last time I answer your posts, since I do not see a point talking to you, a self-appointed expert in seeing higher dimensions, who is incompetent at visualizing a simple cube in 4D. This simple exercise revealed the truth. You claim that you "see up to 8D". Of course, no-one can get into your head and see what you see, but from what you posted above it appears that you are simply deluding yourself.

It is difficult for me to believe that a 3D human can see upto 8D. But where is the citation by 4Dspace which claims that wendy can see eight dimensions? Is this an exaggeration?