Hugh wrote:[...]
"In 4D, a shape can be rotated around a plane."
"It must be understood that in 4D a 3-dimensional cube has neither inside nor outside. All points of a cube are as much exposed in 4D as are the points of a square in 3D."
That's correct. Think of it this way: a 2D being confined to the 2D plane would consider that a square has an inside and an outside: the inside is the area surrounded by its 4 edges, and the outside is everything else in the 2D world. The 2D being cannot reach inside the square and pull out a piece of its interior, without first cutting through its edges and tunnelling all the way into where that piece lies, because 2D space does not afford any directions that could allow you to bypass the edges of the square and the stuff that surrounds that piece of the interior. So from a 2Der's point of view, the inside of the square is not exposed to the outside; you cannot reach in and take something out without first breaking a hole through its edges and digging through until you reach that thing.
Also, this square in 2D can only rotate around a point -- because it's confined to 2D space and has no freedom to undergo any other rotation. So in 2D, rotation around a point is the only possibility.
Now consider the very same square, this time immersed in 3D. Does it have an inside or an outside? Not really... well, it does, in the sense that there's a square-shaped area that sits between its 4 edges. But this area is now no longer an "inside", because now, from 3D, you can touch any part of the area without needing to break open its edges and digging through the surrounding material. You can easily cut out a piece of its interior without touching the rest of it.
Furthermore, a square immersed in 3D can rotate not only in the plane that it lies in; it can rotate around its edges, or its diagonals, or any of a whole bunch of different 3D rotational axes. Such rotations are impossible in 2D.
The square itself hasn't changed; the only difference is that it is now immersed in 3D rather than confined to 2D.
Similarly, when we handle a 3D cube in our 3D space, we find that it has an inside and an outside: you cannot reach inside the cube without first breaking through one of its 6 outer faces. And while 3D space affords us many ways to rotate the cube, there is no way to rotate it "around a plane" -- that's a 3D impossibility. However, if this same cube were immersed in 4D space, then things would be different. What we imagine to be its inside now becomes exposed to someone who has access to 4D. A 4D person could reach into (what we think is) the inside of the cube without passing through its 6 faces. The cube could also now rotate "around a plane" -- in a direction that we 3D people cannot perceive nor reach. The cube itself hasn't changed -- these are only the effects of the extra dimension of space that it can now access.
"Vacuously, in a square there is only 1 square that contains a given edge. In a cube, every edge is shared by 2 squares. In a tesseract, 3 squares meet at every edge. Taken pairwise, squares through the same edge define three cubes. Detecting the three cubes seems akin to shifting a view point when observing the Necker cube."
In the context of the page you linked to, the author here is talking about a particular projection of the tesseract, in which one is trying to identify where each of its 8 cells are. The Necker cube illusion here isn't strictly a Necker cube illusion, but a shift in viewpoint because we're trying to "see" cubes that are oriented in 4D space, which causes them to appear in a strange orientation on the screen because normally in 3D, cubes can't have those kinds of orientations. Their actual orientation is actually fixed; there is no real illusion here other than the artificial effect produced by our inability to see them in 4D.
"I found this observation useful when playing with the applet below. What is it about? Travelling in 4D may have a milder effect on a 3D body than turning it inside out. It may only change its orientation."
According to the context of the original page, the author is talking about the possibility of a 3D object getting flipped into its mirror-image in 4D. This is a purely mathematical device -- and comes from an early realization by Möbius, I believe it was, who realized that one could flip a 3D object into its mirror-image by a rigid rotation if only one could access a 4th dimension of space. This is similar to the idea of taking a 2D figure, say the letter R, and flipping it into Я. You can't do this in 2D, because no 2D rotation can do such a flip. But if you could rip the R out of the page, so to speak, then you could spin it around in 3D into Я and then stick it back onto the page.
This says nothing about the
physical possibility of flipping, of course: it's merely a mathematical device to illustrate the difference between 2D and 3D, or 3D and 4D. If we really
could flip through 4D space, then what happens if we don't perform the full 180° rotation required by the orientation flip, but only a 90° rotation? Then we'd be staring off into the 4th direction, and be in
neither orientation! And what about if we got interrupted before we get to the 90° point, say somebody startled us and we're left stuck at 36° from the 3D hyperplane? Now we'd be staring off into 4D in a diagonal direction, and as soon as we tried to move in any of the familiar 3D directions, we'd leave the 3D universe altogether. Since we have no idea how to navigate in 4D space, we'd quickly get lost and be unable to return to the 3D universe -- because even if we did manage to find where it is, we wouldn't know how to rotate ourselves into the right orientation to fit back in!
There's also the question of what we'd
see while in the
middle of a 180° rotation through 4D that flips our orientation. Clearly, if we really were undergoing such a rotation, then our line-of-sight would be outside the 3D hyperplane between the time that we start the rotation at 0° and the time that we end the rotation at 180°, and so we'd be looking off into 4D space somewhere from 1°, 2°, ..., to ... 178°, 179°. Obviously such a thing would be an absolutely fantastical visual experience of seeing things that don't exist in the 3D universe, and I'd say we'd
totally notice it if this happens! Now I can't speak for anybody else, but at least I've never experienced such a thing before, and I'd say that if somebody
doesn't experience such a thing, then in all likelihood there isn't any 4D rotation taking place, but it's probably something else.
Also, if somebody was looking at you while you're performing a rotation through 4D, they'd see your body morph and warp in an impossible way, and I'd say they'd totally freak out! Moreover, they won't be any
less freaked out after you return to 3D, because they'd see that you've just turned into your mirror-image -- your left hand is now your right hand, and vice versa, and the writing on your shirt has flipped into its mirror-image. (Of course, if I wanted to mess with you, I
could secretly swap my wedding ring to my right hand and swap my shirt with another one that has all the writing custom-printed backwards, and perhaps apply some cosmetics to cover up the scar on the right side of my lip and paint its mirror image on the left side, etc.. But I wouldn't be able to do that Terminator morphing thing that a true 4D rotation could do to my body, and I might have forgotten to switch my shoelaces so the kink in the right lace is still on my right shoe, so I probably won't be able to keep up the deception for very long.
)