Right. I was just saying that the image on the link (which is 2D/3D) could be constructed in real life (3D). As someone has already stated, it’s just a torus (donut) spinning on its coplanar axis.
There’s a visualization aid which I’ve sometimes used: imagine that you have a small sheet of glass which is acting like a TV screen or a laptop display. Anyone who watches TV is probably used to imagining the image is 3-D when it’s really 2-D, right? So pretend that’s the case here: the image on your sheet of glass is portraying a 3-D image in a 2-D space.
Now imagine that you have a whole stack of such sheets, each one displaying it’s own image. That would be a representation of a 4-D image in a 3-D space. If a person on one of those sheets moves around in the first three-dimensions, then that person is confined to his or her original sheet like any other TV character, but if they want to move perpendicular to the first three dimensions then they just have to move out of their starting sheet and into a different sheet in the stack.
You can’t visualize it because the 4th dimension is not part of your world, just as a 2-D Flatlander cannot understand the concept of “perpendicular” to his world (in the *z-*direction) because the 3rd dimension is not part of his. He cannot visualize something rising “up” out of his world (he knows only north-south and east-west), just as we cannot visualize something rising “up” (in a 4-D sense) out of ours (we know only north-south, east-west, and up-down. But we can understand it mathematically.
You can’t visualize it in more than three dimensions, but the concept of perpendicularity works just fine in any number of dimensions that you care to think about.
To really get into this, you need to know about vector spaces. For simplicity’s sake, I won’t give it the axiomatic treatment. If you have two vectors (a, b, c) and (d, e, f) and a real number g, you can define (a, b, c) + (d, e, f) as (a + d, b + e, c + f) and g * (a, b, c) as (ag, bg, cg), then you’ve got a three-dimensional vector space. It just so happens that the line from (0, 0, 0) to (a, b, c) is perpendicular to the line from (0, 0, 0) to (d, e, f) exactly when ad + be + cf = 0. Most notably, any two of (1, 0, 0), (0, 1, 0) and (0, 0, 1) are perpendicular to each other.
In four dimensions, the situation is about the same. Addition and scalar multiplication (the second operation I defined) are done exactly like you’d think, and perpendicularity is defined in a very similar manner; just add in a term corresponding to the product of the fourth entries in each vector. Generalizing to n dimensions works exactly the same way.
This would only work if we ourselves were three-dimensional, which we’re not. Humans, like almost everything with which we interact, are at least four dimensional, since we have a nonzero duration. If a being which treated all four of our dimensions as spatial were to pick you up and rotate you, your left and right would be reversed, but so would your morrowward and yesterward, so you’d be heading into the past instead of the future. Or alternately, if you were rotated through some other extra dimension, then your directions for that dimension would be reversed, though I can’t say if this would have any effect perceptible to us.
Maybe not with a rotation, but it could get that affect by reflecting you in the wyz plane.
Ok, I see it now. You’re right! The thing that was throwing me is that two of the cubes are turning themselves inside out. But if you project a rotating cube on to a 2-D surface, some of the squares will be doing the same thing. For a moment, they may even look like a line, much like, for a moment, some of the cubes in the animation look like a projection of a cube onto a 2-D surface. Thanks!
Didn’t Robert Heinlein write a story about a four-dimensional cube house? It was built as a cruciform shape; then there was an earthquake and the house folded itself up into a tesseract.
“And He Built a Crooked House.” It was mentioned in the other 4-D thread currently running.
We could never make a model that would be anything like a true 4-D object. The reason is that we actually see in 2-D. We see in polar coordinates. An object has a horizontal coordinate, a vertical coordinate, and a distance. We can only tell the distance because we have two eyes which look at objects from slightly different angles. The more different the angle, the closer the object is. Close one eye and we can only rely on context clues. (I think we do that mostly anyway.) So that’s why we can perceive three dimensional objects drawn on two dimensional planes. Depth is the only thing removed. Just the same, a four dimensional object could probably be displayed in three dimensions and be seen by a four dimensional as if it was really four dimensional, but we wouldn’t see it that way because our third dimension is depth which cannot replace an angular measurement. For us to be able to picture a four dimensional object would be like a man who had been blind all his life imagining the color green.
Actually, good sir, it was post #13 of this thread. Unless this thread itself is 4-dimensional and I perceive that it was post #13 but from your dimensional perspective, it looks like a different thread. (I think I just sprained my brain on that one. … 
)
bordelond, your extra conceptual question gets beyond where my younger geeky brain (decades ago when I took time to think about these sorts of things) could get to. I like the questions and need to see if I can reacquire some of my former thinking abilities. Of course, there are better brains than mine that have added good stuff to the discussion that I need to digest as well.
This all is just one of the reasons I love this Board. Where else will you get a comment like
or rational discussion about the complications of time travel.
And it’s included in the compilation Fantasia Mathematica, which I highly recommend for all its offbeat mathematically-inspired tales.
Oops. Sorry 'bout that.
D’oh! (But I’m a ma’am.) Tuesday was not a good day for multitasking.
I’ll second the recommendation of Fantasia Mathematica. I especially enjoyed the story about the island of five colors.
Six if you count the ocean.
Another great book treating dimensions is A.K. Dewdney’s The Planiverse. Dewdney used to run a newsletter about a two-dimensional universe along the lines of Abbott’s Flatland, but trying to be more realistic than having triangles and polygons as beings. Martin Gardner did a column on it. Years later, I think he took the best ideas from that newsletter and turned it into The Planiverse. The book describes the physics, chemistry, and engineering of a 2D world as perceived by us 3D beings. Very weird and interesting. You find that you can’t really appreciate a 2D world as the inhabitants can any more than you can picture a 4D world. 2D paintings are meaningless to 3D people. 2D sculpture can be appreciated, but not in the same way that the 2D inhabitants observe and appreciate it.
2D engineering and physics is odd. You can’t have a being with a digestive tract like ours – it would fall apart. for digestion and circulation, 2D beings have “zipper organs” that allow temporary passage without bisection. 2D circuits can’t be completed around anything or they’ll cut it off – al 2D electricity has to be by battery – power lines would destroy civilization. A 2d “stringed instrument” is really a drum, because that line restricts the air under it. 2D volleyball has an opaque net, because unless the material itself is transparent, you can’t see through it. 2D wars are a collection of single combats, because no more than two soldiers can confront each other at a time. 2D soil isn’t porous – everywhere two grains touch, they completely seal off everything beneath them.
and so on. That they could create a complex and workable 2D biology and culture is pretty impressive. At the end, they start to talk about higher dimensions.
You should have spoilered that!!! (I specifically did NOT mention it.)