Excerpt from http://eusebeia.dyndns.org/4d/vis/01-intro#Is_it_possible_to_visualize_4D said:
Even though we are 3D beings who live in a 3D world, our eyes actually only see in 2D. Our retina has only a 2D surface area with which it can detect light coming into our eye. What our eye sees is in fact not 3D, but a 2D projection of the 3D world we are looking at.
In spite of this, we are quite able to grasp the concept of 3D. Our mind is quite facile at reconstructing a 3D model of the world around us from the 2D images seen by our retina. It does this by using indirect information in the 2D images such as light and shade, parallax, and previous experience. Even though our retina doesn't actually see 3D depth, we instinctively infer it. We have a very good intuitive grasp of what 3D is, to the point that we are normally quite unconscious of the fact we're only seeing in 2D.
You have once again confused sight with perception. Our perception is in 3D, our "sight" is in 2D, and both our eyes sense light, not space. All spacial awareness (length, width, and height) comes from the brain after the image is processed. Before the image is processed it is just a varied collection of colours with no meaning. All spacial awareness is inferred. We cannot see space.
Further interestingly so, all things 3rd dimensional would be really, really sharp to said creature X3
We could just slice through them like a mathematically perfect 2nd dimensional knife through a 3rd dimensional stick of butter!
Yet, all this 4th dimension stuff is giving me a slight headache.
Anyways, I think I'm actually starting to understand it. But I still don't understand what happens if you rotate a 4D object.
Yet, all this 4th dimension stuff is giving me a slight headache.
Anyways, I think I'm actually starting to understand it. But I still don't understand what happens if you rotate a 4D object.
[It's really massive, I have to spoiler the whole thing!]
When we think of rotations, we think of them as along an axis of sorts. But as you see, when we rotate this square in the 2nd dimension, it has no axis of rotation! There's only up/down and left/right, but there is no forwards/backwards for the axis of rotation to go through. It instead has a point which it rotates about.
Rotation is a property that begins in the 2nd dimension, since it takes at least 2 dimensions to rotate in. When we rotate a square in Flatland, there is no other extra dimension to be used, hence the 0D point of rotation. But when we rotate (Using only 2 dimensions) a cube in Cubeland, there's an extra dimension that is then used for the 1D line of rotation.
This further extends into the 4th dimensions. When you rotate (Still only using 2 dimensions) a hypercube, it has 2 free dimensions that are then used for a 2D plane of rotation. (In 5D, a 3D space of rotation o.o )
But instead of trying to describe rotation through the other dimensions, we can instead describe them more directly, as a planar phenomena. (Such a fun word to say!)
The square would rotate along the XY plane, while our cube could rotate in either XY, XZ, or YZ planes. As you see, all the points of the square/cube still rotate along the planes just like they did around an point/axis of rotation. This makes it easier to describe rotation and extrapolate it to the 4th dimension.
In the 2nd dimension there is only 1 principal rotation, in the 3rd there is 3 principal rotations, and in the 4th there's a whopping 6 such planes of rotation!
Now that that's been set, we can move on!
And now let's rotate a square along one of its edges.
This seems pretty reasonable to us, but what a Flatlander would see would be bizarre.
They would see some obscure shape that's morphing from a line, to a trapezoid on changing angles, into a square, back into a trapezoid, and turn inside-out. (To do the same thing on the other side) They would be completely baffled to the nature of such an object, and wouldn't believe us if we said it was only a square!
But what if we rotated a cube on its face in the 4th dimension?
It seems to be morphing from a flat object, into a trapezoidal pyramid of changing angles, to a cube, back, and again turn inside-out. And all the while, the cube has never deformed, since it was rotating along the XW plane.
More rotations if square and cubes if you'd like:
XY - XZ - YZ
XW - YW - ZW
And one last interesting tidbit about rotations. Since rotations are along a plane, (And we only used 2 per rotation) you can create another plane of rotation on a 4th dimensional object. These "double rotations" are called Clifford rotations, and mathematicians have known about them for a long time; it just wasn't easy to visualize.
So here ya go, a cube rotating in the XY and ZW planes!
If we still used the n-D of rotation, this one would have a 0D point of rotation since all the dimensions have been used for rotations. (In 4D, planes generally intersect at a point; weird, I know oWo )
Now we'll be taking to understanding the hypercube and its components.
If you were to project a cube when you look at it vertex first onto a 2D plane, you would see this:
But since our Flatlanders can only see the outside edge one, they'd see a hexagon! (They can't see the inside)
We can try to make this projection of a cube transparent, or put holes in it for doorways so that they can go inside and understand it.
The same thing applies if we project a hypercube when we look at its vertex into the 3rd dimension, except you'd see a rhombic dodecahedron:
As one could conjecture, it isn't the outside that is important, but the inside structure. We are missing a lot of important clues when we omit the insides. In fact, we can't even see the vertex we looked at!
But when we include the inside structure, you can then see the vertex (In yellow) which we oriented this hypercube towards the user.
Just like in 3D when we have 3 faces around a vertex in a cube, in 4D we have 4 cells around a vertex in a hypercube. Below are each of these cells individually highlighted.
All of these cells are perfectly perfect cubes, but because of the visual stretching artifact of projecting it into a dimension below, they don't look all to fine. These are only the front 4 facing cells, the other 4 behind have been omitted for clarity.
And finally, we can go and rotate a hypercube!
Here is a hypercube with one cell painted blue.
And here is the same hypercube rotated slightly.
You're probably surprised, but lets see the same thing on a 3D cube.
For easy comparison:
From cubes/squares they've become trapezoidal pyramids/trapezoids!
And next, just like there is a vertex at 2 lines, and an edge at two faces, in 4D they have a ridge at 2 cells. (Kinda forgot to incorporate this somewhere before >_<'' )
Now, we know where the closest edge is in the rotated cube, but where is the closet ridge in the rotate hypercube? Surprisingly, it's actually "inside"!
And when I say "inside", it is the same thing of when a Flatlander looks at our rotate cube projected onto their plane. The closest edge is inside the drawing, but in reality it just normally on the outside.
How about the farthest ridge on the rotated hypercube?
Rotated cube's farthest edge highlighted for comparison:
Now try to figure out what these 2 cells are analogous to in a cube:
It may not be obvious what it is representative of. We could create an analogy that these are like the top and bottom faces of a cube, but that isn't very insightful since we've already got a pair of those in the hypercube. The cube has 6 faces, but the hyper cube has 8 cells, so we have to simply accept that these 2 cells are a new phenomena (There's that word again!) in 4D. Just like the top and bottom faces of a cube, these are in-between the farthest and closest ridges.
Lastly, here is a hypercube with highlights to assist if seeing how the cells move about; unfortunately I couldn't find any decent gif :<
