4d paper

[papernuke]

Is four dimensional paper a three dimensional box? In the box we (three dimensional beings) put our science projects and all that junk, but can a four dimensional being just draw our science project on their paper? Somehow in it like how we put our things in there?

[quickfur]

Is four dimensional paper a three dimensional box? In the box we (three dimensional beings) put our science projects and all that junk, but can a four dimensional being just draw our science project on their paper? Somehow in it like how we put our things in there?

The way to understand this is to use dimensional analogy, as usual. We shall first consider the 2D->3D case, and then we'll use dimensional analogy to look at the 3D->4D case.

2D->3D

Suppose a 2D being lives on the surface of your desk. It can only see what's on the desk, but not anything above or below. In fact, it can only see the edges of anything you put on the desk. For example, if you lay a piece of paper on the desk, the 2D being will see what it thinks are 4 walls (the 4 edges of the paper). It has no way to actually read what's written on the paper, unless it digs through the side of the paper and "excavate" each letter from the paper. (And even then, it has to be careful not to break the letters into pieces, otherwise it will have no idea what was there originally.)

If the 2D being is to see any "writing" on the paper at all, the best way would be to write it on the edges of the paper, say by making little jagged cuts on the side in some kind of pattern. This is the most natural way a 2D being would write; but for us 3D beings, that is just silly: we write on the surface of the paper, not on the edge.

Furthermore, since the 2D being can only see the outer edges of things, it will have no way to tell the difference between a hollow O and an O with the inside colored in---at least, it has no way to know unless it digs through and finds that the letter is either solid or hollow. Or if it has 2D X-ray to see whether or not there is a hole in the O.

Now, a piece of 3D paper, as we know, is a flat sheet of fiber, and we can easily roll it up, fold it, or otherwise bend it in 3D space. From a 2D being's point of view, however, these seem to be completely impossible feat. If it examines the piece of paper we lay flat on the table, it will see what looks like a completely solid square block. It would not be able to imagine how you could ever roll it up or bend it or fold it without destroying it. The key here, of course, is that having an extra 3rd dimension to move into allows us to roll up the paper without tearing it apart.

3D->4D

Now consider the 3D->4D case. Just as a piece of 3D paper is a 2D rectangle (well, with some small thickness, but we'll ignore that for now), a piece of 4D paper is a 3D cuboid. From our perspective, this is a solid block of wood, and if a 4D being were to lay a piece of 4D paper on our 3D space, we would see 6 solid walls. We wouldn't be able to tell if anything was written "on" it at all---from our perspective, nothing is written on the block, but inside the block. We'd have no way to see what the writing was unless we dug into the block, or use some other kind of probing device like x-rays.

The way we naturally write would be on the faces of the block---but this to a 4D being is like writing on the "edge" of a piece of paper: very silly, from its point of view. Why not write "on" the paper? ("inside" from our perspective).

Also, 4D letters may have shapes that requires a 4D viewpoint to distinguish, for example, a hollow sphere and a solid ball. The difference is obvious to a 4D being, but to us, both looks like a sphere on the outside, and we don't know which letter it is unless we dig through the sphere or probe it with an x-ray to see if it has a hollow interior.

Then, to a 4D being, the piece of 4D paper is completely flexible: it can be rolled, folded, or bent without breaking. To us, though, we can't imagine how a solid block of wood can be rolled without breaking it. And to fold a block of wood in half so that it (effectively) only occupies half its volume... that's just incomprehensible in our 3D understanding. But this is all completely obvious to a 4D being. Again, the key is that in 4D, there is an extra dimension to roll or fold the block of wood into, so we don't have to make any tears to do it.

A further note about boxes: we 3D beings are used to thinking of things in terms of their 2D surfaces. Sometimes, we unconsciously fail to distinguish between a hollow box and a solid box. But from a 4D being's point of view, this is as blatantly different as four matchsticks joined at the tips to make a square, versus an actual square sheet of paper. You can't write anything in a hollow square; the ink would just end up on the desk. But you can write things on a sheet of paper. Similarly, a 4D being can't write anything in a hollow box: the ink would just end up on whatever's "under" the box in the 4th dimension. So 4D paper must be completely solid, in the 3D sense.

[Max88831]

Is four dimensional paper a three dimensional box? In the box we (three dimensional beings) put our science projects and all that junk, but can a four dimensional being just draw our science project on their paper? Somehow in it like how we put our things in there?

Apparently I'm not so sure about the fact that a piece of paper is going to be a box. THis is because no matter how thin the piece of paper is it still will have a height measurement. THis means that the piece of paper in the 3D world is already a rectangle. THat means when this shape is transfered into a 4D place it will not be like a cube. As to what it looks like, I have no idea, I might later but so far I'm not sure. Just keep in mind that paper is already a 3D object it has lenght, width and height (even if it is very thin) and it is able to move on a 3D plane in 3 degrees of freedom making it a 3D object. Now when we draw on a paper (suppose the drawing came alive) the drawing would be a 2D image of a stick man that can only go up and down and left and right on the page giving it 2 degrees of freedom making it a 2D creature. Now I think it would only be logical for a 4D creature or person to be able to draw a 3D picture (in the literal sense, they are able to make these real cubes that they can touch and feel all 6 sides of with only a drawing utensil). And yes you got it right and im pretty sure 4D beings would want to use there own set of scientific advances to learn and build completely different things that if we could get our hands on will definitely have scientists scratching there heads for a long time.

[kingmaz]

Quickfur's explanation was really helpful.

To look at it a slightly different way. You could almost imagine the 4d paper (swock?) as a new sealed pad of 200 sheets of paper in our 3d world.

From the outside this would look like a pad of paper and there would be no way of telling if it was blank without examining each page, which you can't do without ripping off the seal.

3d beings would be surprised indeed to find that 4d beings were capable of writing on each page and even having some characters passing through reams of paper continguously without ripping off the packaging.