Folding N-D Paper

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Folding N-D Paper

Postby ICN5D » Thu Apr 17, 2014 5:09 am

I remember hearing somewhere how one cannot fold a piece of paper more than seven times, or it gets too thick and extremely difficult. I suppose it's more a matter of thickness vs LxW, but it is supposed to be the case no matter how big, or spacious in LxW. So, I was wondering what kinds of effects this has on 4D paper: Could a cubic-sheet of 4D paper also have this limitation, or does 4D offer some additional properties not previously anticipated? Ultimately, a sheet of paper in 4D is an extremely thin tesseract, and nearly cube-like. This allows for three ways to fold it in half. The creases could be parallel to X, Y, or Z, and would end up being a linear sequence of them, in the multiple folding process. Would an extra folding dimension provide more or less foldings, assuming a congruent ratio of thickness to a 3D sheet of paper?
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Re: Folding N-D Paper

Postby Keiji » Thu Apr 17, 2014 5:39 am

If you always fold the paper in the same direction (producing a dragon curve) then dimension doesn't matter; folding an extruded piece of paper is the same as extruding the folded paper. Try thinking about how you would fold a 1D 'sheet' in 2D, then extrude that to 2D paper we know in 3D, you'll get the same as if you just fold a piece of 2D paper in the first place.

Additionally what matters is thickness versus length in the direction you're folding, which does not change through the dimensions.
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Re: Folding N-D Paper

Postby quickfur » Thu Apr 17, 2014 7:41 pm

The reason you can't fold more than about 9 or 10 times or so, is because the thickness of the result is an exponential function of the thickness of the paper. Given a thickness t, foldind the sheet in half results in a thickness of 2t. Fold that in half, and the resulting thickness is 2(2t) = 4t. Fold that again, and the resulting thickness is 2(4t) = 8t. So in general, folding in half n times produces a resulting thickness of 2nt. Since 210=1024, a sheet of paper folded in half 10 times would have a thickness 1024 times the thickness of the paper (or equivalently, have the thickness of a 1024-page book :o ). That's why it's impossible. (Well, it's possible, but you'll need a very large piece of paper to do it! :lol:)

Notice that in the above analysis, there is nothing that depends on the dimension of space. So this result pertains to all dimensions.
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Re: Folding N-D Paper

Postby anderscolingustafson » Thu Apr 17, 2014 10:19 pm

The number of times you can fold a hypersheet of paper depends on its proportional thickness of the paper and how elastic the paper is. In any number of dimensions the thickness will double every time you fold a hypersheet of paper. The number of times you can fold a hypersheet of paper along a hyperplane that is parallel to two of its sides does not depend on the number of dimensions the paper has. As the number of dimensions increases the proportional distance between the corners of a hypersheet also increases so the number of times you can fold a hypersheet diagonally does increase with the number of dimensions.
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Re: Folding N-D Paper

Postby ICN5D » Thu Apr 17, 2014 11:02 pm

Thanks for your feedback everyone! An exponential function of thickness, that's the key. Crazy how 10-fold ( no pun intended ) will be the same as 1024 sheets! Wowee, one would need a hydraulic press to achieve greater than that. But, the cube of paper will probably be so fat, it won't be recognizable as a folded sheet, but a terribly compressed block of paper pulp.
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Re: Folding N-D Paper

Postby quickfur » Fri Apr 18, 2014 12:29 am

anderscolingustafson wrote:[...]As the number of dimensions increases the proportional distance between the corners of a hypersheet also increases so the number of times you can fold a hypersheet diagonally does increase with the number of dimensions.

Here's something interesting. In 2D, folding a piece of paper diagonally is the same as folding it in half, since the paper is just a line segment and there's no difference between folding in half and folding along a diagonal. In 3D, folding a square piece of paper diagonally produces a fold in the shape of an isosceles right-angled triangle, which has the interesting property that it can then be folded in half to make smaller and smaller congruent isosceles right triangles.

However, in 4D, folding a cubical piece of paper diagonally (i.e., vertex to opposite vertex) does not produce a nicely doubled-up fold! This is because the vertices of the cube do not lie on the bisecting plane midway between two opposite vertices. A diagonal fold will end up with a hexagonal folding joint, and two congruent but mismatchingly-oriented bisected cubes stacked together. This means that any further folding of the result will produce a messy stack of folds with unevenly jutting edges, because none of the folds will line up nicely!

There is a way to make nice folds in 4D: by folding not according to the 4D diagonal, but the 3D diagonal, to make folds in the shape of an isosceles right triangular prism. This then is the direct analogue of folding in the 3D case. But then the part about opposite vertices of the n-cube being farther and farther away as the number of dimensions go up, will no longer apply. :o

Now, I'm not 100% sure about the 4D cube (i.e., 5D paper). From what I can tell, it also does not line up nicely: you can fold the tesseract around an octahedral joint bisecting the tesseract, but again the ridges of the two folds don't line up nicely (they each have tetrahedral symmetry, but in dual orientations). I suspect this continues to be the case for all higher dimensions.

So it would appear that 2D and 3D are the only dimensions where n-square paper has nice diagonal folds! :o_o:
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Re: Folding N-D Paper

Postby ICN5D » Fri Apr 18, 2014 1:15 am

Wow, that's pretty wild. See, there are strange things going on! That is something I was looking for. Things like this always seem to point at 3D for being nice and stable, without too much weirdness with extra space. Once and again, 3D is quite possibly a Dimensional Anthropic Principle. Until we discover a working 4D universe, that is...
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Re: Folding N-D Paper

Postby quickfur » Fri Apr 18, 2014 2:57 am

I dunno, I generally don't accept anthropic principles because I think they are tautological circular reasoning nonsense. The thing is, we were born into 3D and grew up in it, so 3D-specific phenomena are familiar to us, so we regard them as "normal", "mundane", and "stable". But if 4D beings exist, someone who grew up in 4D would regard "wild" things like cubical paper not folding up nicely as a fact of life, and not think twice about it.

In fact, I have some notes on a story sketch about a 4D boy who has a dream about 3D, and wakes up boggled by all the weird things that happen in 3D that contradict accepted fact in 4D. The need for bridges and cross-walks, for example, boggles his mind, because ... why would anyone want to deliberately step on a road surface intended for vehicles, when they can just walk around it? Except... they can't, 'cos they're missing a dimension! :lol: And what's all this talk about planets and orbits... how could these people live on these balls of earth with no thought whatsoever about their precarious position? Well, except their planets are actually stable! :lol: The boy's dad, of course, writes off his dream as childish fantasy, because obviously 4D is the most natural dimension for the universe to exist in, since it is the last dimension with regular pentagonal polytopes, and since it is the only dimension in which the rotation group SO(4) is chiral! And, what a deficient dimension 3D is -- only 5 regular polyhedra? How could the laws of nature even operate with such a deficient geometry? Surely 4D is the most logical dimension for space to have! And what's this deal with having to build bridges across rivers? What a laughable idea -- bridges! All that trouble making it balance and bear weight without breaking -- for what? So you wouldn't fall into the water while tempting fate by daring to hang over it in the first place? The work of idle hands and idle minds, I tell ya! :lol: 3D indeed. How could anyone tolerate such a miserable, claustrophobic existence anyway? Idle fiction, is what it is! :lol:
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Re: Folding N-D Paper

Postby ICN5D » Fri Apr 18, 2014 4:50 am

Hahaha, quickfur, you're actually kind of funny. You've been nursing this idea for quite some time, I can tell. You probably live and think in the world of 4D anyways. And, why not? With things like rippling light waves and double rotations, who wouldn't want to visit this place often? All of the strange and bizarre possibilities. Oh, I get it! Look at that crazy backwards universe I thought up. I had to spend a lot of time visualizing how it works. Idle minds indeed...... but constructive, I feel. Though not as much as making toratope renders, which rank very high on my scale of coolness factor. If it wasn't for distractions like sports cars, computer games, or lady friends, I would be making more of them. In fact, it's all I think about when trying to fix and sell bicycles at work. What's the next shape I'm gonna explore? Sadly, though, not enough people want to talk about six dimensional toratopes. But, it's okay, the pictures start the conversations now. Well, that's about it, done with self-indulged splurging.

Have you thought about solidifying the story in full form? I'm sure you have, and we're all sitting around waiting for it :) Sounds very interesting. Reverse analogies are great at getting the idea across. Start on the other side of the fence, and explore the familiar from a different set of eyes. This way, we explore what we already know to be true, but from the vantage point of a skeptical outsider. This tends to have an effect of inherent truth in the protagonist's universe, where the reader may be less likely to second guess what they see happening in 4D land. It is now 3D land that becomes the alien, bizarre world, full of strange possibilities.
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Re: Folding N-D Paper

Postby quickfur » Fri Apr 18, 2014 5:29 am

ICN5D wrote:[...]
You've been nursing this idea for quite some time, I can tell.

How d'you guess? ;) :P :lol:

You probably live and think in the world of 4D anyways.

You wouldn't be the first to tell me that. :D My former roommate used to introduce me to new friends as "the guy who thinks in 4 dimensions". :P

[...] Sadly, though, not enough people want to talk about six dimensional toratopes. But, it's okay, the pictures start the conversations now.

Now that's a good idea. I should start making poster prints of my 4D renders to post on the wall or something. That'll draw questions, for sure. :P Although it does also get me started on my bad habit of giving long rambling lectures about obscure 4D geometry that goes way over everybody's heads and draws yawns and glazed eyes. :P

Have you thought about solidifying the story in full form?

Yeah I've been wanting to write it since, oh, 5 years ago (at least)? And then I got married, and got busy with work, and other more important things, and ... the local time shop closed down, and now I can't purchase any more spare time for my hobbies. :cry:

Reverse analogies are great at getting the idea across. [...]

It's not altogether a new idea, though. Edwin A. Abbott beat me by far in this area with his classic Flatland. Though arguably, coming from a lower dimension isn't quite the same as dropping back down from a higher one.

This tends to have an effect of inherent truth in the protagonist's universe, where the reader may be less likely to second guess what they see happening in 4D land. It is now 3D land that becomes the alien, bizarre world, full of strange possibilities.

It gets better. The original plan was to subtly avoid all direct references to 4D (or 3D, for that matter), and make it seem like a rehash of Flatland, until the reader starts noticing that something ain't quite right with what is being described -- whaddya mean, you can walk around rivers, and whaddya mean, "the" sidewalk of a road? Then it becomes obvious that the narrator is talking about a world rather foreign from what it first appeared to be... until the math teacher starts lecturing about the peculiarity of "geometry in the plane" that permits stable orbital motion. All of a sudden, the seemingly innocuous circumstances surrounding an earlier admonition to "stay on the sidewalk!" takes on a whole new meaning, and the keen reader realizes, wait a minute, that reference to six regular solids wasn't a typo on the author's part...

(Why am I spoiling the plot here? 'cos, at the rate things are going, I'm not holding my breath till the day this story actually gets written...)
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Re: Folding N-D Paper

Postby anderscolingustafson » Fri Apr 18, 2014 6:52 pm

3d is a higher number of dimensions than 2d and there are some things that only work neatly in 2d so a 2d life form might think of 3d as being too weird for life.

2d is the only number of dimensions in which there are an infinite number of regular polytopes in 2d while in all other dimensions there is a finite number of regular polytopes. 2d is the only number of dimensions in which regular polytopes can have any number of sides. 2d is the only number of dimensions in which you can add sides to a regular polytopes and have it remain regular. 2d is the only number of dimensions in which a regular polytope can have any number of sides. 2d is the only number of dimensions in which all polytopes have the same number of corners as they have sides. 2d is the only number of dimensions in which you can distort the shape of a polytope without distorting its sides.

A 2d life form might find it strange that there would only be a finite number of regular polytopes in 3d and that when you add sides to a regular polytope in 3d that its no longer a regular polytope. It might find it strange that polytopes could have a different number of corners than the number of sides in 3d. It might find it strange that you can't distort a 3d polytope without distorting its sides. It might also find it strange that when something spins that it would have two points on its surface that don't move in 3d when in 2d the only part of s spinning object that doesn't move is inside the object.
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Re: Folding N-D Paper

Postby ICN5D » Sat Apr 19, 2014 5:19 am

quickfur wrote: Although it does also get me started on my bad habit of giving long rambling lectures about obscure 4D geometry that goes way over everybody's heads and draws yawns and glazed eyes. :P



You, too, huh? Well, nice to see I'm not alone. I'm addicted to the grokking moments, which leads to more crazy wild things to know. I get so excited about something new, I just want to share it with everyone else. Unfortunately, though, most people don't have a clue what it means, but it's still so amazing. The pessimist says "suffering from a condition", while the optimist says "using your gift". I prefer the latter. Your real friends will like this about you, and invite you over just to hear your tales of the higher world. I firmly believe in living life this way, the right people will then take notice. But of course, my tangential tales will span cosmic creation to planck scale to particle colliders to superconducting fields to whatever else is left. And of course, shapes and dimensions.
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Re: Folding N-D Paper

Postby quickfur » Sat Apr 19, 2014 5:36 am

So far, I've not been able to find an audience for my lectures on the J92 rhombochoron, except here. :P My wife likes looking at the renders I make, but she has no idea what they represent (and isn't interested enough to listen to my long rambling explanation :( ).
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