Quote from iamafractal on December 4, 2007, 11:00 pm

espen180 wrote:

Fine. I did some research and found out that there is a theoretical fourth space dimension. An example of a 4-dimensional shape is the Klein Bottle.

moebius transformations:

http://www.youtube.com/watch?v=JX3VmDgiFnY

well the klein bottle is really neat in that it is a non-orientable surface with euler characteristic of zero(one 'side'), which i guess is in layman's terms a 3 dimensional moebius strip with a twist in the 4th spacial dimension. neat. there are Klein bottles for sale, but actually they can't quite be 100% accurate since we don't know how to twist something in the 4th dimension. so... well its almost what its supposed to be. the 'neck' part should go up into the 4th spacial dimension and meet the body, so that you don't actually get to have a 'hole" where the neck meets the body.

http://video.google.com/videoplay?docid=1545472243306352388

a simpler 4th dimensional object would be a hypercube aka tesseract. if a square is 4 lines, and a cube is 6 squares, then a hypercube is like a cube, with another cube on each side. if you put that into 3 dimensions, you'd have something that looked like a couple plus signs. that is a cube in a center, with one cube touching each of its sides. now if you take all the other cubes, and simply fold them vup into the 4th dimension, such that they all touch, then you have a hypercube. all angles are 90 degrees.

http://video.google.com/videoplay?docid=1545472243306352388

this animation depicts a 3 dimensional "shadow" of a 4 dimensional hypercube. the hypercube that would be projecting this image would not actually be distorting at all, merely rotating around in 4 dimensions, thus creating a 3 dimensional illusion of distortions.

http://www.metacafe.com/watch/624539/di ... hypercube/

there's another one.

http://www.metacafe.com/watch/624539/discover_the_4d_the_impossible_hypercube/

Quote from iamafractal on December 4, 2007, 11:23 pm

Crooked Paul wrote:

If a 4-spacer picked you up and rotated you through that fourth spatial dimension and then put you back down, your handedness would be reversed. Note that you would not be able to feel or detect the change while it was happening. You wouldn't be harmed or "turned inside out" or any nonsense like that. You're just being moved through a dimension you can't see. So, say you got flipped by a 4-spacer. If you are normally right-handed and this happened, you would be left-handed from then on, unless you could convince the being from 4-space to flip you back.

there was a really good sci-fi story, and i think it might even have been made into a movie, or episode of some sci fi series like outer limits or something, where an unfortunate space traveler had exactly that happen to him. he got flipped in the 4th dimension, and when he came back to earth, lots of every day molecules became poison to him.

i can't remember any really good examples, but certain everyday things we need to live become toxic poisons when you switch their handedness. you wouldn't be able to easily survive such a flip.

i do remember in the book, diaspora, where all of society has ended up copying itself into software, and is simulated in a sort of computer that is about a meter cubed, there is an extensive exploration of higher dimensions.

if you were all software, then to be honest, having the same limiting physics rules of our 3D universe would be boring. you could easily up the number of dimensions in your simulation to anything you desired. its good too, because there's much more room to connect things together.

further reading...

http://tetraspace.alkaline.org/books.htm

oh and i found the name of the book with the vup and vown... that would be spaceland... it was a fun romp into the 4th dimension, and a relatively good way to visualize it, not that we can.

Quote from Crooked Paul on December 5, 2007, 1:23 am

Fucking awesome linkage, Shmitz and iamafractal!! (Trust a dude with that name to come through.) It'll take me a while to work through this.

We read a short story in my college class in which the hero was a normal high school kid, a 3D person, but he had somehow trained himself to see and access a 4th spacial dimension. He could do rad things like reach inside sealed containers through 4-space and retreive objects inside. He could walk "through" walls (really he was going around in a dimension only he could detect). It was awesome, but I don't remember what it was called.

@ Shmitz: That tesseract folding demo map is the bomb, the goddamn hydrogen bomb. Can you make it so hitting the button again makes the tesseract unfold into its net again? Or just have it toggle back and forth in a loop once you hit the button. That would be hypnotic. So sweet! Nice work, man. You're a mapping genius.
Made a youtube vid: http://www.youtube.com/watch?v=249AxWw-RbE

The much-much-later hey-waitaminite Edit: Did you do all that math yourself?! and if so, how?

Quote from theprogram00 on December 5, 2007, 2:34 am

Ive been reading this thread for a while but didn't post because although i'm very interested in the 4th dimension, i'm not much of an arguer, and i don't have much to add.

I mainly wanted to say what a wicked map Shmitz made, i found that really interesting and useful.

Paul talked about a book which had someone who could walk through using the 4th dimension. I presume portals therefore work in the same way, and would be linked to each other using a tunnel that wasn't visable to us as it was built solely in the 4th dimension? Therefore we could walk through the tunnel or "portal" and have walked around everything that we normally see.

Quote from Crooked Paul on December 5, 2007, 3:25 am

theprogram00 wrote:

I presume portals therefore work in the same way, and would be linked to each other using a tunnel that wasn't visable to us as it was built solely in the 4th dimension? Therefore we could walk through the tunnel or "portal" and have walked around everything that we normally see.

You're on the right track, but this isn't quite accurate. The character in the story could move and rotate through the 4th spatial dimension without deforming anything. Not himself, not 3-space.

Crooked Paul wrote:

He could walk "through" walls (really he was going around in a dimension only he could detect).

Analogously: Having access the 3rd dimension, we can easily draw a line on a piece of paper (a 2-manifold or 2-space or 2D universe) -- and say there was another line in the way -- just pick up the pen and "hop" over that line and continue on the other side. We have more axes, which means we have more options. The important thing to notice here is that having a third axis is what allowed us to make the path of the pen avoid the "obstacle" in 2-space. We didn't have to deform the paper to do it.

Similarly, the character in the story has an axis on us. So he can go around objects in a way we cannot perceive intuitively. He isn't deforming our 3-space, either, just moving in 4-space.

A more technical way to say this is: In 3-manifolds we have six degrees of freedom, but in 4-manifolds there are eight. In 3-space we get movement along all three axes and rotation around all three axes. But 4-space has four axes -- mo' axes, mo' options -- which means they get movement along all four axes and rotation around all four: eight degrees of freedom.

This allows him to do lots of things we can't grok, all without any deformation.

Portal is different, though. We are folding 3-space in Portal. Chell never leaves 3D space, even when she's in the middle of a portal linkage and there are two of her present. This is analogous to our experiment with the paper folded in fourths. Nothing that you draw on the page can "leap up" out of it, but if you -- a lofty 3-spacer -- fold the entire 2D universe, then Flatlanders in the affected area are "doubled," or they can "teleport" from their local perspective, depending on how you deform the paper.

Make sense?

Quote from iamafractal on December 5, 2007, 7:44 am

Shmitz wrote:

I present 3d cube folding vs. 4d cube folding. Load it up in Portal, look at the labeled diagrams, press the buttons, watch the magic. It is important to note that in both scenarios, the blue "box" is smaller at the end because it's farther away! Likewise, the orange box is bigger because it's closer.

If I had more time, I'd have made it so you could unfold them back and repeat as desired, so just restart the map if you want to do that. Also thought about adding commentary, but that might've been going just a little overboard.

I think it might be possible to use this example to start actually talking about how portals connect two locations through the higher dimensions. ;P

great work! i wish i was so proficient at mapping that i could use it like that to illustrate a point. wow. *applause*

so just to be clear, when you see the cubes folding up into the 4th dimension into a hypercube, you are seeing the distorted 3d shadow of it. in the real 4th dimension, all vertices are 90 degree angles.

also, in your demo here, the 3d box you make as well as the 4d tesseract, you have chosen to include a top and a vop. to keep it simple, one might simply imagine 4 squares around a center square. lift those up into the 3rd dimension, and you have a cube. plop a top on it and its closed.

same with the hypercube. you can take a cube and put a cube on each of its 6 sides. fold them all vup into the 4th dimension, and viola, you have a tesseract/hypercube. plop another cube on vop of that, and you close it. the cube on vop would look really distorted in the 3d shadow, yet it would be perfectly squared up in 4 dimensions.

from there, it becomes at least a little more imaginable to proceed to even higher dimensions. take a tesseract, and attach another tesseract on each of its 24 faces, and then you can fold that wup into the 5th dimension, and viola! you have a 5-cube.

Quote from iamafractal on December 5, 2007, 7:50 am

Crooked Paul wrote:

the hero was a normal high school kid, a 3D person, but he had somehow trained himself to see and access a 4th spacial dimension. He could do rad things like reach inside sealed containers through 4-space and retreive objects inside. He could walk "through" walls (really he was going around in a dimension only he could detect).

in spaceland, the author points out that flat 2 dimensional creatures would likely die if they got lifted into the third dimension, because the new "sides" would become exposed. so to make a 2 dimensional creature survive, they must get a 3 dimensional skin attached on the top and bottom in the z axis to keep them closed as they get lifted up.

that also goes for 3 dimensional creatures such as ourselves. if we were to get lifted up into the 4th dimension, all our insides would become exposed on all the new sides and we'd just fall apart. so somebody would first have to attach some kind of 4d skin to us to keep us together when we enter into the 4th spacial dimension.

Quote from Crooked Paul on December 5, 2007, 12:54 pm

iamafractal wrote:

in spaceland, the author points out that flat 2 dimensional creatures would likely die if they got lifted into the third dimension, because the new "sides" would become exposed....

that also goes for 3 dimensional creatures such as ourselves. if we were to get lifted up into the 4th dimension, all our insides would become exposed on all the new sides and we'd just fall apart.

I remember that from Spaceland, but I didn't want to mix up the two stories' different takes on the material. It's confusing enough as it is. Cool idea, though.

The Fourth Dimension -- of GORE, BLOOD, and VISCERA.

Quote from Adair on December 5, 2007, 2:51 pm

Some of the stuff you all are posting is far over my head, but I just found this short video about moebius transformations:

http://www.ima.umn.edu/~arnold/moebius/

Quote:

Any real numbers can be plotted on a line that runs from negative to positive infinity, but throw in an imaginary component and the line becomes a plane, where complex numbers are plotted on both the real and the imaginary axes. M?bius transformations are mathematical functions that send each point on such a plane to a corresponding point somewhere else on the plane, either by rotation, translation, inversion, or dilation. It may sound confusing, but after watching this simple and elegant explanation of M?bius transformations created by Douglas N. Arnold and Jonathan Rogness of the University of Minnesota, Minneapolis, everything becomes clear. Set to classical music, the video demonstrates the transformations in two dimensions but then backs away and adds a third?placing a sphere above the plane and shining light through it. As the sphere moves and rotates above the plane, suddenly all the transformations become linked, in a way that conveys visually in minutes what would otherwise take "pages of algebraic manipulations" to explain, says Rogness.