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You can take a cross-section of any dimension lower than the space. In geology, you can study a core sample -- drill out a long, narrow, practically one-dimensional cylinder so you can inspect the layers of the larger three-dimensional sphere that is the Earth.
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Hmm, this makes me wonder, do things in lower dimensions even exist in higher dimensions? I mean, a square is technically a bunch of lines put on top of each other, but since lines are infinitely thin, it isn’t really. A square is made of lines, but to draw a line in 2d would be to make a really long thing with a really small width.
…Although, maybe just by definition that’s not possible because it’s trying to represent 1d in 2d T_T. I guess every object in a certain dimension has the dimensions for said dimension, but it just so happens that every object in one dimension contains parts from a lower dimension..?
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It's a non-Euclidean 4D manifold, if you want to get nitpicky about it. I refer you to the filmstrip metaphor above.
[EDIT: The human-visible parts of the world are a non-Euclidean 4D manifold, I should say. Kaluza-Klein theory (no, different Klein) suggests that it's at least 5D, and some theories predict that it's 11D, and others suggest other numbers.]
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I don’t understand what a “non-Euclidean 4D manifold means” and hence I’m not sure how to apply the filmstrip metaphor, except that since a line that moves across time because 2D on paper, but not to the line because it thinks it’s been in the same place the entire time, something 2D and 3D and 4D and all experience the same thing too, and that relates to 4D because time literally adds an axis to everything? As for the other theories, uh, I'm surprised that there are theories that the world
has more than three dimensions!
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Start at the tape mark and draw a line down the middle of it in one direction. Eventually you'll get to the OTHER SIDE of the tape mark and KEEP GOING because you won't have intersected with your starting point yet. Instead, you'll be drawing on what you would have assumed was the other side until you come back around to where you started. Any individual piece of a mobius strip looks like it has two sides, but taken as a whole, it only has one.
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Err, so basically a mobius strip has only one side because both sides got connected..? Although, in 2d a mobius strip couldn’t have twisted in the first place, and now it looks like lines of a rectangle that shouldn’t be intersecting with each other are intersecting in weird places and the point where the strip is taped together is either or a line, or lost?
And then klein bottles have only one side because, whatever “side” means in 4d was connected (what is a side even ‘~’)? So this bottle, in 3d, has some odd dimension where another bottle exists and those two bottles got connected in such a way that makes it look like, in 3d, the spout has mysteriously intersected with the bottle where it shouldn’t be able to, and is attached to the bottom when it shouldn’t be attached there. Probably, at the bottom spout area, breaking into 4d would make the thing make more sense, but humans can’t do that?
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I was going to say that I'd color each cubic face of the tesseract a different color (this part works) and then have each square face of those cubes be a different shade of that color... but you can't do that! Just like every 1-dimensional edge of a cube is shared by two 2-dimensional faces, every 2-dimensional edge of a tesseract is shared by three 3-dimensional faces!
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Could you have stripes or checkers of color where the faces intersected? Would that even help?
I think given how much I don’t understand, I’ll give up on tesseracts at least until/if I learn more. Buuut what’s a face? Anything on the outside of a thing?
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(Now, this assumes you're working with nice ordinary rectilinear Euclidean space... I won't break your brain with spaces that violate that assumption, but know they exist, and know that you live in one.)
No. You can slice them into a lower dimension (cross-section) or you can project them into a lower dimension (a photograph is a 2D projection of a 4D world; a video recording is a 3D projection) but you can't squash them into a lower dimension without losing information about it.
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So the world is 4D o_o? Why is a photograph 2D but a video 3D? Because videos have that extra axis of time?
School question: What is a "net field"? In physics class we had a homework saying:
Draw the electric field (E) vectors at points A, B, C, D, and E. Draw field coming from each charge and then the net field. Draw lengths proportional to the magnitude. A, B, and E are horizontally equidistant between the two charged particles and each charge is equidistant between A and C or D.
http://i.imgur.com/FgSQ4Ay.png
