[Coda]

Originally Posted by Potironette

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..?

They don't exist as solid objects in higher-dimensional spaces. They exist as boundaries. Your table might be three-dimensional, but its surface is topologically two-dimensional. The world in front of you might be three-dimensional, but the image on your retina is a two-dimensional projection.

There are such things, mathematically speaking, as space-filling curves, where you have a one-dimensional object that touches every possible point in a two-dimensional square. They don't serve a whole lot of practical purpose because you can't actually MEASURE it -- the length is infinite -- but their existence serves an important theoretical role in a number of proofs. (One such theorem: There are the same number of rational numbers (that is, fractions) as there are whole numbers, but there are more real numbers than there are rational numbers... but there are as many complex numbers as there are real numbers!)

I don’t understand what a “non-Euclidean 4D manifold means”

That's intentional. XD I picked that phrasing specifically to make the point that the real world is WEIRD and it can't be so neatly described with simple mathematical models.

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?

I'm not quite sure how to interpret what you're saying here. I... THINK you're right? But you're conflating two separate points I was making.

The filmstrip metaphor is just a way to imagine a four-dimensional space: an infinite series of three-dimensional spaces, all linked together.

As for the other theories, uh, I'm surprised that there are theories that the world has more than three dimensions!

Without getting too deep into it, these additional dimensions are usually interpreted as being "compactified" -- sort of rolled up into tiny little circles, so that if you move in that direction you end up where you started.

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?

Bingo.

And then klein bottles have only one side because, whatever “side” means in 4d was connected

Yep.

(what is a side even ‘~’)?

A very good question, and a difficult one to answer. There are multiple meaningful answers depending on what specific thing you're thinking about. The most useful definition would be the topological one, which is the one that you demonstrated using a pencil on the mobius strip -- two points are on the same side of the object if you can connect them with a smooth curve with no breaks or discontinuities. So a sphere has two sides, an inside and an outside, because you can't trace a curve from a point on the outside to the inside.

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?

Pretty much, yes.

The part that doesn't make sense in three dimensions is the place where the handle appears to pass through the wall of the bottle. It SHOULD be a smooth, continuous, uninterrupted object -- there IS no hole in the wall of the bottle! -- but without being able to see it in four dimensions you just see the two parts clipping together.

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?

If you want to get technical:
* Vertex: 0D points formed by the intersection of edges
* Edge: 1D lines formed by the intersection of faces
* Face: 2D surfaces that, taken as a whole in a 3D object, comprise the boundary of the object; in 4D.
AFAIK there aren't formal names for the higher-dimensional analogues. Some people call the cubes that comprise the exterior boundary of a tesseract "cells."

So the world is 4D o_o? Why is a photograph 2D but a video 3D? Because videos have that extra axis of time?

Yes.

School question: What is a "net field"?

It's the resulting field when you add up all of the fields in the area. Just like "net force" is the resulting force when you add up all of the force vectors acting on an object.

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.


Apparently all I needed to do was draw somewhat ambiguous vectors. And somehow apply the inverse square law on each vector--I still need to practice that. I don't understand what "each charge" and "net field" means though. All I think I know is that "electric field vectors" means where and by how much force a proton would go at a certain point.

Remember that picture of the complex vector field I showed you a few posts back? Your result should look like that.

It's actually not ambiguous at all. You don't have to be precise about the magnitudes of the vectors as long as they're approximately right; this is a qualitative exercise. The important part is that they're pointing in the right direction and that you only draw dots for null (that is, zero-length) vectors.

I can try to answer specific questions you may have, but I'm afraid there's not a lot I can give you in general right now without giving you the answers to your homework outright (which I'm not going to do).