[Coda]

Originally Posted by Potironette

A cross section of a 3d object is 2d. A cross section of a 2d object might as well be 1d. Therefore a cross section of a 4d object a probably 3d?
And then therefore both quotes are saying imaginary numbers are plottable in 3 dimensions?

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.

Why three dimensions though? What happened to the fourth one?

That's a good question and I actually second-guessed myself while reviewing my post. The reason it's three dimensions is because you yourself constrained it thus by only considering pure imaginary numbers. So instead of having a plot that's got two-dimensional values over a two-dimensional field, you've got a plot that has two-dimensional values over a ONE-dimensional field. The three dimensions aren't two inputs and one output, it's one input and two outputs.

If it helps visualize, consider a cylinder; its central axis would be the imaginary number line, and at each point along that axis you have a two-dimensional flat cross-section that the value of the function at that point must be located in -- and only ONE such point in that cross-section can be the value of the function, or else it wouldn't be a proper function.

What if something like x³ was plotted in four dimension? Would it not remain a curve?

No, it would be a surface, because the input into the function has two dimensions at that point. A real number to the third power stays real, and an imaginary number to the third power stays imaginary, so that means the output of your function has to accommodate both of those.

Is something like x² any different in the third dimension compared to with a fourth dimension?

Any three-dimensional view on this function is necessarily a cross-section or projection* of the complete four-dimensional graph. If you constrain the values of x to be purely imaginary, as we discussed earlier, then you're looking at a cross-section of the graph: if we consider (x + yi) to be the input and (z + ti) to be the output, then this constraint is looking at the slice of the output that intersects the hyperplane* defined by the function x = 0.

* I discuss projections way down below.

* A hyperplane is a generic term for something that divides the entire space into two regions. In two dimensions, hyperplanes are lines -- if you plot x = 0, then you've divided the space into a region where y > 0 and a region where y < 0. In three dimensions, hyperplanes are planes -- if you stand up with your arms and legs outstretched and imagine the infinite two-dimensional sheet aligned with your body, you've divided the whole universe into "in front of you" and "behind you". In four dimensions, a hyperplane is three-dimensional. This gets quite a bit harder to visualize, so I'll take the usual metaphor of assigning the fourth dimension to time. If you imagine time as a filmstrip, then one frame of that filmstrip -- the state of the entire universe at that point in time -- then you've divided the four-dimensional space into "past" and "future".

I guess f(x) = x where x and y are both imaginary works on in one dimension though?

The identity function (f(x) = x) is elegantly simple. Whatever dimension you consider for the input, the output will have the same dimensionality. The oddball quirks of complex numbers don't come into play.

So...basically every dimension is the other dimension stretched out? Like a point(0) stretches into a line(1) which stretches into a flat thing(2) which stretches into a 3d thing like a cube or a marble or whatever real-life thing(3) which stretches into err..some weird looking object..

Bingo! And every time you stretch it, you stretch it in a direction that's perpendicular to all of the other directions you've stretched it before -- a direction that, by definition, didn't exist before. (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.)

Hmm, the problem for me with klein bottles (I googled it) and tesseracts, is that I'm not sure how to see it as not 3d.
With klein bottles, since it's 2-surfaces in 4-space, I guess maybe the tube thing on the inside is a 2-surface..? (edit: after writing the bottom, I guess a surface is literally anything that has an area but no volume, so a klein bottle is just showing a 2d surface existing in 4 dimensions?) As for what the 4-space is, I'm confused about what makes it a 4-space.

Looks like the light went on about what a 2-surface is!

What makes it a (minimum of) 4-space is that a Klein bottle can't exist in 3-space. If you tried to make one, the surface would intersect with itself -- which is exactly what the Klein bottle says it DOESN'T do.

You can make a lower-dimensional analogue of a Klein bottle quite easily, though: a mobius strip. Take a long, narrow piece of paper, twist it once, and tape the ends together. Now, if you tried to draw this object on a two-dimensional piece of paper, the lines would cross each other. But in three dimensions, you have an extra direction you can move things around in so you can avoid that problem.

With tesseracts, since it's a 4-solid it means that it is a 4d object in 4 dimensions? What even is the 4 dimensions that I'm supposed to look at?

You can't! XD We as humans are distinctly incapable of properly visualizing four-dimensional space. We can fake it moderately well by hijacking time as a dimension, but then you're only ever looking at a three-dimensional cross-section of it at any given time.

(the world is 3d, right xD? Or is it just one way to look at it x'D?)

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.]

Am I imagining that, say, the center of the tesseract would be like this 3d world then lots of other worlds were to be put all around it. And then, to see in 4d would to be able to see all those 3d worlds at the same time xD? Like, if a rubix cube were transparent?

Hmm... Well, not quite. For starters... A square, which is a stretched-out line, has 4 sides, and each side is a line. A cube, which is a stretched-out square, has 6, and each side is a square. A tesseract, which is a stretched-out cube, has 8... and each side is a cube.

You could paint a 3D object on each of the eight faces of a tesseract. If you looked at the tesseract straight along one side, you'd only see that one object. If you looked at it from a corner, you'd see four 3D objects, each at an odd angle. If the tesseract isn't made of glass, you wouldn't be able to see through it; it IS a solid object.

What you described is actually a fairly accurate representation of looking at a three-dimensional object in a four-dimensional space. Let's bring it down a dimension for ease of visualization. If you make a figure with a piece of string on the 2D surface of a piece of paper, you -- a 3rd-dimensional viewer -- can see the whole thing at once, inside and outside. But a 2nd-dimensional viewer living on the paper could only see it along the edge; he could see the colors and texture of the side of the string, but he could never get the whole picture at once.

So yes, four-dimensional superbeings looking at our universe would be able to see your intestines without opening you up. They could also reach in and pull your intestines out of your body without hurting you -- but little ol' three-dimensional you wouldn't be able to see it unless they pushed a loop of it back down into your plane of awareness.

Similarly, you could pick up a loop of that string on the paper and it would disappear from our little Flatlander's line of sight, and you could put the middle of the string down on the paper and the Flatlander could see that part of it but not the part you still held in your hand. If you let go of it, the Flatlander would see that somehow you had managed to overlap two solid objects without breaking them -- something he would have imagined would be impossible!

...does that mean a klein bottle is 4-space for having 3-d things technically stretched out because the middle spout is a surface wrapped around 3d air and so is the outside surface of the bottle?

Aha! Nope! Gonna break your brain here:

Klein bottles DON'T HAVE an "outside" surface! They don't have an "inside" surface, either.

Now, this isn't a general property of closed 2-surfaces. A (infinitely-thin) balloon is a closed 2-surface, and it clearly divides the universe into three-dimensional "inside" and "outside" regions.

But Klein bottles only have one side, not two.

Take that mobius strip we constructed earlier. 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.

Occasionally in video games a 3d character would clip together. That's about as far as I'm understanding what's beyond 3d. If I held up a piece of paper and looked at it exactly from the side, that would be two lines clipping together and I guess if I were 2d I could "see" 3d things that way. But that's not really what 3d is. If two 3d objects had the chance to clip together, that would be looking at a 4d world from the side xD?

Sounds like you kinda independently derived what I was describing above!

Oh, and with tesseracts, I forgot about them rotating. It looks like a klein bottle were constantly having its surface moved around? But not really...

This might not be the most productive avenue of discussion, because what you looked at was a TWO-dimensional projection of a FOUR-dimensional object. Animating it only gives you one more dimension, and it's not any of the right ones because the fact that it's rotating requires that time be elapsing for the four-dimensional object (that is, a fifth dimension)!

I wonder if anyone's made a VR app that lets you inspect a tesseract interactively -- that would give you three spatial dimensions and one time dimension.

I'd be curious to paint a solid tesseract instead of rendering a wireframe one -- color each visible square (there are 24 of them) a different color.

...... huh. I just grew a little.

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!

Just goes to show you, this stuff is hard. XD

...Whenever something comes up that I slightly don't understand there's a chance I'll ask about it depending how much I can think about it. I think I understand a surface is a 2-d thing and a curve is a 1-d thing and just because something is 2-d or 1-d doesn't mean they can't be bent in higher dimensions.
Though, can they be bent into a lower dimension?

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.

[Potironette]

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.

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..?

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.]

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!

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.

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?

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!

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?

(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.

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.


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.

[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).

[Potironette]

I'm not sure if the net field is meaning the net field of each specific point or all the points...although now that I think about it, specific points makes a whole lot more sense.


Is a "charge" the imaginary protons which each point is?

EDIT: I should probably mention that we went over the above in class--so I'm worried about understanding the wording of the question should another one pop up somewhere on a quiz or test or something. I'm not certain about the 1/9th thing, but based on memory and based on the whole if something is a certain distance away from something else, the force is 1/r² it seems right.

[Coda]

You got B, A, and E right.

You're right that the repulsive force is 1/9 at D compared to A, but you've overlooked the fact that the left side one has 3x the charge, so your answers for points C and D are wrong.

Actual homework answers, don't open unless you think you've got it right:

A: 3 units left - 1/9 unit right = 2 8/9 units left
D: 3/9 units right - 1 unit left = 2/3 units left

[Potironette]

Hmm..The electron exert a force of 1 to the right at A and the proton a force of 3 to the right at A. Then the proton's forced exerted at D would be 3*(1/9) or 1/3 to the right, while the electron would be 1 to the left so I made a mistake there and D should have a vector 2/3 to the left?
Err, why am I right about the repulsive force being 1/9 at D compared to A..? Did I miss something else?


Then for C, the proton should exert 3 to the left while the electron should exert 1*1/9 or just 1/9 to the right. So, for C it should actually be 3 - 1/9 to the left?
But then, that's what I did before too :/
But I also did think about the fact that the proton has 3x the charge of the electron.

[Coda]

You're right because that's the PROPORTION of the strength between the two points. At D it is 1/9 the strength it is at A. It's just the case that the strength at A is 3.

I suppose I might have misunderstood your notes.

[Potironette]

I did get D wrong from forgetting about the 3*1/9 part though, so thanks for that!

[Potironette]

What exactly is the "magnitude of an electric field"?

It's fuzzy, so the thing reads: A uniform electric field points to the ceiling. A small object of mass 4.0 mg and charge 0.90 nC is placed 2.0 m above the ground and does not accelerate, but merely levitates at 0.20 m above the ground.
They ask me to draw a free body diagram and find the magnitude.

I guessed that since it was levitating, I guessed the magnitude of the field would be the normal force balanced against the force of gravity..but that would mean most of the information I was given wasn't used :/. I'm not really sure I'm understanding the question.

[Coda]

The magnitude probably means the magnitude of the vector at that point in the field.

I think the only information you wouldn't be using is the position. You need the charge and the mass and the orientation of the field. And I think that the position is meant to be a red herring -- part of the problem is to identify that you DON'T need it because it's a UNIFORM field (you WOULD need it if it were a point source directly above the object, due to the inverse square law).

That said, I actually don't understand why they gave two different positions, unless it's trying to suggest that it fell some distance before finding equilibrium, but that... doesn't make sense if the field is truly uniform.

EDIT: Oh, it should be noted. The magnitude of the electric field is NOT the force it applies. It's measured in newtons per coulomb.

[Potironette]

Oh, I should have asked also: what is a uniform field? As in a big infinite line of field..? Why doesn't that follow the inverse square law? A point source means field coming out of one point..right?

What exactly is "magnitude" of an electric field?

[Coda]

Uniform means it doesn't vary. The strength is the same everywhere. It doesn't follow the inverse square law because it says it doesn't. :P (In the real world, a uniform field is an approximation of the field between two oppositely-charged flat plates held at a fixed distance from each other.)

The magnitude (sometimes called "strength" or "intensity") of an electric field is, as I said, force / charge. It measures how much force the field can exert per unit charge.

[Potironette]

Two oppositely charged plates held at a fixed distance from each other make a field that doesn't really vary :o?

Also, I feel kind of bad for asking this question, but since I really don't understand it: What is a charge? I keep hearing/doing things with this thing "charge" and I actually don't get it.
Also, when I write out stuff like mass is 4.0 x 10^-6 kg, I write m = 4.0 x 10^-6 kg, but what's the symbol for charge?



EDIT: Oh, I guess the symbol for "charge" is Q or q. Except Q is "the charge that's affecting" and q is "the charge that is being affected"

EDIT2: I found an equation on my equations sheet that says
For a uniform E-field*: ΔV/d
I think the thing did fall after all...maybe, but the wording's weird? I'm going to try plugging things into this.
EDIT3: Except there isn't time nor acceleration..so did the thing even move ><
EDIT4: I don't think that works..assuming my free-body-diagram is fine I'm going to guess it's fine to find the force, then plug everything into F_e = |q|E
*What is an "E-field"?

[Coda]

Well, think about it. If you're an electron, then the closer you get to the positive plate, the harder it'll push you away, but as you get farther away from it you're also getting closer to the negative plate, so it starts pulling harder to pull you towards it. The forces balance out in a way that can be modeled as uniform field strength. (In reality, it's never perfect, especially near the edges.)

The customary variable for charge is q.

Charge is basically just the electromagnetic analogue of mass. It's just a property that matter has that influences how strongly fields act on it, just like more mass means more gravitational force. It measures the imbalance between protons and electrons, but since we're not dealing with individual ions here we've got a unit for macroscopic charges called the coulomb instead of counting electrons. (After all, you don't count the protons and neutrons in a macroscopic object to determine its mass.)

[Potironette]

Ohh, that makes sense. The fact that both charges on both sides were moving things in one direction had blanked to me.

Thanks for that explanation of charge! I think I need to acknowledge its existence better.

[Coda]

Well, good luck with your homework. I'm off to bed.