[Coda]

Yep, the middle part has all of its electrons right where they want to be so it resists further movement.

Is it that the sunlight hits the undoped silicon --> an electron now leaves that depletion zone and a hole is created there(?) (this part I'm confused about)

The hole is simply the absence of an electron that's supposed to be there if it were in an equilibrium state. So yes, if an electron gets forced out by a photon, it leaves a hole behind where it used to be.

and somehow it's the bottom p-type layer that now has a hole, maybe because it moved an electron up to the depletion zone because electrons prefer to move in one direction

Bingo.

as the electrons from the N-type layer are attracted to the P-type layer?

Through the circuit (on the left side of the picture), yes.

Why is it that there are two arrows coming out of the "Photon Absorbed in Depletion Zone Electron-hole Creation"?

One arrow represents the path of the electron. One arrow represents the path of the hole left behind. Nothing fancier than that.

And..the photon just bypasses the N-layer and goes to the Depletion zone :o?

If it helps with understanding, consider that the primary component of window glass is silicon, and you see photons pass right through THAT all the time.

Photons with too low of energy to dislodge an electron end up just passing right through. Higher-energy photons are more likely to interact. Some energetic photons will get unlucky and hit an electron near the surface of the panel, sending an electron spinning around in the region where it's already happy to float around, and nothing significant happens except the panel gets a little bit warmer (like EVERYTHING does when you shine light on it). But since silicon is still somewhat transparent at those slightly-higher energies, some photons aren't going to bump into anything until they're deeper into the material, and those are the ones that make electricity.

Your intuition might be saying this already, so let me validate it: Yes, solar cells are actually very inefficient, and most photons that land on them don't get converted into electricity. You need huge arrays of thousands of solar cells to generate enough energy to power a house. But we don't really care all that much about that inefficiency, because those photons weren't going to be doing us much good anyway.

Unfortunately, I don't even know what a computer chip does nor where it is :/. Um, what is a "chip"?

"Chip" is slang for "integrated circuit," of which the most important one in your computer is the CPU (or "processor"). They're the (usually black) rectangular things stuck to circuit boards, and they're full of micro-printed silicon wafers to create complex circuits. I would recommend against digging any deeper into this at the moment because it spirals VERY VERY QUICKLY into mindboggling complexity. Suffice to say that modern technology is printing individual details only a few atoms wide and we're very very close to hitting fundamental quantum-scale limitations on how small we can make things and still have them work.

[Potironette]

I'll not ask about the last bit on computers, especially since I don't understand most of the words there ("integrated circuit," "CPU"/"processor," "circuit board," "silicon wafer," "fundamental quantum-scale limitations" [though for this one I guess I can just take away that modern technology is getting parts smaller and smaller and extremely close to some point where it can't work any smaller]).


I guess solar panels are only worth it with lots of space and lots of solar panels xD?
...do solar panels connect to the same power outlets that receive electricity from powerlines or whatever else is generating power from outside? Because if they do, doesn't that mean installing solar power means both a lack of access to those power lines and that those sockets can't be used if it's been cloudy for many days and power stored in batteries ran out (would running out happen? Although I guess this depends on how many solar panels a particular household has)?

[Coda]

Originally Posted by Potironette

I'll not ask about the last bit on computers, especially since I don't understand most of the words there ("integrated circuit," "CPU"/"processor," "circuit board," "silicon wafer," "fundamental quantum-scale limitations" [though for this one I guess I can just take away that modern technology is getting parts smaller and smaller and extremely close to some point where it can't work any smaller]).

Your takeaway there is correct.

Just a few bullet-point-sized definitions to help you round out some basic knowledge:

* "Silicon wafer" is exactly what it sounds like -- a thin sheet of silicon.

* An "integrated circuit" is a bunch of microscopic electronic components made from a single piece of silicon. It's called "integrated" because before they were invented you'd have a bunch of small parts (usually roughly cylindrical in shape) that you'd have to wire together.

* A "CPU" is basically the brain of a computer. Every computer has to have one. To give you an idea of just how complex they are nowadays, the CPU in an iPhone 7 has 3 billion individual components printed into it (imagine how small an iPhone is -- and the CPU isn't even the only chip in it!), and the CPU in an Xbox One has 5 billion.

I guess solar panels are only worth it with lots of space and lots of solar panels xD?

That's not completely true. It's all a question of what you're doing with it. If you have a small load that needs to have power no matter where it is, especially if it's going to be deployed somewhere you can't easily route mains power, then even a small solar panel is useful. Solar-powered traffic signals are pretty common, for example.

But yes, if you want to power something large, you need a lot of them. For large-scale solar electricity generation, it's more efficient to use mirrors to reflect sunlight to heat up water to spin a turbine.

...do solar panels connect to the same power outlets that receive electricity from powerlines or whatever else is generating power from outside? Because if they do, doesn't that mean installing solar power means both a lack of access to those power lines and that those sockets can't be used if it's been cloudy for many days and power stored in batteries ran out (would running out happen? Although I guess this depends on how many solar panels a particular household has)?

They do, although there's another stage that has to go in between -- solar panels generate direct current like a battery instead of alternating current like spinning-magnet generators do, so a device called an inverter is necessary.

But after that, yes, they do connect to the same outlets. However, it's actually not particularly difficult to combine multiple sources of electricity, so houses with both solar panels and city services electricity just have a power regulator that draws from the solar panels when the batteries have sufficient charge and smoothly transition over to city services otherwise. In especially sunny places such as southern California, solar panels can actually generate more power in the course of a day than the house will actually use (or at least, more than the batteries can store), so those houses can actually sell their generated power back to the electric company.

[Potironette]

Oh wow, I didn't know that there were traffic lights that used solar power! Nor did I know that mirrors were used for power.
What happens to the traffic lights if there's a long storm? Or do they store enough in batteries to make it through..? Actually, do traffic lights even use that much power?

How does the solar power make its way to the electric company..? Does it even?

[Coda]

I'm talking about single little warning flashers, not crossing lights. I would imagine those things draw so little power that they could run for a week without a recharge. I've also seen roadside speed indicators ("YOUR SPEED: ___ MPH") with a solar panel that I would guess to be about... oh, half a square meter? Big compared to the rest of the sign but not enormous. If one of those were to run dead, it wouldn't cause any trouble, so it's not a problem if it's cloudy for a few days.

It doesn't actually have to make it all the way back. Like I said, it's not all that hard to combine sources of electricity, so just pushing more current into the line instead of pulling it out is enough to share that energy for other nearby buildings. The power meter keeps track of how many kilowatt-hours you draw in compared to how many you push out and you get billed for the difference or get credit to your account.

[Potironette]

Oh! So there are power meters xD! Come to think of it, I've not thought about the fact that I didn't know how power bills were kept track of--or any other bills, for that matter.

little warning flashers...roadside speed indicators

Ohh, that makes a lot more sense.

[Potironette]

Why does this equation: F_e = (k|q₁||q₂|)/r²
Have absolute values around q₁ and q₂? Wouldn't it be easier to have if something turns out negative they attract and if something turns out positive they repel? Or is that actually more of a hassle than it is useful?

Why does that equation use "r" for distance?

[Coda]

To be honest? I don't know. I have a GUESS (using the absolute value means you could use vector quantities instead of scalar ones) but nothing concrete. Maybe it's just a question of consistency.

r is for radius; you can model such effects as a spherical field emanating from one of the point charges and then reason about the other point charge based on its distance from the center.

[Potironette]

Thanks for the replies! For the first thing I was just curious about it. For the second one, someone mentioned not remembering what the teacher said about it so I think I missed that in class ^^;;

For math I had a homework question that asked for a sketch of: r(x) = (3x² + 6)/(x²-2x-3)
When there's an imaginary root, do I just ignore those? (I ended up simplifying the equation to (3(x² + 2)) / ((x-3)(x+1)) for the sketch)

[Coda]

Yeah, you can more or less ignore them when you're graphing.

As for WHAT an imaginary root is...

The real numbers aren't the only kinds of numbers you can work with using standard algebra. There's an extension of the real numbers called complex numbers: a complex number is the sum of a real number an an imaginary number.

Complex numbers allow you to have meaningful values for some things that the real numbers fail at. The most significant use is also the fundamental definition: The imaginary number i is equal to the square root of -1. And just as all real numbers are multiples of the basis 1, all imaginary numbers are multiples of the basis i, and so all complex numbers can be expressed in the form a + bi.

The square of any complex number is a real number. Unlike real numbers, though, where x² is always positive, the square of a complex number may well be negative.

The norm of a complex number (a + bi) is sqrt(a² + b²). This is an extension of the notion of absolute value: you'll notice it's equal to the absolute value for numbers without an imaginary part.

So what is an imaginary root?

Well, complex numbers work just fine and dandy in polynomials. You can multiply and add complex numbers just fine as long as you remember the identity i * i = -1. So if a real root is a solution to p(x) = 0 where x = a + 0i, then an imaginary root is a solution to p(x) = 0 where x = 0 + bi.

[Coda]

Let's consider p(x) = x² + 1. We know there are two roots, but none of them are real. But what about imaginary roots, now that we know about them?

Well, let's pull out our old friend the quadratic formula. a=1, b=0, c=1, so:

x = (-0 +/- sqrt(0² - 4*1*1))/(2*1)
x = +/- sqrt(-4) / 2
x = +/- 2sqrt(-1) / 2
x = +/- sqrt(-1)
x = +/- i

So x = i and x = -i are the imaginary roots of the function!



Now, if you want to think about what complex numbers LOOK like... It gets complicated!

If real numbers are one-dimensional, then complex numbers are two-dimensional. If functions of real numbers can be plotted in two dimensions... functions of complex numbers are plotted in FOUR. That's not easy to imagine on its own!

But you CAN think of it as a FIELD.

Imagine the real number line. If you were to map a function over it, then each point on that line would have a corresponding number for the value of the function at that point.

For complex numbers, imagine a plane. If you consider f(x + yi) -- that is, the real part of the number along the x axis, and the imaginary part along the y axis -- then you can evaluate the function at any point on the plane, and the result is a two-dimensional vector. For example, this is a graph of the function f(x) = (x + 2)(x - 2), if you allow x to take on complex values:



With a graph like this, the roots are the places where the length of the vector is zero. The real roots are places where this is true on the x axis. The imaginary roots are places where this is true on the y axis. There could also be complex roots: Consider the function f(x) = x⁴ + 1. We know it has to have four roots, but we can obviously see that none of them are real... Let's solve it anyway!

(Lots of algebra, put in a spoiler for brevity)

First off, let's try to factor x⁴ + 1 some.

Let's say y = x², so we can rewrite this as y² + 1. And we already figured out the roots of that one up above, so we know that this factors out to (y - i)(y + i), which is (x² - i)(x² + i).

Now we have to factor each of those factors... One of those is a=1 b=0 c=-i and the other is a=1 b=0 c=i. The quadratic equation comes into play again: (Going ahead and taking out the b terms)

a=1 b=0 c=i
x = +/-(sqrt(-4*1*i))/(2*1) = +/-sqrt(-4i)/2 = +/-2sqrt(-i)/2 = +/-sqrt(-i)

a=1 b=0 c=-i
x = +/-(sqrt(-4*1*-i))/(2*1) = +/-sqrt(4i)/2 = +/-2sqrt(i)/2 = +/-sqrt(i)

So... what's sqrt(i)? Well... Complex numbers are of the form (a+bi), and we know that sqrt(i) * sqrt(i) = i by the definition of square roots, so we put these two things together to say:

sqrt(i) = a + bi
i = (a + bi)²
i = a² + 2abi + (bi)²
i = a² + 2abi - b²
0 + i = (a² - b²) + 2abi

Oh wait a sec. We've got a real number plus an imaginary number on both sides there! And since you can't turn a real number into an imaginary number via addition (you already know that you always get a real number when you add two real numbers) then that means that we can break that up:

0 = a² - b²
--and--
i = 2abi

i/(2ai) = b
1/(2a) = b

0 = a² - (1/(2a))²
0 = a² - 1/(4a²)
0 = a² - 1/(4a²)
--multiply by 4a²--
0 = 4a⁴ - 1
--factor using (x² - 1) = (x + 1)(x - 1)--
0 = (2a² + 1)(2a² - 1)

Since each of these factors must each equal zero, let's consider the second one, since we know the first one is going to end up with an imaginary root again and that's just going to run us all the way back to the top.

0 = 2a² - 1
1 = 2a²
1/2 = a²
sqrt(1/2) = a

Oh hey, that looks useful!

1/(2a) = b (from earlier)
1/(2sqrt(1/2)) = b
1/sqrt(4/2) = b
1/sqrt(2) = b
1/2 = b²
sqrt(1/2) = b

So that means sqrt(i) = sqrt(1/2) + sqrt(1/2)i!

Plugging our answer back into our earlier result of +/-sqrt(+/-i)... Well, that means we need sqrt(-i), but that's sqrt(-1 * i) = sqrt(i)*i = sqrt(1/2)*i + sqrt(1/2)*i*i = sqrt(1/2)*i + -sqrt(1/2).

Therefore, the four complex roots of f(x) = x⁴ + 1 are +/-sqrt(1/2) +/- sqrt(1/2)i. No real roots! No imaginary roots! But if you take (x - sqrt(1/2) - sqrt(1/2)i)(x - sqrt(1/2) + sqrt(1/2)i)(x + sqrt(1/2) - sqrt(1/2)i)(x + sqrt(1/2) + sqrt(1/2)i) and multiply it out, sure enough you get back to x⁴ + 1 like we wanted!

[Potironette]

Admittedly, a lot of that went over my head, but just to clarify...

>An imaginary number is anything with "i"
>A real number is anything without "i"
>A complex number is not an imaginary number nor a real number but a mix of the two?

The norm of a complex number (a + bi) is sqrt(a² + b²). This is an extension of the notion of absolute value: you'll notice it's equal to the absolute value for numbers without an imaginary part.

Err..what do you mean by the "norm"? Is it the magnitude or something? Like the imaginary number is on the y-axis (I don't get why the y-axis but for some reason internet always uses it? Buuut they're likely just ignoring the y-axis and replacing it with a separate imaginary axis?) and then there's a point, say (3, 4i) and the magnitude is the same as sqrt(9 + 16) which is 5?
But I don't understand how the absolute value fits in..

So what is an imaginary root?

Uhh, so what indeed? All I know about it is that i is the square root of -1, so I suppose it's all the multiples of the square root of -1?

Well, complex numbers work just fine and dandy in polynomials. You can multiply and add complex numbers just fine as long as you remember the identity i * i = -1. So if a real root is a solution to p(x) = 0 where x = a + 0i, then an imaginary root is a solution to p(x) = 0 where x = 0 + bi.

When I have imaginary roots, and I make a graph, it it that I won't even see the imaginary root graphed?


As for the second post, I'm currently wondering what it means.
First, imaginary roots exist...
Real numbers are one dimensional because they're on a number line? And then having two number lines as the x and y axis they're...plotted as two dimensional??
Also, if real numbers are one dimensional and are plotted in two dimensions, then imaginary numbers should be one dimensional and plotted in two dimensions too! Maybe..?
Complex numbers are a combo of real numbers and imaginary numbers put together. How that works, I'm guessing, is by having a figure with four axis, two for imaginary and two for reals and somehow one point can be made from that? But how that works gets into the difficult visualization part.

But you CAN think of it as a FIELD.

It's possible to think of it as a "field." What is a field..?

For complex numbers, imagine a plane. If you consider f(x + yi) -- that is, the real part of the number along the x axis, and the imaginary part along the y axis -- then you can evaluate the function at any point on the plane, and the result is a two-dimensional vector. For example, this is a graph of the function f(x) = (x + 2)(x - 2), if you allow x to take on complex values:

From this point forward I'm confused.
What is f(x + yi) and what is the vector referring to?
In the graph picture, I guess the x-axis turned into an axis that for complex numbers because of the problem with how difficult it is to visualize three dimensions? But that can't be it, surely? I'm definitely guessing here. And besides, why have just one axis take on complex values? Why not both? Is that even possible?
As for the last bit I still need to follow it on pen and paper, but essentially it's possible for something to just have complex roots!


Not math question: What is a "pacifying issue"? I read somewhere that someone thought that, for sure, the government was thinking (1970) that Earth Day or something environmental was a "pacifying issue." Does it mean an issue the government wants to just ignore..? Thinks will just die after a while?

[Coda]

A complex number is a real number (which might be 0, so all imaginary numbers are complex numbers) plus an imaginary number (which might be 0i, so all real numbers are complex numbers).

Yes, norm = magnitude = absolute value.

Imaginary numbers are put on the y axis by convention purely because the real number line is usually drawn horizontally. It doesn't actually matter, but the convention makes it possible for people to look at a graph together and agree on what's being plotted.

Real numbers are one dimensional because they're on a number line? And then having two number lines as the x and y axis they're...plotted as two dimensional??
Also, if real numbers are one dimensional and are plotted in two dimensions, then imaginary numbers should be one dimensional and plotted in two dimensions too! Maybe..?

Yes, if you look at JUST imaginary numbers, then they are indeed one-dimensional. However, the main reason you can't plot a function using imaginary numbers in two dimensions is because if you multiply two imaginary numbers together, you get a real number, so you wouldn't have a way to plot it. You COULD meaningfully plot it in THREE dimensions, and you'd get a single curve, not a surface.

Complex numbers are a combo of real numbers and imaginary numbers put together. How that works, I'm guessing, is by having a figure with four axis, two for imaginary and two for reals and somehow one point can be made from that? But how that works gets into the difficult visualization part.

Yes, you're completely right. This is sometimes graphed using color to represent some of the axes (often hue for one of them and brightness for the other), or it can be roughly sketched using the picture like I showed above.

It's possible to think of it as a "field." What is a field..?

You're familiar with electric fields, magnetic fields, and gravitational fields already. You can correspond every point within those fields to some useful value, such as "the force an electron would feel at this point". The same is true here: You can correspond every point on the xy plane with the value of the function at that complex number, and draw some representative examples, like in the picture I showed.

In fact, the example I showed could be interpreted as a magnetic field, with a bar magnet laid upon the x axis. You can see the vectors tracing out curves that go from one pole of the magnet to the other. There's an actual physical realization of this -- if you put a sheet of paper over such a magnet, and sprinkle iron filings over the paper, they will actually align themselves with that picture!

From this point forward I'm confused.
What is f(x + yi) and what is the vector referring to?

Just like you're used to seeing f(x) referring to a function over the real numbers, f(x + yi) is referring to a function over the complex numbers. The vector refers to the value of the function at that point -- a vector pointing straight right might represent 1+0i, one pointing straight up might represent 0+1i, one pointing up-right (and a little longer than the other two) might represent 1+1i, et cetera. The length of the vector corresponds to the norm (= magnitude = absolute value) of that complex number.

In the graph picture, I guess the x-axis turned into an axis that for complex numbers because of the problem with how difficult it is to visualize three dimensions? But that can't be it, surely? I'm definitely guessing here. And besides, why have just one axis take on complex values? Why not both? Is that even possible?

No, one axis isn't representing complex numbers. It does indeed require BOTH axes. The x axis is the real part of the input to the function, and the y axis is the imaginary part. That's why it's written f(x + yi).

As for the last bit I still need to follow it on pen and paper, but essentially it's possible for something to just have complex roots!

Yep, that's correct. They can still be called imaginary roots, from the perspective of them not being real roots, but complex roots is technically more precise.

Not math question: What is a "pacifying issue"? I read somewhere that someone thought that, for sure, the government was thinking (1970) that Earth Day or something environmental was a "pacifying issue." Does it mean an issue the government wants to just ignore..? Thinks will just die after a while?

It's an issue that is brought up to attract focus to something that they think will bring peace to the public. Earth Day wasn't there for the purposes of ACTUALLY protecting the environment. It was created to make people feel good about thinking about the environment.

By calling it a "pacifying issue" the implication is that it's distracting away from more controversial issues.

[Potironette]

Ohh, so f(x) means, the function where real number x is put into it, whereas f(x + yi) means the function where a complex number is put into it, in which the real number is on the x axis and the imaginary numbers are represented by the y axis?

Yes, if you look at JUST imaginary numbers, then they are indeed one-dimensional. However, the main reason you can't plot a function using imaginary numbers in two dimensions is because if you multiply two imaginary numbers together, you get a real number, so you wouldn't have a way to plot it.

So..basically for things like y=x² you can't plot imaginary numbers because when every x is an imaginary, x squared would be a real and thus not on a graph where both axis are imaginary..? But then y = x would be graphable and maybe y = x³ would be too..? But then there is a way to plot it--I think I'm misunderstanding something :/

You COULD meaningfully plot it in THREE dimensions, and you'd get a single curve, not a surface.

Err, what's the difference between a curve and a surface? In fact, what is a curve and what is a surface?


Thanks for the definition of the pacifying issue! I wasn't seeing it while googling it. The person speaking about it was saying that the environmental issue was not just a pacifying issue but in fact serious, and I'd been confused about what he meant.

[Coda]

Originally Posted by Potironette

Ohh, so f(x) means, the function where real number x is put into it, whereas f(x + yi) means the function where a complex number is put into it, in which the real number is on the x axis and the imaginary numbers are represented by the y axis?

Yup, bingo.

So..basically for things like y=x² you can't plot imaginary numbers because when every x is an imaginary, x squared would be a real and thus not on a graph where both axis are imaginary..? But then y = x would be graphable and maybe y = x³ would be too..? But then there is a way to plot it--I think I'm misunderstanding something :/

You can of course find specific examples that you would be able to graph. (And yes, I think x³ works, because i³ = -i. The graph ends up being upside down relative to the real-valued version.)

Geometrically, this means you're taking that four-dimensional graph and taking a cross-section of it.

Err, what's the difference between a curve and a surface? In fact, what is a curve and what is a surface?

Whee, this starts getting into some crazy abstract geometry, and I've gotta be careful in defining this or else we're going to dive into another rabbit hole. XD

Okay, so... A point is a zero-dimensional object. No matter how many dimensions of space you have, a point in it is infinitely small.

A curve is a stretched-out point. It's infinitely thin, but it has a defined length. Imagine an infinitely thin piece of wire. It can bend around in however many dimensions you have available in the model -- in a one-dimensional system (for example, the real number line) all curves are straight line segments, but in a two-dimensional system (a graph), curves can bend around as long as they stay flat (because if they didn't stay flat then they wouldn't fit in the two-dimensional space). In a three-dimensional system (space), a curve can bend around in all three dimensions, et cetera. We say that a curve is a one-dimensional object embedded in an n-dimensional space.

A surface is a stretched-out curve. It has a defined area, but it's infinitely shallow, so it has no volume. Imagine a rubber balloon -- you can stretch it and bend it into any shape you want, but the rubber itself is still two-dimensional. It gets hard to visualize taking this into four dimensions, but you CAN bend it in four dimensions, and that's what counts here.

Anything higher gets called a hypersurface or an n-surface as long as it's embedded in a space with at least one dimension higher than itself. If it's not, then it's called a hypersolid / n-solid (or just a solid, if it's in three dimensions), because it's completely space-filling. (Yes, you could call a two-dimensional object in a two-dimensional space a 2-solid.)

I'm... pretty good at higher-dimensional geometry. >.> I have a reasonably easy time visualizing 2-surfaces embedded in 4-space. Klein bottles are fun. So are tesseracts. (Though tesseracts are 4-solids, not 2-surfaces.)

[Potironette]

You COULD meaningfully plot it in THREE dimensions, and you'd get a single curve, not a surface.

You can of course find specific examples that you would be able to graph. (And yes, I think x³ works, because i³ = -i. The graph ends up being upside down relative to the real-valued version.)

Geometrically, this means you're taking that four-dimensional graph and taking a cross-section of it.

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?
Why three dimensions though? What happened to the fourth one? What if something like x³ was plotted in four dimension? Would it not remain a curve? Is something like x² any different in the third dimension compared to with a fourth dimension? I guess f(x) = x where x and y are both imaginary works on in one dimension though?

--

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..

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.

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? Am I imagining that, say, the center of the tesseract would be like this 3d world (the world is 3d, right xD? Or is it just one way to look at it x'D?) 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? ...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?

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?

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...


...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?