[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.