Does topology distinguish between real and imaginary dimensions?

[Feynstein100]

Let's say we have four 3D spaces: (x, y, z) , (x, y, iz) , (x, iy, iz) and (ix, iy, iz), with i being the imaginary unit. Now, let's say we have a donut in each of these spaces. Geometrically, the donuts are different objects, have different equations and different properties (I think) but would they be considered the same object in topology?
I want to say no because topology deals with connectedness and if the dimension is different, then that should mean that objects in different spaces would be connected differently. However, I'm not sure. Could someone weigh in perhaps?

 

[martinbn]

I would say that topology needs "I" level thread not "B", and I don't think your questions can be answered at the desired level. More importantly you cannot ask them correctly at "B" level. It would be very difficult to unwind you post and point out all the inacuracies. For example what exactly are the four spaces in you first sentence?

 

[Feynstein100]

I would say that topology needs "I" level thread not "B", and I don't think your questions can be answered at the desired level. More importantly you cannot ask them correctly at "B" level. It would be very difficult to unwind you post and point out all the inacuracies. For example what exactly are the four spaces in you first sentence?

Thank you for the reply. I've changed the level to "I" according to your suggestion.
The four spaces I mentioned are just 3D versions of what would be the real plane (x,y) and the complex plane (x,iy). I was gonna go with the 2D version but then I wouldn't get the donut So I decided to make them 3D. But yeah, other than that, they're just 3D planes, if that makes sense.

This is what I had in mind when I meant "space", with an increasing number of the axes being imaginary.

 

[martinbn]

Thank you for the reply. I've changed the level to "I" according to your suggestion.

I wasn't suggesting that. I was suggesting that you don't have the background for what you want to ask.

The four spaces I mentioned are just 3D versions of what would be the real plane (x,y) and the complex plane (x,iy). I was gonna go with the 2D version but then I wouldn't get the donut So I decided to make them 3D. But yeah, other than that, they're just 3D planes, if that makes sense.

The real plane and the complex plane are the same topological space. So your four spaces are really just one.

 

[Feynstein100]

I wasn't suggesting that. I was suggesting that you don't have the background for what you want to ask.

Oh

The real plane and the complex plane are the same topological space. So your four spaces are really just one.

Well, that seems to be my answer. Topology doesn't distinguish between real and imginary dimensions. It's only the number of dimensions that matters. Hmm but why?

 

[martinbn]

Oh

Well, that seems to be my answer. Topology doesn't distinguish between real and imginary dimensions. It's only the number of dimensions that matters. Hmm but why?

There is no such thing as imaginary dimension. Where did you get it from?

 

[Feynstein100]

There is no such thing as imaginary dimension. Where did you get it from?

Wait, there isn't? What about the complex plane then? One of the axes there is imaginary, right?

 

[WWGD]

I'd say too, that it depends on the choice of embedding of the Torus/Donut. Are you using the "Standard" $S^1 \times S^1$ ? Maybe you want to identify the embeddings up to , what , homotopy, isotopy, etc? But I agree with Martin that it's not clear what imaginary axes are. Ultimately, ix, iy, iz would be interpreted as the respective $x,y,z$ axes rotated by $\pi/2$. Is that what you meant? If so, rotations are homeomorphisms ( here automorphisms), which would thus have no effect in the underlying topology of the space(s). I mean, is , say , ix,y,z, the space resulting of rotating the x-axis by $\pi/2$, while leaving the other axes fixed?
So, ultimately, not sure your question is well-posed enough to be given a meaningful answer.

 

[WWGD]

My idea was that if such axes existed, then a number would be represented by them as, e.g., ax+byi +cyi +dz, etc.
Not aware of any such thing.

 

[Mark Harder]

As I understand topology, any two objects that can be transformed to each other by a continuous function are 'identical' in the sense that their images can be superimposed on each other. If you can write a function that maps one of your tori one-to-one onto another one, then the two are topologically identical.

 

[WWGD]

As I understand topology, any two objects that can be transformed to each other by a continuous function are 'identical' in the sense that their images can be superimposed on each other. If you can write a function that maps one of your tori one-to-one onto another one, then the two are topologically identical.

True, though you need a bit more, your functio must be both onto ad its inverse must be continuous as well.

 

[Hornbein]

The real plane and the complex plane are really the same thing. It's just different notation. Complex numbers are very useful for cyclic phenomena like wave forms, but anything you can do with complex numbers can also be done with real numbers. It might be clumsy and verbose, but it can be done.

 

[Feynstein100]

My idea was that if such axes existed, then a number would be represented by them as, e.g., ax+byi +cyi +dz, etc.
Not aware of any such thing.

Wait then how do you make sense of the complex plane? Isn't i different from real numbers and thus should be treated as such? I thought the whole point of the complex plane was to illustrate that real and imaginary numbers are independent of each other. Which is why they are represented by different axes. It's kind of what an axis means, I think? That things on it are independent of other axes.

 

[Feynstein100]

The real plane and the complex plane are really the same thing. It's just different notation. Complex numbers are very useful for cyclic phenomena like wave forms, but anything you can do with complex numbers can also be done with real numbers. It might be clumsy and verbose, but it can be done.

I'm having a hard time wrapping my head around this. I've been treating them as different my entire life

 

[Frabjous]

The real plane and the complex plane are really the same thing. It's just different notation. Complex numbers are very useful for cyclic phenomena like wave forms, but anything you can do with complex numbers can also be done with real numbers. It might be clumsy and verbose, but it can be done.

I thought they showed that real and complex hilbert space formulations of QM could give different results a couple of years ago.

 

[martinbn]

I'm having a hard time wrapping my head around this. I've been treating them as different my entire life

Topologically they are the same. Why do you think they are not?

 

[WWGD]

Wait then how do you make sense of the complex plane? Isn't i different from real numbers and thus should be treated as such? I thought the whole point of the complex plane was to illustrate that real and imaginary numbers are independent of each other. Which is why they are represented by different axes. It's kind of what an axis means, I think? That things on it are independent of other axes.

Well, you have your Real and Imaginary axes, snd you can represents a number as$a+ib$ So there's a correspondence between the Complexes$a+ib$ and the Real , imaginary axes. But Im not aware of any mathematical object that's written as a function, combination of terms of the form $a*ix+b*iy+c*iz$, nor any of the other combinations you propose. Similarly, polar/cylindrical/spherical coordinates would correspond to axes indexed by $(r,\theta);( r ,\theta , \rho)$, etc.
Edit: So , you have a correspondence between coordinate systems and axes. Im not aware of any coordinate systems that warrants your choice of axes.

 

[Pikkugnome]

I think real and complex plane are are both two dimensional spaces and geometrically the same. The difference comes from their algebraic structures, which are different. Two dimensional vector space does not have the same multiplication as complex plane has and vice versa, you don't see dot or cross products defined for complex plane either.

 

[WWGD]

I think real and complex plane are are both two dimensional spaces and geometrically the same. The difference comes from their algebraic structures, which are different. Two dimensional vector space does not have the same multiplication as complex plane has and vice versa, you don't see dot or cross products defined for complex plane either.

Yes, Complex numbers are an Algebra, while $\mathbb R^2$ is not, but rather, as you stated, a vector space, albeit one with a topology, i.e., a Topological Vector Space.

 

[Feynstein100]

I thought they showed that real and complex hilbert space formulations of QM could give different results a couple of years ago.

That sounds really interesting. Could you maybe share a link for it? I'd like to learn more

 

[Feynstein100]

Topologically they are the same. Why do you think they are not?

Wait then how do you make sense of the complex plane? Isn't i different from real numbers and thus should be treated as such? I thought the whole point of the complex plane was to illustrate that real and imaginary numbers are independent of each other. Which is why they are represented by different axes. It's kind of what an axis means, I think? That things on it are independent of other axes.

Because of this ^

 

[Feynstein100]

Well, you have your Real and Imaginary axes, snd you can represents a number as$a+ib$ So there's a correspondence between the Complexes$a+ib$ and the Real , imaginary axes. But Im not aware of any mathematical object that's written as a function, combination of terms of the form $a*ix+b*iy+c*iz$, nor any of the other combinations you propose. Similarly, polar/cylindrical/spherical coordinates would correspond to axes indexed by $(r,\theta);( r ,\theta , \rho)$, etc.
Edit: So , you have a correspondence between coordinate systems and axes. Im not aware of any coordinate systems that warrants your choice of axes.

That sounds interesting but I didn't really understand it. Would you mind elaborating a bit? Sorry

 

[Feynstein100]

I think real and complex plane are are both two dimensional spaces and geometrically the same. The difference comes from their algebraic structures, which are different. Two dimensional vector space does not have the same multiplication as complex plane has and vice versa, you don't see dot or cross products defined for complex plane either.

Wow, this sounds very interesting. I didn't know spaces could have algebraic structures too. Would you mind telling me more about it?
Also, this opens new possibilities. Like, could 2 objects that are the same geometrically be different algebraically?

 

[Feynstein100]

Yes, Complex numbers are an Algebra, while $\mathbb R^2$ is not, but rather, as you stated, a vector space, albeit one with a topology, i.e., a Topological Vector Space.

Hmm so what's the difference between an algebra and a vector space? Sorry I'm a total noob

 

[martinbn]

Do you know what topology is?

 

[Frabjous]

That sounds really interesting. Could you maybe share a link for it? I'd like to learn more

Here you go. The technical papers are referenced if you want to go deeper.
https://physics.aps.org/articles/v15/7

 

[vela]

Wow, this sounds very interesting. I didn't know spaces could have algebraic structures too. Would you mind telling me more about it?

What do you think a vector space is, for example? Have you studied abstract algebra?

Also, this opens new possibilities. Like, could 2 objects that are the same geometrically be different algebraically?

@Pikkugnome gave you an example in his/her post.

 

[Feynstein100]

What do you think a vector space is, for example? Have you studied abstract algebra?@Pikkugnome gave you an example in his/her post.

Idk a vector space is like a field but for vectors?

 

[Pikkugnome]

Wow, this sounds very interesting. I didn't know spaces could have algebraic structures too. Would you mind telling me more about it?
Also, this opens new possibilities. Like, could 2 objects that are the same geometrically be different algebraically?

I don't know much maths. I see it like this. You can have a flat plain without anything else. You can imagine geometric objects on it: circles, lines, ... . If you add coordinates to the plane, then you can say about the objects more. They have coordinates now and might reveal new things about the objects you didnt notice before. You define distance to the plane. Then you can say that points of a circle are equidistant from one coordinate point the center of the circle. I like to think that new structures you define gives you an ability reveal new things about the plane or its objects.
Real plane and complex are to me two different points of view of the same thing. You just try to look it in a different way.

 

[Office_Shredder]

Idk a vector space is like a field but for vectors?

Are you a high school student? It's great to be enthusiastic about learning more complicated math, but skipping ahead to concepts you don't understand at all will harm your development. If you really want to understand this question you should start by learning linear algebra, and what a metric space is.