Does topology distinguish between real and imaginary dimensions?

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?

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?

martinbn said:

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.

Feynstein100 said:

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.

Feynstein100 said:

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.

martinbn said:

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

Oh

martinbn said:

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?

Feynstein100 said:

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?

martinbn said:

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?

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.

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.

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.

Mark Harder said:

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.

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.

WWGD said:

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.

Hornbein said:

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

Hornbein said:

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.

Feynstein100 said:

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?

Feynstein100 said:

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.

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.

Pikkugnome said:

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.

Frabjous said:

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

martinbn said:

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

Feynstein100 said:

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 ^

WWGD said:

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

Pikkugnome said:

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?

WWGD said:

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

Do you know what topology is?

Feynstein100 said:

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

Feynstein100 said:

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?

Feynstein100 said:

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.

vela said:

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?

Feynstein100 said:

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.

Feynstein100 said:

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.

Maybe if the OP can give us a few examples of what s/he wants to do, we can figure out and suggest something.

Heuristically, "real" and "imaginary" dimensions ought to be the same because "imaginary" dimensions are those involving a real number times the square root of negative one in each dimension. But the designation of something as positive or negative is a purely arbitrary choice of units.

For example, if you are dealing with electromagnetic charge, the fact that an electron has a negative one charge, and a proton has positive one charge is purely an arbitrary assignment of directions on the real number line and could have been different with a different arbitrary assignment of units. So, nothing "physical" should depend upon how positive v. negative units are assigned.

ohwilleke said:

Heuristically, "real" and "imaginary" dimensions ought to be the same because "imaginary" dimensions are those involving a real number times the square root of negative one in each dimension. But the designation of something as positive or negative is a purely arbitrary choice of units.

For example, if you are dealing with electromagnetic charge, the fact that an electron has a negative one charge, and a proton has positive one charge is purely an arbitrary assignment of directions on the real number line and could have been different with a different arbitrary assignment of units. So, nothing "physical" should depend upon how positive v. negative units are assigned.

Thank you for the analogy. I found it really helpful. However, I think you misunderstood me. In the proton vs electron example, I'm not arguing that one should be labeled positive and the other negative. That's completely arbitrary and simply a matter of choice. What I am saying, however, is that the proton and electron have different charges and thus should be treated as such. To me, saying that real and imaginary numbers are same is like saying the proton and the electron have the same charge. Or worse, that they're the same particle. Might as well take an apple and an orange and say they're the same fruit. I don't care what you call each fruit, as long as you acknowledge that they're different fruits and give them different names.
I'm sorry, I'm feeling quite frustrated. I recently learned that I have autism and one of the consequences of having autism, apparently, is that I can distinguish between things where non-autistic people cannot. I treat things individually instead of making generalizations. I thought that in a forum full of physicists, most of whom are likely autistic like me, I'd find sympathetic ears. And yet, I feel just as different and alien as always.
Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.

Feynstein100 said:

Thank you for the analogy. I found it really helpful. However, I think you misunderstood me. In the proton vs electron example, I'm not arguing that one should be labeled positive and the other negative. That's completely arbitrary and simply a matter of choice. What I am saying, however, is that the proton and electron have different charges and thus should be treated as such. To me, saying that real and imaginary numbers are same is like saying the proton and the electron have the same charge. Or worse, that they're the same particle.

The thing is that if you reverse the positive and negative charges of the proton and electron, then equations involving the square root of proton charges will now be in the imaginary plane, while equations involving the the square root of electron charges will now be in the real plane.

As you correctly posed the question in the first place, it isn't that real numbers and imaginary numbers are the same. But, they have the same topology.

You can, in general, map points in the real plane to the imaginary plane on a one to one basis. You can likewise map points in the imaginary plane to the real plane on a one to one basis.

Feynstein100 said:

Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.

I'm sorry to disabuse you of the notion that you are seeing deeply into mathematics. You are seeing far less than those of us with mathematical training. You lack a basic understanding of abstract mathematical concepts. Your question about "real" and "imaginary" dimensions misses the point about how real and complex numbers relate to each other mathematically.

The real plane ##\mathbb R^2## and the complex plane ##\mathbb C## are isomorphic when considered as real vector spaces. I.e. vector spaces over ##\mathbb R##. In this case, we are simply considering a complex number (vector in the complex plane) as an ordered pair of real numbers with no further structure or properties.

The complex plane, however, is usually considered not as a real vector space, but an algebraic field. That is to say, we define multiplication of complex numbers, which (as @Pikkugnome has pointed out) distinguishes ##\mathbb C## from ##\mathbb R^2##.

Note that a set in mathematics can be given algebraic and/or analytic structure. This means that the same underlying set can be the basis of different mathematical objects. Algebraic generally refers to addition and/or multiplication and analytic refers to properties of some fundamental notion of distance or length. Ultimately, leading to the notion of analytic topology underpinned by the notion of a collection of open sets.

It's perfectly acceptable, therefore, to see both ##\mathbb R^2## and ##\mathbb C## as built on the same underlying set of ordered pairs of real numbers. The difference is that ##\mathbb C## is endowed with a rule for multiplication that produces a closed field.

If you introduce the same multiplication rule for ##\mathbb R^2##, then you do not create a new, distinct mathematical object, but simply ##\mathbb C## in a different notion. This is called an isomorphism. And, to claim that this somehow avoids introducing complex numbers misses the point. You have created an object with the same properties as the complex numbers and have the complex numbers in all but name.

In fact, an interesting and sometimes useful isomorphism of the complex numbers is to a subset of real ##2 \times 2## matrices:
$$a + ib \leftrightarrow
\begin{bmatrix}
a& b\\
-b&a
\end{bmatrix}$$Finally, if we define distance between two points in ##\mathbb R^2## by the usual Euclidean metric:$$|(x_1, y_1) - (x_2, y_2)| \equiv \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$And we define the distance between two compelx numbers as$$|z_1 - z_2| = \sqrt{(z_1 - z_2)(z_1 - z_2)^*}$$Then, purely as simple topological spaces, the two are isomorphic.

However, when we look at complex-valued functions of a complex variable, we move into complex analysis, which has no analogue in real analysis or multi-dimensional real vector spaces.