Topology & modelling 4D objects in 3D

[quickfur]

Now that I stopped laughing at our misunderstanding... actually, my understanding of you. The fault lies entirely with me My apologies! So, the "4Der eyes" is the projection of 4d->3d? Granted, it is still not delivered to us in its full glory, because, after that, it has to be projected 3d->2d. Now I understand what you meant!

Thank you for all our efforts and all your time

OK, now that I calmed down a bit, I have to say it's also partly my fault for taking your words at face value, instead of giving you the benefit of the doubt and trying to figure out what you were trying to say. So I also have to apologize for my rude reactions.

Now I see where you're coming from... you're trying to do 4D->2D projection directly (which jives with your previous statement that you want to see 4D as a 3Der instead of a 4Der -- now it all makes sense to me). Initially I brushed that off as an unhelpful approach -- probably because I'm too biased by my past experience, because when I initially studied 4D, there was a point where I had an epiphany that just as we 3Ders see 3D as projections into 2D, so also a native 4Der would see 4D as projections into 3D, and therefore to visualize 4D "as a 4Der would", all it takes is for us to know what is the projected 3D image in the 4Der's eye (which in theory is no problem for us, because we're so facile with 3D constructs), and from thence infer the 4th dimension as the 4Der would do. Ever since then I've always approached the subject from this angle of 4D->3D projections, and I've always considered the 3D->2D part merely as a clutch so that it can be displayed on the 2D screen.

Your approach is different, in that you're trying to tackle the 4D directly as a 4D->2D projection. I had my doubts about the usefulness of such an approach, but after some reconsideration, I have to say that perhaps the idea is not as untenable as I had preconceived. Remember the Flatland question that you posed earlier, about the part where the Square can see the entirety of his 2D world laid bare before his eyes from a 3D vantage point? I had passed it off as an error on Edwin Abbott's part, since the Square doesn't have 3D eyes, and therefore can't possibly see what a native 3D being can see. However, I missed the fact that just because the Square can't see 2D the way we see 2D, doesn't necessarily mean that Abbott's statement about the entire 2D world being laid bare before the Square's eyes is untrue. For, if you think about it carefully, now that the Square has been displaced into 3D, his eye therefore resides at some 3D coordinates outside his 2D world. That means that now, every point of the 2D world has an unobstructed path to his eyes, so that he must be able to see all of it. Of course, due to his retina being merely 1D, he would not see this grand view the same way we 3D beings see it; instead (assuming that all of that light falls on his retina) he sees it all collapsed into a single line, with every slice of the 2D world superimposed upon each other in the tiny space of his 1D retina.

Similarly, if we, as a 3D being, were to be somehow displaced into the 4th dimension, then in theory every point of the 3D world would have an unobstructed path to our eyes, and therefore all of it must become visible to us simultaneously. Of course, our eyes having only a 2D retina, we can't see this grand view as a native 4Der can (a full 3D array of pixels, if you think of her retina as a 3D array of light-sensitive cells); instead, we see everything superimposed upon each other into a single 2D image. Since the image is 2D, every plane therefore must be either clockwise or anticlockwise, from our point of view, so what you said totally makes sense after all.

(Of course, the possibility of this "direct" 4D->2D vision is based on the assumption that the light from outside the 3D hyperplane our eyes lie in will somehow be correctly focused onto the retina -- but it's a bit pointless to argue over such technicalities. For our purposes here, we might as well just consider the eye as a single point in 4D space, which captures whatever light passes through that point. The important thiing is that this is a direct 4D->2D projection.)

What is most curious, of course, is that under such a direct 4D->2D projection, what we'd see is in fact exactly the same as the images I've been rendering: everything is transparent... or appears to be so, not because they're actually transparent, but because light from multiple points in 4D space is falling upon the same cells in our 2D retina. So although I have been thinking in terms of the 3D image that forms in the 4Der's retina when she looks at some 4D construct, the 2D images themselves can also be interpreted as a direct 4D->2D projection -- and the transparency is even automatic! And the math bears this out: a 4D->3D projection from some viewpoint V can be represented as a 5x5 homogenous matrix, and a subsequent 3D->2D projection can be represented as a 4x4 homogenous matrix, but the 4x4 matrix can be extended to 5x5 (by added another row/column with a 1 at the lower right corner), and the two multiplied together to yield a "direct" 4D->2D projection matrix that corresponds to some modified 4D viewpoint V' (the modification is usually just a change in orientation of the 4D camera, without changing its location or what it's pointing at, with the effect that the projected image is rotated in some way).

[4Dspace]

Thank you quickfur for your reply

Last night I was so excited about some realization, that I could not sleep. And that is about the insight I got from this rendition. Thank you so much for doing it! In retrospect, from my POV, our misunderstanding was the most serendipitous event, because otherwise I would have never seen it and never even suspected such a thing. I wonder if you would too see what I see. Take a look at it again:



Do you see a pulsating "something or other"? Especially, if the object were a sphere and not a cube, the projection would trace in time an aspect of a torus in the form of a lying 8. It starts with a loop on one side, then twists, as it turns edge-wise to our POV, and then traces another loop of the 8.

And as the 3d object in our view rotates, what we also see is pulsation, tick-tack, tick-tack...

And so, my realization was that, some transformations 4D->2D may give an impression that the object in question is 4d, or that "we see" something pulsating, while in fact, that's merely a 3d object rotating in 4D. The mostly omitted, sort of, 3D projection serves as a "folding screen" in between.

I feel that this is such a precious little piece of the puzzle I am trying to solve. Thank you for it!

To me it also hints at some symmetry in reflections between (3d in 4D) -> 2D and (4d in 4D) -> 2D

[quickfur]

[...] Last night I was so excited about some realization, that I could not sleep. And that is about the insight I got from this rendition. Thank you so much for doing it! In retrospect, from my POV, our misunderstanding was the most serendipitous event, because otherwise I would have never seen it and never even suspected such a thing. I wonder if you would too see what I see. Take a look at it again:



Do you see a pulsating "something or other"? Especially, if the object were a sphere and not a cube, the projection would trace in time an aspect of a torus in the form of a lying 8. It starts with a loop on one side, then twists, as it turns edge-wise to our POV, and then traces another loop of the 8.

And as the 3d object in our view rotates, what we also see is pulsation, tick-tack, tick-tack...

Not quite sure what you mean, are you talking about a sphere whose left side, say, is on the origin? Then as it rotates, it will alternately appear on either side of the YZ plane. Or do you have something else in mind?

And so, my realization was that, some transformations 4D->2D may give an impression that the object in question is 4d, or that "we see" something pulsating, while in fact, that's merely a 3d object rotating in 4D. The mostly omitted, sort of, 3D projection serves as a "folding screen" in between. [...]

That may well be the case.

I'm also reminded of the shapes of atomic orbitals, which is a topic that greatly interests me. When I was younger, I was always confused by the apparent non-homogenous appearance of the d orbitals, because the d_z² orbital has two lobes and a torus-like ring, instead of four lobes like the rest of the d orbitals. Later on I began to realize that the torus-like shape is caused by the wave function taking on complex number values, and in this particular case, the shape of the wave function intersects the space of real coordinates at a "different angle" from the other lobes, thereby producing a torus-shaped intersection instead of the expected four-lobe shape. In other words, you can think of the orbital as a higher-dimensional construct (each complex number can be regarded as a 2D vector, so the orbital exists in 6D, though it's non-Euclidean in this case) having both a four-lobe shape and a toroidal shape; what we observe is just the lobe-shaped (or torus-shaped) cross-section of it.

All kinds of interesting things can be rationalized as higher-dimensional constructs interacting with our 3D world.

[4Dspace]

Not quite sure what you mean, are you talking about a sphere whose left side, say, is on the origin? Then as it rotates, it will alternately appear on either side of the YZ plane.

Yes. Somehow the animation makes me see that already.

I'm also reminded of the shapes of atomic orbitals, which is a topic that greatly interests me. When I was younger, I was always confused by the apparent non-homogenous appearance of the d orbitals, because the d_z² orbital has two lobes and a torus-like ring, instead of four lobes like the rest of the d orbitals. Later on I began to realize that the torus-like shape is caused by the wave function taking on complex number values, and in this particular case, the shape of the wave function intersects the space of real coordinates at a "different angle" from the other lobes, thereby producing a torus-shaped intersection instead of the expected four-lobe shape. In other words, you can think of the orbital as a higher-dimensional construct (each complex number can be regarded as a 2D vector, so the orbital exists in 6D, though it's non-Euclidean in this case) having both a four-lobe shape and a toroidal shape; what we observe is just the lobe-shaped (or torus-shaped) cross-section of it.

All kinds of interesting things can be rationalized as higher-dimensional constructs interacting with our 3D world.

That's very interesting. So, what is your idea of a structure of an atom? Do you believe it is actually 6d? Have you tried to reconstruct the structure?

I personally have a hunch that the Standard Model, with all those "particles" which are also "forces" is a primitive reduction of n-D geometry into labels. Is it as if for whatever strange reason, "they" cannot recognize the underlying geometry (perhaps because they cannot conceive that space may actually have more spatial dimensions than the 3 that appear -? Or maybe they can't "see" -?) And so they replace concrete, purely-geometric elements of this structure with labels. Say, instead of a plane in 4D, with a well-defined relationship with the subspace in which it appears, it is called "a quark" and the directions of this plane (i.e. its "sides") they called "up" or "down", etc.

In retrospect, such approach does appear idiotic, does not it?

[4Dspace]

The other thing I thought of as an interesting project, re this:

I'm also reminded of the shapes of atomic orbitals, which is a topic that greatly interests me. When I was younger, I was always confused by the apparent non-homogenous appearance of the d orbitals, because the dz2 orbital has two lobes and a torus-like ring, instead of four lobes like the rest of the d orbitals.

Have you tried to do projections of simple objects, like spheres and cubes, onto various subspaces, perhaps skipping some intermittent projections at times, like you did above? What shapes will appear? Because from this, one could reconstruct "the real thing", i.e. the underlying reality. It may turn out, for example, that the shape of the dz2 orbital is a specific projection of a "sphere" from n-space to (n-x) space. And n ain't that big of a number.

Basically, it's a n-D puzzle. The n-D is a difficult part, but the puzzle does not have that many pieces, and the mathematical relationships between most of them are already known. The "forces" and "functions" are actually projections of POVs from one subspace onto another. As far as the real thing is concerned though, there is a couple of rules, like, "time ticks differently in each subspace (with different n)" and subspaces interact along the plane that separates them. That's the most important one. So, when "action" takes place on this plane, two... timelines? chain of events? spring up, one going into one subspace and other going into another.

But speaking of the structure, orbitals is actually one of its most complex parts. It is the "glue" that ties nuclei to the 3d of EM and also the "rollers" on which they glide on the surface of 3d EM. That's the picture that emerges, for me at least.

[quickfur]

[...]That's very interesting. So, what is your idea of a structure of an atom? Do you believe it is actually 6d? Have you tried to reconstruct the structure?

The 6D is really just complex 3D (the space of 3-element vectors where the elements are complex numbers). This is just a product of the wavefunction of the electron having a phase that takes on complex values. As for the actual structure, who knows... quantum physics is a very weird beast. Personally, I don't believe point particles actually exist; they are just waves of some kind.

I personally have a hunch that the Standard Model, with all those "particles" which are also "forces" is a primitive reduction of n-D geometry into labels.

You should be a string theorist.

Is it as if for whatever strange reason, "they" cannot recognize the underlying geometry (perhaps because they cannot conceive that space may actually have more spatial dimensions than the 3 that appear -? Or maybe they can't "see" -?) And so they replace concrete, purely-geometric elements of this structure with labels. Say, instead of a plane in 4D, with a well-defined relationship with the subspace in which it appears, it is called "a quark" and the directions of this plane (i.e. its "sides") they called "up" or "down", etc.

In retrospect, such approach does appear idiotic, does not it?

Which is why string theory exists. Last I heard, string theory requires up to 11D to fully describe everything currently known in physics. If you wanna know how to visualize dimensions that high, talk to wendy. She can do up to 7 and 8 D like the back of her hand, and more with a bit of math. That's still way out of my depth.

[quickfur]

[...] Basically, it's a n-D puzzle. The n-D is a difficult part, but the puzzle does not have that many pieces, and the mathematical relationships between most of them are already known. The "forces" and "functions" are actually projections of POVs from one subspace onto another. As far as the real thing is concerned though, there is a couple of rules, like, "time ticks differently in each subspace (with different n)" and subspaces interact along the plane that separates them. That's the most important one. So, when "action" takes place on this plane, two... timelines? chain of events? spring up, one going into one subspace and other going into another. [...]

I don't know where you got your idea of the standard model being "primitive" from, but I'll have you know that in modern quantum physics (not the stuff from like 20 years ago which made it into the older textbooks) you're not dealing with just some finite dimension, but with projections from an infinite-dimensional Hilbert space into a finite subspace. Things get kinda ... interesting ... when do you that, to say the least.

[4Dspace]

Last I heard, string theory requires up to 11D to fully describe

They make several mistakes in their assumptions, IMO. One is that those additional dimensions actually exist at any given point in space, and that they are "wrapped" or "small" or whatever, just to "explain" why we don't "see" them. But space is a dynamic structure that gravitates toward a certain number of dimensions in its "normal" state, but give it enough pressure, and all sorts of new dimensions will spring up. So those higher Ds happen only at pretty high energies.

See, even though I too think that they describe the structure of space itself, string theorists themselves do not recognize this fact. They speak not of space. They speak of strings that make up stuff is space.

[4Dspace]

The 6D is really just complex 3D (the space of 3-element vectors where the elements are complex numbers). This is just a product of the wavefunction of the electron having a phase that takes on complex values. As for the actual structure, who knows... quantum physics is a very weird beast. Personally, I don't believe point particles actually exist; they are just waves of some kind.

I see that you don't see. Your approach to geometry is purely mathematical. Thus you had such difficulty recognizing that each space is separated into 2 sides, 2 directions. And that is basic. No wonder you think that all spaces are basically the same. You are one of those poor blind quantum guys that keep doing everything backwards, cause they don't see what it is they are computing

Take that integer that pops up in the end of Schrodinger equation. Have you every asked, Why integer? What it means?

[4Dspace]

I don't know where you got your idea of the standard model being "primitive" from, but I'll have you know that in modern quantum physics (not the stuff from like 20 years ago which made it into the older textbooks) you're not dealing with just some finite dimension, but with projections from an infinite-dimensional Hilbert space into a finite subspace. Things get kinda ... interesting ... when do you that, to say the least.

I checked out your infinite-dimensional Hilbert space. That's infinite mess. You are going in the wrong direction. That's the direction of increasing complexity. "You" think that smaller things make up larger ones. But you have not considered that smaller things could be the consequences of "things" elsewhere.

I give you an analogy. Imagine a speedboat cutting through the silvery glade of an alpine lake, at dusk. See the wake it leaves behind, fanning out, the waves dispersing ...eventually the glade evens out. Now, "you" detected some of its waves and you want to understand what's going on (cause you did not see the speedboat). And the waves... they disperse into paisley wiggles, smaller and finer ..and then disappear. Do they? "You" think that if only you can figure out, what those tinniest wiggles are --why, they comprise the larger wiggles--- if only you could include them into your calculations, then you will be able to compute the whole thing so much better.

Because "you" believe that smaller things comprise larger ones. And once you figure them out --you" believe-- you will understand the "large things" they make. But...

Have you noticed? The event was: a speedboat cutting through the glade of a lake. The consequences were the wiggles.

Why?

Surely, "your" computational power is such, that you can compute the whole thing backwards. But think how much easier it is to compute straight on. Starting with the boat.

You modern-day "physicists" are nothing but glorified calculators Blinded by your phenomenal ability to compute, you don't see what you compute.

[quickfur]

[...]You modern-day "physicists" are nothing but glorified calculators Blinded by your phenomenal ability to compute, you don't see what you compute.

You're talking to the wrong audience here. I'm not a physicist.

[wendy]

I have a physics degree. It's because of it that i see 4d and higher.

[4Dspace]

That's great wendy! Are you in the Strings camp or SM? (oops, instead of the Standard Model, it may come up like sado-maso, which it is, come think of it, no? )

[wendy]

Metrology is þe name of þe game, which naturally leads to dimensions.

The metegloss is larger and older than the polygloss (which was written over a christmas break, mainly, with a jug of wine and a litre of bread).

[4Dspace]

Metrology is þe name of þe game, which naturally leads to dimensions.

The metegloss is larger and older than the polygloss (which was written over a christmas break, mainly, with a jug of wine and a litre of bread).

As I said before, what wendy says is way, way over my head.