A common mistake in the 'Flatland' analogy

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

A common mistake in the 'Flatland' analogy

Everywhere I've read about how 2D beings could infer 3D by looking at a 2D projection of a 3D object, I was then shown such a projection seen from... above in the third dimension, like this image:

Then, implicitly or explicitly, the author tells me that this is the image that the 2D being would see and interpret. But wait... there is a little problem. '2D humans' cannot see their flat world from a 3D point of view, they only have a 1D retina which gives them a 1D view of their world.

This is (imprecisely drawn by the hand) how they would see this projection if they were at the pink dot's position:

(assuming that the blue color was some transparent air and that the edges of the cube are not drawn, only the vertices.)

If they had two eyes, they could feel some depth, feel how far from them each vertex and edge is but this is the best vision they can have of a 3D to 2D projection!
Right so, it's not only hard for them to infer a 3D vision from this, it's... impossible. At least it seems to me. Even for us who are used to 3D, we will never see a 3D cube from this line. Even if we can rotate the cube and see the 2D depth with parallax.

Now what about us? If my correction is correct and if the analogy is valid, it means that the projections of 4D objects that us 3D beings can see from inside our 3D world with our 2D retinas look very different than if they were seen from 'above' in the 4th dimension. We see these projections from a very bad point of view, making them totally not insightful. They will not let us get that feeling of seeing in 4D that we are after.

Ovo
Dionian

Posts: 18
Joined: Sun Jul 22, 2012 9:44 am

Re: A common mistake in the 'Flatland' analogy

Ovo wrote:it's not only hard for them to infer a 3D vision from this, it's... impossible.

Add onto this the fact that the actual line that they are looking at is only 1D, it doesn't have any thickness to it at all, as is shown in the picture... it only has length, no height... it's infinitely thin...

Hugh
Tetronian

Posts: 688
Joined: Tue Sep 27, 2005 11:44 pm

Re: A common mistake in the 'Flatland' analogy

Sure, they have no such thing as height in their mental visualization.
Ovo
Dionian

Posts: 18
Joined: Sun Jul 22, 2012 9:44 am

Re: A common mistake in the 'Flatland' analogy

Ovo wrote:Now what about us? If my correction is correct and if the analogy is valid, it means that the projections of 4D objects that us 3D beings can see from inside our 3D world with our 2D retinas look very different than if they were seen from 'above' in the 4th dimension. We see these projections from a very bad point of view, making them totally not insightful. They will not let us get that feeling of seeing in 4D that we are after.

Nice pictures Ovo.
I tend towards that thought too. I think all 3D pictures of 4D will try to subvert our minds to thinking of 4D with a 3D mindset.
That is a lot of the reason that it has been a journey for us all here towards achieving a greater feel for understanding 4D.

Still, depicting 4D is a little easier for us 3Ders than it is for 2Ders to depict 3D.
They only have the use of colour, translucence, and a single line.
We can at least show an implied inside absent of lines; which they can't depict.

Still, we have to be careful. It is always very easy to fall back into 3D thinking; and hard to truly separate from 3D.
gonegahgah
Tetronian

Posts: 468
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: A common mistake in the 'Flatland' analogy

It is very true that a 2Der, looking from just a single POV at the projection image of a cube, will have practically no clue at all as to how to visualize 3D. The same applies to us, looking at a 2D image of the 3D projection of a 4D object: if we just look at the thing as a 3D object, it will provide no information at all as to the nature of the 4D object in question.

The 2Der, however, can in theory explore that 2D image of the cube -- walk around it and look at it from various angles, maybe step inside it (if we construct it such that some of the edges have openings through which the 2Der can pass), measure the angles the walls make with each other, etc.. Eventually, the 2Der will form some kind of mental picture of the "floor plan" of the image: how the quadrilateral rooms are connected to each other. This is not too dissimilar to our own experience, when we walk into a new building without a map (say it's a single-storey building), we can walk around the corridors, explore the rooms, make note of how the rooms are connected to each other, and then form in our mind a mental picture of the building's floor plan, without ever seeing a map of the place.

This is why I've always advocated separating the viewpoint used for 4D->3D projection, and the viewpoint used for the 3D->2D projection to show the image on the computer screen. The former is the "real" viewpoint; the latter needs to be dissociated from the former so that we can freely let it vary, and thus acquiant ourselves with the "3D floor plan" of the image that resulted from the 4D->3D projection. Staring at a single 2D image and imagining that that it will somehow magically make us "suddenly see 4D" is misleading.

Furthermore, once the 2Der is acquianted with the "floor plan" of the 2D projection of the cube, it still needs to know how to interpret this floor plan. For, if it treats it merely as a 2D floor plan, then that is not a cube at all; it's just 3 oddly-shaped quadrilaterals stuck to each other. Imagining that 3 oddly-shaped quadrilaterals stuck together is somehow equal to a cube is, obviously, wrong. The 2Der can only begin to have some understanding of the cube, if it understands not just the "floor plan" of the 2D image, but the significance of it: the unusual angles at which the walls of the quadrilateral rooms meet is an indication of the 3rd dimension. If this crucial point is missed, then the 2Der's understanding of the image will only ever be that of an oddly-shaped 2D object, not the real thing, the 3D cube.

Similarly, when we look at projections of 4D objects, it is very tempting to identify the compelling 3D shapes that are immediately apparent to our eyes as being the actual 4D object itself. This is wrong. That 3D structure we see before us is not the 4D object at all; it is only a projection image. So our first order of business is to explore this 3D structure: discover what it is made of -- perhaps some polyhedral complex with some faces, edges, vertices, etc.. Then we form in our mind a "3D floor plan" of this structure, how these pieces are connected to each other. But if we stop here, then we have merely understood a 3D structure; there is no 4D. We must proceed to the next step, which is to understand the significance of various aspects of this 3D structure. For example, the fact that some of the polyhedral "rooms" in the structure may look like squished or otherwise distorted cubes. We must understand what this represents: it is the result of an undistorted cube being seen from an angle.

In other words, to make any sense of projection images of 4D objects at all, we must learn how to interpret them. They must be interpreted not as merely some complicated 3D shape with funny angles; each feature must be interpreted in terms of what it tells us about the 4D object used to make the image. For us 3D beings, we must consciously interpret the image in this way; otherwise our brain's instinctive 3D-centric interpretation slips in and we form a wrong idea about what the 4D object really is.
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

P.S. I would also like to point out that in Ovo's image, transparency is used to show the back of the cube. This device is helpful for us 3Ders, who have the benefit of being able to see all of a 2D image at once, but a 2Der would find it utterly confusing, because now the various quadrilateral images of the cube's faces are self-intersecting. If visibility culling is employed, then there will be no self-intersection (provided, of course, that the target object is convex), and thus the 2Der at least has a chance to see the quadrilateral shapes of the projected square faces -- explore it like a floor plan, as I mentioned.

The same thing goes with 4D->3D projections: most such projections that I have seen online do not employ visibility culling -- I suppose the idea is that we want the most accurate image possible, and so we include the images of facets that lie on the far side of the object. However, the result is that the images of the 4D object's facets (cells) now self-intersect, and from our disadvantaged 3D viewpoint, the projection image is far from being comprehensible. I mean, even we 3Ders already have enough trouble discerning the position and shape of the dodecahedron's faces when shown this image:

Can you imagine the confusion on the part of the 2Der if it had to look at this image from the 2D point of view?

Things improve significantly if we employ visibility culling:

From the 2Der's point of view, this image is still a rather complex one, but at least now it can explore the "pentagonal rooms" that constitute the image without being confused by self-intersections, and thus form a mentally-coherent "floor plan" of the thing. Then it at least has a fighting chance of learning to infer 3D from this floor plan.

The same goes for us when we look at projections of 4D objects. Visibility culling is almost mandatory, because otherwise the resulting self-intersecting image will be utterly impenetrable from our disadvantaged 3D viewpoint. I do not claim, of course, that visibility culling will solve all the problems related to 4D visualization; but at least, it does remove one significant obstacle to making projection images comprehensible to us 3Ders -- self-intersections. Once there are no self-intersections, the task of locating the various volumes that make up the 4D->3D projection image becomes significantly easier. It's still challenging for complex objects, but much more tractable than if self-intersections were present.
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

I've been considering if it would be possible to depict projection to some degree in the shadow objects in the rotation method so I've taken note of what you have mentioned here as it may be important, if it is possible to do so, to give the 3D viewer in the 4D world a simpler projected image that will resolve to a 3D image when brought into our plane. Thanks QuickFur.
gonegahgah
Tetronian

Posts: 468
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: A common mistake in the 'Flatland' analogy

Quickfur, I'm glad you agreed with my first post and I agree with what you added. Though my main point was about the fact there is misleading or incorrect information everywhere when it comes to this analogy and - it's a bit embarassing to say - your website contains this mistake many times.
Let's see in detail:

1) On the Cross-sections page, you almost explicitly say multiple times that the 2D beings see those 2D images with circles, squares, etc. So, no, they only see a 1D projection of these images. It would help if you added the missing step showing that those 2Ders only see a line, which give them much less hindsight on the projection that us 3Ders have when we see it from farther in the 3rd dimension with our 2D retinas.

2) On the Projections (2) page, in the "Rotating a Cube through 4D" section, we read:
We shall now use dimensional analogy to investigate the perspective projection of a 3D cube as it gets rotated through 4D. This will greatly help us understand projections of 4D objects later on.

We'll start by taking a look at a 2D square rotating in 3D:

Notice how, from a purely 2D perspective, the image of the square only appears as a square when viewed face-on. When viewed from an angle, it appears not as a square but a trapezoid. Its internal angles appear to be changing, and its outer edge appears to be lengthening and shortening as it rotates through 3D. However, we know that the square isn't actually changing its internal angles or the length of its edges; it just appears that way because of foreshortening in perspective projection.

Now let's take a look at the analogous situation of a 3D cube rotating in 4D, and see if we can make sense of it:

Here, you are comparing a
3D to 2D projection to a
4D to 3D to 2D projection (with some angle in the 3D projection)
and this makes it a wrong and misleading analogy.
The projection of the square should be a 3D to 2D to 1D projection, with some angle on the 2D part. It would look as a line with semi-transparent segments overlapping, one fixed segment, a second segment squashing inside-out, and two segments joining the extremities of the first ones to each other.
I wouldn't intuitively see a square rotating in 3D from this projection, which would make me understand why I can't get the 4D feeling by looking at the cube projection either. And this is important!

3) In Projections (3):
Look at the projection of the 3D cube again. What part of the image do you automatically focus on? Your attention naturally falls on the central region of the image, where 3 of the cube's edges meet at the corner that's facing you. In fact, your attention so spontaneously centers itself on this central region, that you are usually unaware that this view of the cube has a hexagonal envelope! But if you were a 2D creature, your viewpoint would be rather different: what catches your attention first would be the hexagonal envelope, and it would be tempting to identify the hexagonal envelope with the cube. However, the real point of interest lies inside the envelope.
Here again, you don't make clear that the 2Der only sees a 1D projection. Moreover, it sounds like you use "viewpoint" in the psychological sense of the word, so I read that "the 2D creature would see exactly that image but doesn't interpret it as we do", which is obviously incorrect.

4) In Interpreting 4D projections (1), same problem.

Aside of this, your site has the most coherent guide to 4D visualization I've read and it helped me a lot. So I hope you can correct it for future readers!

__________________________

My second point was to insist on the fact that we will never get the automatic 4D feel by looking at these 4D to 3D to 2D projections, even with training. Directly infering the highest-dimension view from a double projection is something our brain has never done and is probably incapable of.

My quest is over now.
Ovo
Dionian

Posts: 18
Joined: Sun Jul 22, 2012 9:44 am

Re: A common mistake in the 'Flatland' analogy

Ovo wrote:Quickfur, I'm glad you agreed with my first post and I agree with what you added. Though my main point was about the fact there is misleading or incorrect information everywhere when it comes to this analogy and - it's a bit embarassing to say - your website contains this mistake many times.

Hi Ovo,

Thank you very much for your corrections to my website. I see that I have failed to make explicit to the reader that the 2Der does not see the 2D image in its entirety; that was my mistake. I was thinking more in terms of the "floor plan" that the 2Der might acquire after some study of the images, rather than what they might see at first glance. I'll have to rework these pages to make this point clearer.

[...]Here, you are comparing a
3D to 2D projection to a
4D to 3D to 2D projection (with some angle in the 3D projection)
and this makes it a wrong and misleading analogy.
The projection of the square should be a 3D to 2D to 1D projection, with some angle on the 2D part. It would look as a line with semi-transparent segments overlapping, one fixed segment, a second segment squashing inside-out, and two segments joining the extremities of both to each other.

Correct, there is an implicit assumption here that the 2Der, who is presumably well-versed with inferring the shapes of 2D objects based on their 1D projections, will be able to tell that these line segments represent a morphing trapezoidal figure. Just as we, looking at the 4D -> 3D -> 2D projection, can perceive the apparent morphing of the cube between various frustum-shaped forms -- even though what we see isn't any of these forms directly, but a 2D projection of them.

I wouldn't intuitively see a square rotating in 3D from this projection, which would make me understand why I can't get the 4D feeling by looking at the cube projection either. And this is important!

Yes, thanks for pointing this out. This is a very important point to avoid misleading the reader into thinking that staring at these 2D images will somehow magically make them see 4D. I shall have to rework these pages to make this point clear.

3) In Projections (3):
[...]But if you were a 2D creature, your viewpoint would be rather different: what catches your attention first would be the hexagonal envelope, and it would be tempting to identify the hexagonal envelope with the cube. However, the real point of interest lies inside the envelope.
Here again, you don't make clear that the 2Der only sees a 1D projection. Moreover, it looks like you use "viewpoint" in the psychological sense of the word, so I read that "the 2D creature would see exactly that image but doesn't interpret it as we do", which is obviously incorrect.

Right, again, the implicit assumption here is that the 2Der, upon studying the images from various angles, will determine that it has a hexagonal envelope with some internal structures -- though it will not actually see this structure directly as we do.

I guess a repeated oversight that I have made was to assume that 2Ders and 3Ders alike, upon looking at the respective 3D->2D or 4D->3D projection images (or rather, some kind of physical rendition of them), will reconstruct in their mind a mental model of the image in question. This mental model then serves as the basis for the further inference of the additional dimension -- not the actual image seen by the eye, as the reduction by 2 dimensions is too severe to be of direct utility. But I can see now how this can be confusing for the reader, since this assumption was never made clear, and it does have far-reaching consequences.

[...] Aside of this, your site is the most coherent guide to 3D visualization I've read. So I hope you can correct it for future readers!

Thanks again for taking the time to send me the corrections!

[...] My second point was to insist on the fact that we will never get the automatic 4D feel by looking at these 4D to 3D to 2D projections, even with training. Directly infering the highest-dimension view from a double projection is something our brain has never done and is probably incapable of.

I agree that we will never get the automatic feel, as you put it; our brain is not trained to infer 2 additional dimensions, after all. My thesis has always been that one must form a mental model of the 3D image first, using these 2D images as an aid; it is from the 3D mental image that the 4D inference can take place. The 2D images themselves are clearly inadequate, as you say.

This actually touches an interesting point that I've been mulling over for years: on my website, I often make use of contrasting colors, etc., to help bring out the salient features of the 3D projections, but in my mind, color has never been a part of the actual visualization process. When contemplating the structure of the 120-cell, for example, I have always seen it in my mind as an assembly of grey and silver dodecahedral volumes assembled into a roughly spherical shape. The actual images shown on my website, of course, make use of many colors, transparencies, thin cylinders for edges, spheres for vertices, etc., mainly because the complexity of it makes the structure hard to discern in a 2D image otherwise. But I regard these as merely tools to communicate the 3D structure of the projections to my brain. I do not know how to directly represent what I perceive in my mind as the projection of the 120-cell into 3D, since we can't physically take in the entire view of it through our physical eyes which only see in 2D. Even if I were to build a physical model of the thing, I could at best only show the edges and perhaps faces, but not the volumes directly (and even then, the limitation of our 2D sight still applies -- we don't have the advantage of 3D sonar mapping that whales purportedly have, to perceive the entire structure in all its 3D glory). This is quite a major obstacle.

I was hoping that, perhaps, I could take advantage of our brain's instinctive, and often unconscious, inference from 2D->3D, so that by presenting 2D images of the 3D image, the reader might be able to reconstruct the appropriate 3D model in his mind that I'm trying to communicate. But it's becoming clear that this approach is fraught with peril for all the reasons you pointed out. Perhaps I was suffering from a case of seeing what I want to see, rather than what's actually there. In my eagerness to convey what I see in my mind's eye, I made the mistake of assuming that others will form the same mental picture based on what amounts to an inadequate representation of it on the computer screen. I see the full 3D thing when I look at the 2D image, because I already know what to "see", so to speak, but this cannot be said of my readers, unfortunately, since the 3D isn't actually there in the 2D image.
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

I see the full 3D thing when I look at the 2D image, because I already know what to "see", so to speak
Wait, you can make a 3D mental picture of a cube as seen from the 4th dimension? You can picture seeing the full volume and all faces of a solid cube at once? Like we can see the full surface and all edges of a square on a 2D picture?
Or do you simply mean that you understand that the squashed thing you see is a cube?
Ovo
Dionian

Posts: 18
Joined: Sun Jul 22, 2012 9:44 am

Re: A common mistake in the 'Flatland' analogy

Ovo wrote:
I see the full 3D thing when I look at the 2D image, because I already know what to "see", so to speak
Wait, you can make a 3D mental picture of a cube as seen from the 4th dimension? You can picture seeing the full volume and all faces of a solid cube at once? Like we can see the full surface and all edges of a square on a 2D picture?
Or do you simply mean that you understand that the squashed thing you see is a cube?

Well, not exactly as a 4D being would see the cube, no, but I can picture the volume of the cube as a, well, cube-shaped volume. I do not perceive it as something flat or squashed. I'm working from the 3D model that our brain makes when we look at 3D objects, you see. I don't really see the 6 faces as distinct entities, though they are implied by the cubical "blob" that is the volume of the cube. When I look at the vertex-centered tesseract projection, for example, I let my mind construct the 3D model of the rhombic dodecahedral envelop with the internal edges, then I focus on the 4 parallelopiped volumes that constitute the image, then imagine them as separate volumes that touch each other. Then I apply some dimensional analogy to picture the foreshortening as indicating 4D depth, and in doing so the vertex at the center of the 4 parallelopipeds "sticks out" as though closer in the 4th direction, and the parallelopipeds become as though they were "slanting away" from the vertex into the 4th direction. So in this way I can vaguely discern the curvature of the tesseract as seen from a 4D viewpoint.

Using the same process, with some effort, I can vaguely begin to discern how about 45 dodecahedra fit together to form a 3D "surface" that one may say is an approximation of how a 4D being perceives a 120-cell.

Tesseract-like shapes are the easiest to visualize, though. The 120-cell family polytopes are just so incredibly complex that it is very difficult to picture them in all their details all at once (even just the visible half of them). I still have not successfully "seen" the 600-cell, at least, not accurately or in any significant detail.

The way I picture these things is by imagining that I'm sitting at the center of the projection image, and these cells are all around me in 3D, and then "distance" myself (as though observing myself from a distance) while keeping a full consciousness of where they are relative to my "other self". Thus, I can more-or-less maintain a sort of "3D map" of these cells, while simultaneously observe them as though from a distance. Then I note their foreshortening caused by the 4D->3D projection, which indicates depth in 4D. With some effort, I can then have little glimpses at how they curve "into" the 4th direction.

Of course, I'm not claiming that this is the same thing as a 4Der who actually sees with a 3D retina with all the light and shade and color in full 3D glory. But I think it's a pretty close approximation. Or at least, it may be as far as we can get without actually having a 3D retina to see with.

Also, I don't think I can claim that it's 100% visualization, it's probably a combination of some proportion of 3D visualization (with the help of the 2D images), some dimensional analogy (a bit of 4D depth inference), and some geometric knowledge, that I consciously attempt to synthesize into a coherent whole. One obviously cannot blindly rely on such a convoluted (and arguably error-prone) process to understand 4D, so I try to hone my skill at it by visualizing different 4D objects and comparing the results with known mathematically-proven facts. I have found that it's surprisingly effective, for all of its limitations and caveats. I have managed to independently rediscover many of the tesseract family of uniform polytopes, as well as the bitruncated 24-cell. (I'm hoping one day to tackle 5D visualization, though I fear that may be too much even for my rather vivid imagination--if I can say so myself.)
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

The way I picture these things is by imagining that I'm sitting at the center of the projection image, and these cells are all around me in 3D, and then "distance" myself (as though observing myself from a distance) while keeping a full consciousness of where they are relative to my "other self".
Hm, it sounds like a good way to do it. Of course when I try I can't help but see some sort of 3D scene of a person elevating above a mangled shape... But I'm sure that with a lot of effort, training and theoretical knowledge, one can partially escape this 3D prison, as you say you do.
Ovo
Dionian

Posts: 18
Joined: Sun Jul 22, 2012 9:44 am

Re: A common mistake in the 'Flatland' analogy

Ovo wrote:
The way I picture these things is by imagining that I'm sitting at the center of the projection image, and these cells are all around me in 3D, and then "distance" myself (as though observing myself from a distance) while keeping a full consciousness of where they are relative to my "other self".
Hm, it sounds like a good way to do it. Of course when I try I can't help but see some sort of 3D scene of a person elevating above a mangled shape... But I'm sure that with a lot of effort, training and theoretical knowledge, one can partially escape this 3D prison, as you say you do.

Well, I don't actually see myself, it's more like I'm simultaneously staring at the 3D assemblage of cells from the outside and "sensing" their presence from the center. (I'm sure that didn't make any sense, it's kinda hard to describe.) And I'm not sure how much is an actual escape from our 3D prison, and how much is just an internalization of learned 4D depth inference rules. For all I know, I might just be deluding myself. (It works surprisingly well for relatively simple 4D shapes, though!)
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

All of you guys are forgetting one thing: Isomorphism, and fractalization (more so than Isomorphism) . The 2D person would have a much harder time looking into the 3rd dimension than we do into the 4th for the very reason that they only have 2 dimensions below them, so they have a lot less to work with (one dimension's new axis can contain MASSIVE information, that's why were here, isn't it?). We can look at...

0D -> 1D *
1D -> 2D *
2D -> 3D
*2D can use these "conversions"

...and from that we can create the same effect from those "segments" that we can use to visualize the 4th dimension, using this massive information contained in the 2D -> 3D extrusion.

I do agree with you guys that we have the tendency to look at 4D with a 3D mindset, but we could probably with much work look at 3D with a 4D mindset (if we could properly fractalize the mindset and information), and while not perfect, it would be much more helpful.
Higher_Order
Dionian

Posts: 16
Joined: Tue May 22, 2012 11:44 am

Re: A common mistake in the 'Flatland' analogy

Not sure what you mean by "fractalizing", but I've trained myself to work with 4D->3D projections, and, for all its limitations, it works surprisingly well in terms of visualizing 4D. The technique requires that you consciously interpret the images not as 3D, but as 4D. It takes some effort to break out of the brain's 3D-centric habits, but once you get past that initial barrier, dimensional analogy is extremely helpful in understanding what it is you're looking at.

For example, our 3D-centric habit causes us to always focus on the nearest part of an object's surface, which generally speaking falls on the central area of our field of vision. To correctly understand 4D, though, one has to suppress this tendency, and to focus on the center of a 3D projection image (which from our disadvantaged 3D viewpoint is not on the surface and not the nearest point, but the "inside"). Then one has to recognize that this central region, which our 3D-centric viewpoint tells us is "inside the object", is actually not inside the 4D object, but is actually the nearest part of the 4D object's surface (nearest to the 4D viewpoint, that is). What our 3D-centric viewpoint tells us is the closest part of the projection image is actually just a part of its periphery, and so should be treated as such, instead of drawing undue attention. Just as our 3D sight spontaneously centers itself upon the central part of the 2D projection image, so the 4Der would focus on what is to us the "inside" of the projection image, rather than the surface of the projection image, which to the 4Der is just the periphery.

Another example is how various 3D shapes appear distorted in 4D->3D projections; it takes some effort to train one's mind to interpret the distortions as perspective foreshortenings rather than actual, physical distortions (which they are not).

After learning some key principles along these lines, I found that many simple 4D shapes are actually well within the grasp of "intuitive" visualization. Some of the more complex shapes require more effort, but in principle, it's easier than it sounds. Of course, there's always the caveat that our 3D POV does suffer from some inherent limitations, so we'll never truly see a 4D object as a real 4Der would. But we can get close, for the simple cases. It helps a lot to look at many projection images and consciously train oneself to interpret them in a 4D way (instead of the default 3D-centric way).
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

The mistake is taking the picture for the real thing. It's only a guide. You have to put the solid in it, the same way you put the solid into photographs.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 2010
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: A common mistake in the 'Flatland' analogy

In simplest terms, what I meant by fractalizing was that you can look at what a 1D person could do to correctly visualize a 2D figure, and a 2D person to correctly visualize a 3D figure, and then repeat what they do on the new scale for the new axis and hopefully visualize 4D. Its the process of finding whats being repeated in different scales, then applying it to a new scale to reach further.
Higher_Order
Dionian

Posts: 16
Joined: Tue May 22, 2012 11:44 am

Re: A common mistake in the 'Flatland' analogy

Higher_Order wrote:In simplest terms, what I meant by fractalizing was that you can look at what a 1D person could do to correctly visualize a 2D figure, and a 2D person to correctly visualize a 3D figure, and then repeat what they do on the new scale for the new axis and hopefully visualize 4D. Its the process of finding whats being repeated in different scales, then applying it to a new scale to reach further.

I believe that is called "dimensional analogy".
quickfur
Pentonian

Posts: 2873
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: A common mistake in the 'Flatland' analogy

...oh ok, I've never heard that term before. Sorry for making this much more complicated than it needed to be...
Higher_Order
Dionian

Posts: 16
Joined: Tue May 22, 2012 11:44 am