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.
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.
Sorry for making this much more complicated than it needed to be...