Ovo wrote:it's not only hard for them to infer a 3D vision from this, it's... impossible.
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.
Here, you are comparing aWe 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 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.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.
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.
[...]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.
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):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.[...]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.
[...] 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!
[...] 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.
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?I see the full 3D thing when I look at the 2D image, because I already know what to "see", so to speak
Ovo wrote: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?I see the full 3D thing when I look at the 2D image, because I already know what to "see", so to speak
Or do you simply mean that you understand that the squashed thing you see is a cube?
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.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".
Ovo wrote: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.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".
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.
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