gonegahgah wrote:[...]On the other hand, if they were trying to depict a 3D cylinder in their world, all they could do would be to draw part of a shaded circle edge and draw a line from this to part of a shaded circle edge above. Obviously, this is a fairly poor representation of a 3D cylinder. Perhaps one of the artists here could show what I mean?
gonegahgah wrote:[...]
I just want to clarify something. I used the term 'volume' (with quotes for that reason though it bares explanation) to generally refer to each levels comparative term (ie. area, volume, hyper-volume).
Also, just quickly on the gravity - though its not the main point I'm thinking of - at the 'surface' the 4D gravity would have as units L4; whereas at the surface the 3D gravity would have as units L3. (L is used to refer to length; 4D 'surface' is hypersurface or volume). A number raised to 4 gives a much greater value then the same number raised to 3. So gravity is a magnitude greater on a 4D surface. It is only once you move away from the surface that the gravity drops off at the much greater rate for the 4Der. So on the surface a 4Der would weight much more than a 3Der, which is where we tend to think of something's weight.
But back to the main thing that I think that is important and it is exactly the page that you have provided. We may think that we are giving some impression of what a 4Der is seeing through drawing hyper-perspective but I hope the following shows how fraught that is.
The first image is our psuedo image of a cube. It gives us a sense of volume and a sense of its 3Dness. No problem for us. But when we represent a 4D object in 3D with your projection method it is akin to drawing a cube like the right hand diagram. The front face of the cube is shown as the four frontmost diagonals, the rear face as the four rearmost diagonals, and the 4 side faces are shown as each respective group of side facing diagonals. Suddenly, from this image, we lose the whole sense of the true volume of a cube.
The same basically goes for depictions of tesseracts. All we are showing basically, and as you refer to of course, is the surface (or hypersurface) of the tesseract. Just as for my right hand diagram - where the volume is hidden in the cube diagonals; or connecting congruences (shown as lines) between the 'faces' - the hyper-volume for tesseract depictions is actually hidden in the lines between the hypersurface cubes that are depicted.
gonegahgah wrote:[...]
The diagram on the right is simply a representation of a cube removing the space in between; just as we are forced to leave out the hyperspace of the tesseract when depicting it in 3D. The only way to remove the space inbetween for a cube is to suck out the insides. Using this metaphor, this then (if we leave the corners where they are) turns the cube into an 8 pointed star where each point is a line to the centre. Each 'face' of the cube is then represented only by a set of 4 adjacent lines joining at the centre.
[...]This is then where I say that not only are the faces of the cube represented by sets of adjacent lines radiating out from the centre but also these lines represent also the volume or 'bulk' of the cube; because there is nothing else in the diagram to represent them.
And this also occurs when we depict a tesseract in 3D. The drawn 'cubes' represent - not well - the hyperfaces of the tesseract. But they don't represent it's 'insides' or 'bulk'.
Instead each of the lines connecting each cube to each other cube actually represents the 'insides' or this 'bulk'.
Unfortunately, in 3D I can't depict a cube with square faces but with no space given to the insides (except as I have done). Unfortunately in 3D we can depict a tesseract with cube 'faces' but with no space given to the insides.
The right hand diagram is meant to give us an impression of how different a cube looks to us if we try to draw it in the same way we are attempting to draw tesseracts. What we see a tesseract's cube 'face' like is nothing like what a 4Der sees those same cube 'faces' looking like. Just as the 2Der sees the sides of a square completely differently to how we see the sides of a square.
That is also why I was trying to depict that the connecting 'lines' as actually representing more than they appear to represent by making them not look like normal lines. Instead of just representing lines connecting the cube 'faces' together; they also represent the 'insides' of the tesseract; just like the lines in my anorexic cube do for its insides.
So basically, the rightmost picture is more towards what a cube would look like if we tried to draw it the same way that we are drawing tesseracts in 3D.
gonegahgah wrote:Hi quickfur. I can only make a quick comment at the moment. I'll try to get back later. I just wanted to mention that, which you may say anyway; not sure, whereas we see a cube as taking up volume in our space; a 4Der sees a cube as a flat object. I'll refer to gravity again about this soon but just thought I'd drop that in to test the water. Sorry to dash off without any specifics.
gonegahgah wrote:[...] our 3D tesserat distorts the flat cube 'faces' into 3D cubes [...]
quickfur wrote:What I am saying is that we first do the 4D to 3D projection mathematically, and have the computer represent that 3D image in full, and then give us a way to explore that 3D image however we like -- look at it from different angles, cut out some faces, slice it, whatever, but the goal of that is to reconstruct, in our mind, the full 3D image projected from 4D. When we form a clear mental picture of that 3D model -- which our brain is perfectly capable of doing because we understand 3D -- then we can see exactly what the 4Der sees when she looks at the 4D object.
quickfur wrote:Well, I didn't say it was easy.That's why we need to train the mind to interpret these images as 4D projections, so that the apparent morphing is interpreted as 4D rotation. if you look at it in 3D-centric way, then of course it makes no sense, it's only a strange morphing object. You have to consciously interpret it as 4D rotation for it to make sense.
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