pat wrote:In fact, assuming the x points east-south-east in your pictures and y points north and z points into the screen (north-east), the animation of a cube in the y-z-w space moving in +/- x would also show off the fact that cubes are really "flat" (degenerate) in 4-d. The cube will start at one frustum, move across until it flattens in the middle and start expanding into the frustum on the other side.
That moment of flat will be echo-ed in the 3-d case with one of the other scanning directions where the "square" will flatten to a line before expanding into the trapezoid at the other side.
pat wrote:Actually, I was thinking having both the animations you have now and the ones you had before would be best... though it would involve more text...
I do love the ones you have now.... they're really cool.... and they don't just look like expanding/contracting cubes.
The cross-hatching can wait.... that's awesome though that you're using xfig/transfig to generate these. Bonus! It'll be tough to cross-hatch a cube in a decent way though with xfig... maybe I'll play with it sometime.
Thanks! Any comments on my pages? :wink: I'm kinda stuck on what the next chapter should cover... any suggestions?
bo198214 wrote:Thanks! Any comments on my pages? :wink: I'm kinda stuck on what the next chapter should cover... any suggestions?
Yes, I have a suggestion. In the chapter hidden surface removal, the pictures would be more clear if the 3d-behind-lines would be broken, so there would be only one brain-interpretation (only say Necker-cube). I would like to see: If one 4D-opaque object is 4D-before another object, then it cuts out the other object in the 3d-projection, like this.
quickfur wrote:Hmm. In that chapter, I only used convex objects as examples, so all of the figures should be correct. (Unless you're referring to another chapter?)
The only problem is that the program I wrote for generating those figures isn't advanced enough (yet) to calculate these cut-outs. :-?
bo198214 wrote:quickfur wrote:Hmm. In that chapter, I only used convex objects as examples, so all of the figures should be correct. (Unless you're referring to another chapter?)
No, no, no, dont get me wrong. I think the figures are correct. But if you draw the edges of a 3d cube in cavalier perspective on a paper there are two ways you can interpret the drawing as 3d. (no time to make jpgs ... ) You can swap the back face to the front and vice versa (sometimes this effect is also used in pictures which you can interpret in two ways).
To force perception of only one interpretation, you draw the front lines over the back lines, meaning you draw a back line broken where the front line cross it. And this technique would also avoid double interpretation in your 4D pictures.
You can draw a Necker cube by drawing the first crossing forcing the one interpretation and drawing the second crossing forcing the other interpratation. So it looks paradox. Btw. how about a chapter about 4d Necker-Cube? :wink:
The only problem is that the program I wrote for generating those figures isn't advanced enough (yet) to calculate these cut-outs. :-?
Yes, unfortunately programming (and executing) this is very extensive, so also I hadnt done this in my 4D Building Blocks (though it is planned). By the way what are your mysterious scripts that generate the pictures?
I saw this adanaxis demo (a 4d first person shooter study). Unfortunately *this* doesnt run correctly on *my* computer. The author Andy Southgate also wrote the tesseract trainer and posted on this forum with nick southa. Perhaps he has to solve the same problem. But he doesnt seem to look at this forum regularly. Would only be nice not to do the work trice ...
bo198214 wrote:Hmm, i am not fully convinced that we talk always about the same thing :wink:
So I had to make some pictures despite.
I took your original image of the tesseract, then broke the lines according to their covering relation (as I would interpret it), and at last removed the 4d-hidden surfaces.
You have to admit that your picture
always looked a bit odd, because the back face seems to be concave and the blue inner vertex seems to be outside. And because you use perspective projection there is only one 3d interpretation possible (I think) and I would say you took the wrong 3d interpretation. Except this everything was correct.
Dotted lines are limited to distinguish only between totally back or totally front. The broken line technique is finer by showing which line covers the other, though its no magic bullet either.
Maybe its illegible only in the flat view, when you see it 3d it should not be that confusing. But I am anyway against 3d back face removing when viewing 4d shapes. I want to see the picture like a 4D being with only one eye (at least with that amount of information).Unfortunately, if you add in the back edges as well, it makes for a rather messy tangle. It's not so bad for the 4-cube, but when you deal with things like the 24-cell, it's pretty illegible.
never tried the shooter, even though I read about it when he posted it here. I think it may have something to do with the fact that it only runs on Windows...
bo198214 wrote:Ok, the perspective projection is only unambigous if one knows that it depicts a cube (as I assumed that there are 2 cubes in your picture).
Hmm, what regards the rotation ... I really have difficulties to see it rotating in 3d, sometimes the interpretation even swaps the rotation direction .
A parallel eye, or red/cyan picture would help a lot.
The only problem with these techniques is, that not anyone can benefit from it. Some people only can see crossed eye, some only parallel eye, some nothing of that, and some dont have read/cyan glasses.
And it becomes very cumbersome if you would provide four versions of every picture.
At least that way one would only loose 1 dimension.Maybe its illegible only in the flat view, when you see it 3d it should not be that confusing. But I am anyway against 3d back face removing when viewing 4d shapes. I want to see the picture like a 4D being with only one eye (at least with that amount of information).Unfortunately, if you add in the back edges as well, it makes for a rather messy tangle. It's not so bad for the 4-cube, but when you deal with things like the 24-cell, it's pretty illegible.
So when you use povray I beg you not to use opaque surfaces. There should be something usable like toned glass or so (as in the tesseract trainer). For convex hull I hope you read my article.
never tried the shooter, even though I read about it when he posted it here. I think it may have something to do with the fact that it only runs on Windows...
Never heard of "WINE"?
Though I tried it with windoze, but there seemed some problems with showing text, maybe font or so, so it was not usable...
But one question remains, you have a picture of a 4d necker-cube
Did you have a look at this article? A 3d necker cube can considered that way:
There are two interpretations of a transparent cube, the Necker-cube is a mixture. And now I want to see your 4D-Necker-cube!
pat wrote:A bit of pedantics on definitions.... a Necker cube is just a wireframe drawing of a cube with the ambiguity left intact. Quickfur, what you're calling a necker cube is really an "Impossible Cube".
In your sequence of four drawings, you have (from left-to-right) a Necker cube, a cube, a cube, and an impossible cube.
What you're hoping for is an Impossible 4-D Cube, not a 4-D Necker Cube (as you've already got lots of those).
quickfur wrote:pat wrote:What does the blue mean again?
Just that it lies inside the 3D projection envelope. I guess it's redundant here, 'cos povray's lighting makes it clear that they are inside.
iNVERTED wrote:quickfur wrote:pat wrote:What does the blue mean again?
Just that it lies inside the 3D projection envelope. I guess it's redundant here, 'cos povray's lighting makes it clear that they are inside.
Make that object no_shadow and it'll make it much clearer.
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