Teragon wrote:It seems like what you call the 0° type is what I called the 90° type (which angle is the name referring to?).
And what you call the 90° ain't what I called the 0° type, but some more complex version with a tumble in it.
Teragon wrote:I must say that I can't really follow your new animations, although they're made very nicely, because the only thing that makes sense for me to define the orientation of a surface with a circular cross section is the surface normal vector which is not shown here.
Teragon wrote:Anyway this "tumbled" band is quit some interesting object.
Teragon wrote:It ocurred to me that even a common Moebius strip in 3D has kind of a tumble in it, if you trace out the orientation of the surface normal throughout the course of the strip in an outer coordinate system. The special thing about the 4D version is, that the tumble occurs even in local coordinates, i.e. if you move along the loop without doing the twist and observe the orientation of the surface normal relative to yourself. So what the time is for a tumbling object in 3D, is a spatial dimension for this kind of tumble!
Teragon wrote:This tumbled strip even seems to have a handedness build in, even though the righthanded version may be transferred into the lefthanded version by a small deformation.
Wow! I just discovered a whole new type of Moebius strips in 4D, so there may be three distinct families of Moebius strips in 4D. Originally I thought that it's impossible to derive Moebius strips from torispheres, because three loop directions need at least 5 dimensions, but now it seems there is some special way to do cut it, so you get just 2 loops with one direction each. It's a really bizarre and complex object.
Teragon wrote:Anyway this "tumbled" band is quit some interesting object.
Teragon wrote:Wow! I just discovered a whole new type of Moebius strips in 4D, so there may be three distinct families of Moebius strips in 4D. Originally I thought that it's impossible to derive Moebius strips from torispheres, because three loop directions need at least 5 dimensions, but now it seems there is some special way to do cut it, so you get just 2 loops with one direction each. It's a really bizarre and complex object.
gonegahgah wrote:It dawned on me from your depictions that we could have a basic 3-twist Klein Band or "Moebius 2-torus" that intersects our 3D plane as a Moebius Band.
There would only be a left-handed and a right handed variety I guess? Plus the n+½ varieties of course.
With this a path of lines would intersect our 3D-plane and each of these lines would connect to the rest of the circle cross-section that extends out into the 4th dimension.
This would be the simplest harmonic Klein Band in 4D? Is that correct?
gonegahgah wrote:There might be a scary number. That's why I'll stick to just a basic one. Even it has 360° of varieties! Though I'll certainly be interested in what you find.
Teragon wrote:Could you explain what you mean by "3-plane Klein bands" in distinction from "4-plane" Klein bands or what you mean when you say, the object "only really changes" in 3 dimensions?
gonegahgah wrote:Are you happy with a 4D-shape that passes through our 3D-plane appearing to us only as a Mobius Band?
These equations would seem to suggest that it is okay: w = r * sin(u), x = (R + r * (cos(u) * cos(v / 2))) * sin(v), y = (R + r * (cos(u) * cos(v / 2))) * cos(v), z = r * (cos(u) * sin(v / 2))
Do these equations make sense?
gonegahgah wrote:Also, the shape described by those circles above goes from flat on the ground at the front to on it's side at the back.
I believe that this means that we should be able to conclude that it is a Klein Band.
If that is all correct then each cross-section around the ring is only changing its one orientation axis and that is within 3D.
So that's why I referred to it as a 3-plane Klein Band.
I have to go get ready for tomorrow. Otherwise I'd like to set out a similar table for one of my other Klein shapes.
What it does show for them is that both axis are changing and so the red and blue axis - which are the rotation - are changing through the whole 4D space.
So that's why I referred to them as 4-plane Klein Bands.
Does that all sound correct? Is that a suitable reason to call them such or would there be a better descriptor?
I'll look at this more soon...
gonegahgah wrote:Well I figured that if we were keeping a plain slice philosophy then what can be the only results for the middle slice?
The only result would be that we would morph (for the different varieties) between seeing a Klein Strip and a simple path ring.
gonegahgah wrote:I've come to the conclusion that they would appear no different to the 4Der than a 3D-donut not matter what we do.
Teragon wrote:That's a lot of written words. For now this is just a reply to your first post. In the mean time I've updated the images in my previous to posts. I'd hadn't hosted them somewhere they were safe. You might want to have a look at them again.
Teragon wrote:It’s in the nature of slicing that not all of the information about an object is obtained from a single slice. You have to move or rotate the slice through the object to get all of it.
Teragon wrote:What I prefer to do is making a projection into 3D. What you get then is one 3D image of your object seen from one specific angle.
Teragon wrote:I've written a program to visualize flat objects in 4D that way. These are objects that correspond to wires in 3D. (Working on a program that can do solid objects in 4D too.)
Teragon wrote:4D Objects are projected onto 3D just as 3D objects are projected onto 2D when we make a foto. The 3D image is then projected onto the plane of the cumputer screen.
Teragon wrote:In order to get a correct perception of the image we have to make ourselves aware of the 3D shape and also how the interior looks like (flat objects don't have an interior, but solid objects do).
Teragon wrote:To get a feel about it, here's just a common 3D Moebius strip, rotating through four dimensions:
Teragon wrote:The shading helps to get the shape of the image, while the colors code the distance to the beholder (and to the volume of projection). You can also see that the closer the individual parts of the object get the bigger the appear. The shape of the 3D-image alternates between a Moebius band with all points at the same distance (object lying in the three lateral dimensions, which constitute the field of vision of a 4D being) and a totally flat sheet with one close end covering the far end for a moment. After one half revolution back and front change their roles.
Teragon wrote:gonegahgah wrote:I've come to the conclusion that they would appear no different to the 4Der than a 3D-donut not matter what we do.
More precisely, a 4D beeing would see a twisted torus. With the difference that the torus is flat to it and what looks like the interior for unversed 3D beings is actually the surface.
Teragon wrote:The 90°-object is the more symmetric one, as all the directions the surface normal points at look identical. The surface normal is always pointing ouside of the loop. It just came to me that this means that you could rotate the object in the plane of the loop by some angle, then rotate it by the same angle in the plane perpendicular to the plane of the loop and retain the exact same shape! That means in the same way a torus has a rotational symmetry (=invariance under rotations), the 90°-Moebius-Spheritorus has a double-rotational symmetry (=invariance under double rotations).
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