Teragon wrote:I think there are no invariant planes to be found, because the toroidal rotation involves a deformation and is not a solid rotation. More precisely, the invariant directions are dependent on the location on the strip.
Teragon wrote:You have to keep in mind that such a turn in 4D involves an exchange of ana and kata. You can change left and right in 3D too if you turn head over heals. In this case up and down are exchanged. For us 3D beings a 180° through ana/kata would seem like a mirror operation, if that is what you mean.
Teragon wrote:I thought we were talking about invariant planes.
Teragon wrote:I think there are no invariant planes to be found, because the toroidal rotation involves a deformation and is not a solid rotation. More precisely, the invariant directions are dependent on the location on the strip.
Teragon wrote:If a 3D object is rotated through a 2D plane, a 2D beeing inside of it may not decide if it's going to the left side or to the right side, but
- it's able to tell if its sense of rotation changes if the object looks different on the left side and the right side, i.e. the 2D plane is not a mirror plane of the object.
- it's able to tell if the object has been flipped by 180° if the rotational axis is not a "mirror axis" of the object, because two directions in the plane will changes parts.
gonegahgah wrote:I'm guessing that that means that you don't think it is a valid representation?
gonegahgah wrote:Teragon wrote:If a 3D object is rotated through a 2D plane, a 2D beeing inside of it may not decide if it's going to the left side or to the right side, but
- it's able to tell if its sense of rotation changes if the object looks different on the left side and the right side, i.e. the 2D plane is not a mirror plane of the object.
- it's able to tell if the object has been flipped by 180° if the rotational axis is not a "mirror axis" of the object, because two directions in the plane will changes parts.
Agree. Those two statements are very similar aren't they (just in case I'm reading them wrong)?
gonegahgah wrote:We can see for this that the invariant plane is x/w plane through their middle of the shape (is that correct Teragon?).
gonegahgah wrote:That's all good. Now we get the 4Der to spin to a certain distance while maintaining up and forward.
After this right is pointing in a different direction than before and we have to redraw our image in relation to that.
The interesting thing is that this is very like flipping a 3D image for a 2Der. The reason is that all points maintain their distance from the y,z plane like they do when we 'flip' x-wise in 3D.
gonegahgah wrote:In our 2D world we can't rotate toroid like because the inside of the toroid is shorter than the outside.
If you tried that with a donut the inside would split as it stretches to the outside and the outside would squash as it tries to go through inside.
In a 4D world this may not be the case and there may be a path that a donut can follow the allows the inside to remain short while the outside maintains its longer length.
So, I'm wondering if you could actually rotate a donut through its centre along all its length at once in 4D without stressing the donut at all.
It doesn't effect the topic at hand but I am curious... What does everything think about that? Is that last idea very very wrong?
Teragon wrote:Really don't know what's going on here.
gonegahgah wrote:The same would occur for us if we were able to magically transfer our whole body from left to right.
Suddenly our brain would see what was on the left as being on the right and vice-versa (this is nothing to do with VRI in case anyone is watching).
Sadly we can't do that but a 4Der can. Also, if we were in a 4D space we could do it to by rotating our 3D slice around 180°. We could flip our body 180° effectively so that left is now right and right is now left.
gonegahgah wrote:A Mobius Strip in our world has two versions: clockwise and anti-clockwise.
Is it possible that a Klein Strip in a 4D world actually has a whole 360° of different Klein Strip versions?
Is that what my animation represents? Does it represent each of the different possible Klein Strips that we can observe (or a 3D slice of)?
gonegahgah wrote:* (Just from above) I was interested to notice in the first depiction that the y-line at the front rotates until it is the w-line at the back, and the w-line at the front rotates until it is the z-line at the back. So neither of those form the Mobius strip by themselves. It is the rotations of the circle cross-sections in-between that go to form the Mobius strip that we see. So the total Mobius Strip seen is a series of different angled diametre lines from differently angled circles of which only the diametre line can be seen. Cool hey?
Teragon wrote:... although I don't know where the morphing effect comes from as the colors represent the distance in the w-direction and I'd expect the w-coordinate to be constant.
Teragon wrote:Really don't know what's going on here. I'd like to take a look at the code you used, if that's possible, gonegahgah.
Teragon wrote:Yeah, but it's a matter of definition if you say "y goes to w and w goes to z", or "y goes to z and w stays at w". The unique feature of the surface is the surface normal vector.
Teragon wrote:As I have looked at the transformation of the object between the front and the back side, it wouldn't fit together with a toroidal rotation.
Teragon wrote:As I understand it, this kind of Moebius band has only one version, because it looks the same in w- and -w-direction. Therefore you can rotate this Moebius band into it's mirror image.
Teragon wrote:You're sort of right too that there are different possibilities to curve this Moebius band in 4D, as I describe them in the last paragraph. Those different orientations are transferable into each other with a toroidal rotation however, but only as long as every direction normal to the loop remains smaller than the radius.
gonegahgah wrote:Here are the parametric equations that produce the animation:
x(u,v,t) = (R + r * (sin(v / 2) * sin(u + t) * cos(t) * cos(v / 2 + π/2) + cos(u + t) * cos(v / 2))) * sin(v)
y(u,v,t) = (R + r * (sin(v / 2) * sin(u + t) * cos(t) * cos(v / 2 + π/2) + cos(u + t) * cos(v / 2))) * cos(v)
z(u,v,t) = r * (sin(v / 2) * sin(u + t) * cos(t) * sin(v / 2 + π/2) + cos(u + t) * sin(v / 2))
0 ≤ u ≤ 2π (30 steps), 0 ≤ v ≤ 2π (90 steps), 0 ≤ t ≤ 2π (120 steps)
gonegahgah wrote:One thing I realised (I think) is that you may be able to rotate around one axis even in 4D (the x-axis at the back). Is that possible? But still, it's getting complicated...
gonegahgah wrote:Teragon wrote:As I understand it, this kind of Moebius band has only one version, because it looks the same in w- and -w-direction. Therefore you can rotate this Moebius band into it's mirror image.
I'm not sure what you mean Teragon? A clockwise Mobius Band will look like a clockwise Mobius Band when you turn it over. It won't ever look like its mirror image (except perhaps in 4D by rotating it through itself).
There are definitely two types of Mobius Band in our 3D world: clockwise and anti-clockwise.
gonegahgah wrote:I figure it's possible - in our mind - to rotate the w,y,z-axes together at the back around the x-axis and that represents the different positions where the front W,Y can twist around to; giving a full 360° of possibilities.
Being 4D these might not really have a clockwise/anticlockwise orientation. After all is 'up' able to be distinguished as clockwise or anti-clockwise to both 'left' and 'forward' at once?
I'll draw a picture when I get home if that doesn't make much sense...
gonegahgah wrote:Even though a Klein Band is flat in 4D I think it might still be a combination of these occurances because of its twist.
How does that sound too, Teragon?
Teragon wrote:I think I know now how to interpret the animation. It shows a transition between the different twist directions, as you identified them, but it's a projection onto a 3-plane like the retina of a 4D being! It's a 3D image of a Moebius strip as a 4D being would see it. Altough rarely used this is my favorite form of representation. It would be possible to shade it in a way that it becomes evident what is more close to the observer and what is further apart (I'll do that).
Teragon wrote:You're right with assessment that there is a range of possible twists in 4D spheritoric Moebius strips. Still I can only repeat myself that these objects do not have a clockwise and counterclockwise orientation. Therefore only 180° of possible twists are left.
Teragon wrote:I'm dreaming about an animation where the whole 4D object is parametrized and rotations are made via rotation matrices. The w coordinate is then replaced by a value for the color shading, while the backside is not shown. Even more natural would be defining the shading according to the angle of the surface normal towards a virtual light source. This would be maby the most natural representation of a 4D object since it comes most close to how the process of seeing would work in any number of dimensions. Such a project is beyond my capabilities at the time.
Teragon wrote:Even more natural would be defining the shading according to the angle of the surface normal towards a virtual light source. This would be maby the most natural representation of a 4D object since it comes most close to how the process of seeing would work in any number of dimensions.
Teragon wrote:Green is the mean distance, the more red the closer to us and more blue the further apart (or the other way round). It's easiest to understand if you compare it to a 2D Moebius strip and draw the parallels.
Teragon wrote:I'm dreaming about...
gonegahgah wrote:Hmmm, I'm not 100% convinced of this yet... but give me a chance to explore it further... You might have guessed why I was drawing the clock faces...
gonegahgah wrote:I remember now one of the things I wanted to ask you in my previous post that was [ctrl][whatevered] out of existance...
There is a motion they describe in normal space called tumbling (out of control) I think. From what I understand it is where an 3D object is attempting to rotate along two axes at once.
Apparently it is very undesirable in space or in a plane.
I was wondering if the same motion in 4D could leave one axis stationary? Or is that impossible?
gonegahgah wrote:What is a six flat sided solid called where all the faces can be irregular?
Teragon wrote:Irregular hexahedron maby.
Teragon wrote:I wonder, when you form a 4D Moebius strip by twisting a 3D paper strip and glueing both ends together, if there is a most stable configuration the strip wants to take up or if every twisting direction takes the same amaount of bending energy.
Teragon wrote:Let me take up your example with the clock. A 4D clock may be described by four coordinates: +x pointing in "3-direction", +z pointing in "12-direction", +y pointing out of the front side of the clock and +w pointing at the ana side of the clock. Plus we suppose xyz plane is a mirror plane of the clock. Now we make a mirror image of the clock with respect to the plane xzw as shown in your figure. Everything is the same except that the front and back side of the clock are exchanged with the face pointing in -y direction. Now we make a turn around the xz-plane of the clock, leaving the the 3 and the 12 where they are. The front side goes to +y again, while ana goes to kata. Since ana and kata are the same, the mirror image of this object is topologically ident to the original object.
gonegahgah wrote:The blue is more in the ana direction and the red is more in the kata direction.
Though it would be nice to see it in the shape I created and not the pinched torus...
Teragon wrote:That's the difficulty with 4D objects. They're described by 3 parameters, but common visualization packages allow for 2 of them. I guess it would be quite different to map the 3 parameters to the 2 that are relevant in the 3D slice. Another drawback is that it's not an efficient form of programming to create every object and every rotation by hand.
Teragon wrote:gonegahgah, as I said, this is just a normal rotation around an obligue axis. Two rotations combined give another rotation around a different axis. There is only one rotational axis at a time in 3D. Are you familiar with matrix calculus?
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