Teragon wrote:Take your time.
Agree with your definitions. So a "strip" in 4D is a 3D object?
Teragon wrote:The "thickness the Klein surface needs" was maby an inapproprate wording. It's clear that surfaces need a certain thickness in order to sustain even their own weight. What I actually meant is, in 4D you'd have to balance on a (quasi-)2D surface like on a tightrope in 3D. And in 4D, orientability of 2D surfaces is somewhat different than in 3D, because a normal vector may not be defined (only a normal plane). Imagine balancing around a loop on a 2D Klein surface in 4D. You may always stay on top of it. Reaching the same place again, how do you know you've switched side? The only thing that will have happend is that left and right (the directions in which you balance out) have switched.
Teragon wrote:In order to walk comfortably you need also a certain extension in the second lateral direction. Now the object hase a normal vector pointing always in the vertical direction, the surface has now an upper side and a clearly distinct bottom side. So it's a path on which left and right flip sides within one circle without you turning headfirst.
Teragon wrote:If the 2D Klein surface in the middle of this 3D promenade was extended vertically to a wall, you'd be able to cross the wall just by walking along (in this case a bridge is needed). Such a 3D wall has an open, non-orientable surface, just like a Moebius strip in 3D.
gonegahgah wrote:Hi Teragon,
Sorry for taking awhile. I've been trying to reconcile the following:
I realised that when we rotate into the 4th dimension that the Klein Strip would also appear to rotate in the opposite way with an in-place twist; and not just appear to inflate and deflate.
However when I add that to the equation I appear to get the above...
Secret wrote:Teragon wrote:The "thickness the Klein surface needs" was maby an inapproprate wording. It's clear that surfaces need a certain thickness in order to sustain even their own weight. What I actually meant is, in 4D you'd have to balance on a (quasi-)2D surface like on a tightrope in 3D. And in 4D, orientability of 2D surfaces is somewhat different than in 3D, because a normal vector may not be defined (only a normal plane). Imagine balancing around a loop on a 2D Klein surface in 4D. You may always stay on top of it. Reaching the same place again, how do you know you've switched side? The only thing that will have happend is that left and right (the directions in which you balance out) have switched.
Consider the following:
Since the klein strip is a 3D nonorientable manifold, we can locally define a vector that is normal to this 3D "surface" (chorix as defined in Polygloss, hypersuface in 4D) at all points. Now if you imagine translating this vector along the klein strip following the black arrow, you will reach the "pinched region" which you will then flip (as looking from our 3D vantage point, that part of the surface dived into the other part, thus seems as if it is moving inside. However inside and outside to a 4Der is not bound by the 2D surface/manifold, rather it is bound by a 3D manifold. To a 4Der, the 3D manifold is basically "thin" in the 4th direction, thus they define the "front" and "rear" of this 3D surface based on the normal vector pointing towards or away in 4D from the surface, analogous to how we define up and down on a 2D plane) (NB the dark blue thing is the analogous situation for 3Ders walking on a moebius strip (assuming they don't fell over)
Secret wrote:EDIT: About the wall thing, no the 4Der cannot leave the loop unless he/she jumped, because buidling such a wall with necessary make it extend both anwards and katwards, because it flips when it loop through the klein strip's twisted region, thus enclosing the klein strip completely (enclosing as in analogous to a cylindrical wall in 3D enclosing access to the region in the middle). however such structure must have a small region of self intersection in 4D space, because the klein bottle is already 4 dimensional thus all avaiable directions in 4D that allow extrusion without any obstruction is being used up
To check whether you will collide into something in 4D, remember that if we assume gravity is in the 4th direction, then a 4Der has only 3 degrees of freedom to walk around
Secret wrote:The klein strip is derived from a klein bottle in that you fill in the middle, empty region of the klein bottle, similar to the 2D paper middle of a moebius strip. So it is basically a moebius cylinder
Teragon wrote:Hey gonegahgah,
I'm just starting to follow your animations, so I can't tell if something's wrong.
Still not sure if we're talking about exactly the same object, because for a 3D surface closed in two directions, open in one, I see no reason why it should intersect itself. When I watch the moving cross-section, the strip seems to be closed in 3 directions instead of two.
Secret wrote:Can you try rotate this about the xz plane as shown, I am suspecting the green line region will expand into a circle and the olive region to contract into a line in the projection/3D rotation slide?
ICN5D wrote:You could also combine both the translation and rotation together, for the animation. The key is to fully translate (side-step) from one side to another in 4D, at certain increments of a rotation. I usually make five scans, at 0, 22.5, 45, 67.5 and 90 degrees. See here for what that looks like with toratopes.
The function that does this is:
(x*sin(t)+a*cos(t))
(x*cos(t)-a*sin(t))
where 'a' is translate, and 't' is rotate. Not sure how to fit that in a parametric function, since I work with implicit forms.
gonegahgah wrote:I think, as you are starting to get some feel of, one of the tricks is that a Klein Bottle is basically a 3D object whereas a Klein Strip is a 4D object...
All those renders I have made look like shells with hollow insides.
However what we see as insides of the Klein object in the renders are not insides at all. As I say I'm not 100% just what that 'space' is just yet and will have to ponder it...
Though it is starting to come to me...
The Klein Strip, just like a Mobius strip, has no insides as you are thinking of them. It is not a tunnel.
The entire Klein Strip path is exposed to the air; just as is the case for a Mobius strip.
Teragon wrote:As I understand it there is an inside of the Klein strip, in the same way as there is an inside of a Klein bottle, because of the intersection (even though the 3D surface has only one side). To get inside from the outside you would have to remove part of the surface, just like in the case of the Klein bottle. If there was a hole to the outside, you could see it in several cross sections (3D hole), or the surface would not intersect itself (which is impossible for a closed 3D surface in 4D... the Moebius strip in 3D is an open surface). What you see as inside in the animation are slices of 4D-air.
Teragon wrote:What I don't understand yet is, and here may be some crucial point, the rotation shows an object that intersects itself in a Moeubius strip, while in the sliced bread animation, the intersection obviously traces out the area of a circle. In the first case it seems like in two directions the surface flips sides, whereas in the second case it seems like only one direction. Where is the middle of the object along the dirction the cuts and taken, where is the start?
Teragon wrote:In any case it's a nice object. Quite different from the Moebius strips I've described. When I've got all the clues together I'm going to give a systematic summary of the different onesided surfaces in 3D and 4D.
gonegahgah wrote:It is interesting that if we keep tipping the Klein strip for them to walk around the tunnel then they would take two loops to get back to the beginning.
They could draw some graffiti at their start point and walk around one loop and wonder who cleaned away the graffiti.
The could then continue to walk around the loop again and suddenly find that their graffiti has magically returned. Cool!
gonegahgah wrote:Teragon wrote:What I don't understand yet is, and here may be some crucial point, the rotation shows an object that intersects itself in a Moeubius strip, while in the sliced bread animation, the intersection obviously traces out the area of a circle. In the first case it seems like in two directions the surface flips sides, whereas in the second case it seems like only one direction. Where is the middle of the object along the dirction the cuts and taken, where is the start?
I'm not quite sure what you mean Teragon?
The rotational one shows a full 360° of views of the front of the Klein Strip while we stand in one spot.
The side-step one only shows only the one view rotation at 0° and we are moving ourself sideways into the 4th dimension while maintaining this 0° orientation.
The side-step one involves sideways movement (ana/kata) while the other involves standing on one spot and rotating.
Teragon wrote:That tipping thing is an interesting aspect. I think that there is no way to pass between inside and outside (if there is no extra hole), not even for a 4D being. It would be visible in the sequence of cross sections. Otherwise the surface would have to have a 2D edge.
Teragon wrote:That was an important information! I had interpreted it as a cross section instead of a projection before, now I can make the question more clear. The moving cross section shows an object reminiscent of a spheritorus:
Teragon wrote:Essentially, if you exchanged hight (z) for the invisible dimension (w) in this depiction, you would find a simple torus as a cross section.
Teragon wrote:If you exchanged x or y with w, you'd see two ellipsoids or a sphere and a circle.
Teragon wrote:The rotation is more complicated to understand for me. What confuses me is that I can see no invariant plane in the object. Any (single) rotation should have its invariant plane, but in the animation there's only the part that's always flat staying invariant. The segment at the opposite side should also stay the same.
Teragon wrote:Same here, it's usually easier to learn and get food for though when there is exchange with like-minded people
Teragon wrote:Ok, the right object is flat like a sheet of paper in 4D. You have applied so much efford. Essentially I just wanted to know, how the invariant plane of the rotation is oriented and where the origin of the rotation is. If I don't understand one basic point, it's inefficient to go through all the further details, maby everything else would become clear by itself if just this point was clarified.
Teragon wrote:...3D plane (actually I prefer this term over volume in 4D)...
Teragon wrote:...in the samer manner the object would keep its shape in one plane. This is my point. There is no invariant plane in the animation!
Teragon wrote:Meanwhile I learned a lot about Moebius bands in 3D, 4D and 5D and and found a conceptional way to visualize some of their properties. I'm going to make some figures when I have time.
gonegahgah wrote:First off, thank you for your patience. I thought initially that you were asking me questions to learn but I now realise that I am more the learner in our roles.
gonegahgah wrote:Now, I could depict the world to the left in the up direction along with to the right in the down direction. I could alternatively depict them the opposite way around as left in the down direction with right in the up direction. It wouldn't make any difference to the 2Der as they have no sense of where left and right actually are.
gonegahgah wrote:This also appears to be the case for the spinning cube in a 4D space. Except that this time there are 360° of interchangeable axis; and not just left and right. In other words you can make left be any of 360° of sideways with right being opposite to this and ana/kata being perpendicular. In the next moment you can then change left to being any other angle in that 360° of sideways with the other axis moving in correspondence.
There is no preferential left/right in 4D so it makes no difference to us. We can change left/right to being any of the 360° of opposing sideways available - with ana/kata being perpendicular - and we can do this at anytime.
I am wondering if this relates to my animation and if this supercedes invariance? Mind you I don't know that yet but hopefully?
Could I get your thoughts on that? I'll think about it more as well...
gonegahgah wrote:So perhaps, in our example, the one invariant would be the Mobius strip at the centre which doesn't seem to change at any change of the base sideways axis angle (although it is hidden in the bulk most the time).
How is that sounding Teragon?
Teragon wrote:gonegahgah wrote:Now, I could depict the world to the left in the up direction along with to the right in the down direction. I could alternatively depict them the opposite way around as left in the down direction with right in the up direction. It wouldn't make any difference to the 2Der as they have no sense of where left and right actually are.
This is only the case if the object is actually symmetric with respect to the plane of the 2D being, i.e. if the left side is the mirror image of the right side.
The object that's rotated through 2D space may also be asymmetric or the projection may repeat itself 3 times or more with a 360° turn. It depends on the rotational symmetry of the object with respect to the rotational axis. By rotating a higher dimensional object through a 3D cut, I can learn about its symmetries.
gonegahgah wrote:The key thing is that the rotation can be up or down but with only one corresponding to left or right at a time. It doesn't matter. The 2Der is not able to say which way is the preferred left or which way is the preferred right.
And if need be it is easy to flip this usage...
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.
Things up high will still be in that direction and things further in the distance will still be further in the distance.
The only thing that will have changed is that we are now seeing everything in reverse in our sideways direction.
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.
Again not sure what your point is, I thought we were talking about invariant planes. Concerning your deliberations I would say:
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.
Teragon wrote:One the lefhand side the surface normal vector lies fully inside our cross section. What we see as flat region is actually a cross section orthogonal to the surface, with part of the edge as a boundary on both sides. Going in any direction along the strip, the normal vector turns into the hidden w-direction. What I have shown is just its component in the (x,y,z)-direction. The directions involved in the twist here are just up/down and ana/kata. On the right side the normal vector points wholey into the w direction, so we see a tangential cut through the 3-surface now. In return we see the whole volume of the surface and the whole area of the edge now.
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