Traversable Klein 'bottle' paths

[gonegahgah]

Here's a depiction of the different orientations the back cross-section can have for the different varieties of 4D Klein Strip.

The blue diametre rotates between the z-axis and the y-axis. The red diametre rotates between the w-axis and the z-axis.
The parametric equations for these are:
w = sin(u) * cos(t)
x = 0
y = cos(u) * sin(t)
z = (sin(u) * sin(t) + cos(u) * cos(t))

Note: Although it might look like similar to a 3D tumble it isn't. In a 3D tumble there are two rotations but each changes the axis of the other's rotation while they spin.

Now if I can only work out how to connect the front of the Klein Strip to these rear cross sections...

[Teragon]

Ok, gonegahgah. Don't really get what is going on here. But it's a movement in 3D, definitely. 3 coordinates are involved here (x=0).

[gonegahgah]

That's cool. I'll show the working one day soon.
I realised I could use the same formula on different axes to render something similar to what I used to have.

In this I've added the w-axis to the z-axis. Otherwise this should be one version of the Klein Strip.
If I could have rainbow colours for the w-axis (and still use the x-axis which is otherwise only used for stepping around the ring) then it would be better.
Anything you see bulging in the x-axis should actually be bulging in the w-axis.
Anything bulging in the y-axis and z-axis should be.

This is only one version of 360° of versions. I should be able to do another version (90° out from this one). I'll do that tomorrow.
Then somehow I need to work out a general version to cater for all the combinations...
Anyhow, the parametric equations are very similar to the last one:
w = r * sin(v) * cos(u)
x = R * sin(v * 2)
y = (R + r * (cos(u) * cos(v) + sin(u) * sin(v))) * cos(v * 2) [Note: I actually had "cos(u) * cos(v) - sin(u) * sin(v)" in this when I rendered it; that just seems to turn it into the 270° version]
z = r * sin(u) * cos(v)

In the case of the depicted version I combined w and x as: x = (6 + 2 * sin(v) * cos(u)) * sin(v * 2)
It is interesting to see the extra kink at the back on the right hand side...
You can see that I only do it for half the Klein Strip and I don't draw the second side. I should actually try that to see if it makes any difference?

I realised (now) that you are correct about the 'insides' being the walking surface Teragon.
Just as we walk on the consecutive line cross-sections of a mobius strip; the 4Der walks on the consecutive circle cross-sections of the mobius strip.

[gonegahgah]

I did a wireframe animation and noticed that something was wrong.
So I redid my working diagram this time rather than trying to coble it together from the diagram I had done for the rear aspects.
That produced a variation on the formulas; but alack that still didn't fix the problem.
It took me awhile and it finally dawned on me that I wasn't really adding w to x and therein lay the problem...
The formulas are now fixed and that mysterious kink is now gone; and it now more resembles the original depictions I made.

The wireframe is showing a different view then the filled image. This view seems to best show how the circle cross-section of the path rotates the entire way.

And, here are the corrected equations:
w = (r * cos(u) * cos(V))
x = (R * sin(V * 2))
y = (R + r * (sin(u) * cos(V) + cos(u) * sin(V))) * cos(V * 2)
z = r * sin(u) * sin(V)
In the animation I have put x = (r * cos(u) * cos(V)) + (R * sin(V * 2)).
Again if we could have rainbow colours then any bulge to the x-left at all parts of the ring would range to purple and any bulge to the x-right would range to red.
So the left side and the right inside would look purple ranging to the left inside and the right side looking red.

[gonegahgah]

Even that had an error in it. I noticed as I was trying to do the Klein 0 type.
Here are the Klein 0 type and 90 type looking down at them on the ground:

That little bulge at the front are the backside of the connecting planes. In some respects it looks better just drawing the circles...
I'll add the animations of these when I get home as well as the equations.

And here they are:

The left is the Klein 0 type again and the right is the Klein 90 type. I've shown these from the best angle to show off the rotation of the cross-sections.
The left one is shown with PHI 90 and PSI 310, the right one is shown with PHI 0 and PSI 50.

Here they both are from the same angle to compare the difference between them:

If you look closely at the left one you can see that the blue line goes from sideways (w-axis) to vertical (z-axis) and the red line goes from forwards (y-axis) to sideways (w-axis).
If you look closely at the right one you can see that the blue lines goes from sideways (w-axis) to forwards (y-axis) and the red line goes from forwards (y-axis) to vertical (z-axis).

Here are the equations for the left one:
w = (r * (-sin(u) * sin(v / 2) + cos(u) * cos(v / 2)))
x = (R * sin(v))
y = (R * cos(v)) + (r * sin(u) * cos(v / 2))
z = r * cos(u) * sin(v / 2)
In the animation I've made x = (R * sin(v)) + (r * (-sin(u) * sin(v / 2) + cos(u) * cos(v / 2)))

Here are the equations for the right one:
w = (r * cos(u) * cos(v / 2))
x = (R * sin(v))
y = (R * cos(v)) + (r * (sin(u) * cos(v / 2) + cos(u) * sin(v / 2)))
z = r * sin(u) * sin(v / 2)
In the animation I've made x = (R * sin(v)) + (r * cos(u) * cos(v / 2))

[Teragon]

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.
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.

Anyway this "tumbled" band is quit some interesting object. 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!

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]

Some of this may seem trivial, but I guess it's not bad to start by making the concepts very clear in 3D...

Moebius 2-torus ("Moebius strip")

Full name: n-Moebius 2-torus
Family: Moebius 2-torus
Cut of a full 2-torus, twisted ½+n times
Line extruded, twisted ½+n times and closed to a loop
Cross section: Line
Open directions: 1
Closed directions: 1
Twisted directions: 1
Chiral



The figure shows how a full torus is cut in order to obtain a Moebius 2-torus. You can imagine the red plane tracing out the whole loop by revolving around the black axis at the center. The position of the red surface on the torus can be described by one angle. On the surface one dimension is reserved for the loop, indicated by the red arrow. For every position on the loop there are two directions left. The open direction of the surface, which is cut by the thick black line, and the normal direction of the surface, indicated by the arrow in orange*. The arrow in orange may point in any direction inside the red plane and the direction it points to may change in the course of one revolution around the loop. This is exactly what happens with a Moebius strip. Within one revolution of the red plane through the loop, the surface normal vector does one 180°-turn, the thick black line traces out a Moebius strip.

A 180° turn is just the simplest possiblity. The figure below shows the simplest three versions of the Moebius 2-torus (1-Moebius 2-torus (180°), 2-Moebius 2-torus (540°) and 3-Moebius 2-torus (900°)).



We may go one step further. The red plane in the first figure of this post defines the lateral plane of the strip, which of course depends on the position on the loop. So the orientation of the surface normal vector inside the lateral plane is defined by just one angle and the surface may be color-coded with respect to this angle:

[img]"http://www.picshome.com/en/download.php?id=00342DFB1[/img]

Now, for the schematic representation of the Moebius strip one coordinate is redundant. We can just draw a circle and color code the orientation of the surface in the lateral plane (with a certain width of course, so that we are able to see it). Starting from the blue site and going around the loop clockwisey one time, we reach the complementary color (red in this color scheme) and thus arrive at the opposite side of the surface.



Colors can also help us to visualize the actual structure of the Moebius strip - as a 4D being would perceive it.



Here the strip lies inside the 3 dimensions perpendicular to the line of sight. Every point on the Moebius strip is the same color and therefore lies at the same distance to the beholder. Keep in mind that the observer is nowhere in the 3-space the object is plotted in, but outside in the color direction. The pictures below show the Moebius band inclined by 90° towards the beholder. The object looks now flat, but it has a depth. The redder, the more close to the observer, the bluer the further afar. The right version each represents exactly the same perspective for the 4D being, but a different one for us as we can't perceive the whole 3D perspective at one.



*(Edit:) If the concept of a normal vector is unclear, just consult https://en.wikipedia.org/wiki/Normal_%28geometry%29 or some other basic explanation.

[gonegahgah]

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.

My apologies Teragon. I went from starting the u-value stepping from the red line in my original animations to starting the u-value stepping from the blue line; so this has shifted everything 90°.

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.

Looking this up I still only have a bare idea what a surface normal vector is but hopefully the following animations will help. They are the 0° (formerly 90°) and 90° (formerly 0°) from front on:

This shows the x-direction to left-right, the y-direction into the screen, and the z-direction as up/down. The x-direction basically follows the middle of the ring so I've also added the w-direction to this.
The thinner section (or cross-over point) of the Band is the front for both and the cross-section travels left along the front around to the fatter section at the back where it travels right.
You'll see for both that the blue line starts off horizontal (because it is w=-2...2,y=0,z=0) and the red line starts off pointing into the screen (because it is w=0,y=-2...2,y=0).
The left image shows: blue line rotating in the vertical from being full w-sideways at front to vertical at back; and red line rotating in the horizontal from pointing into the screen y-wards at front to being full w-sideways at back.
The right image shows: blue line rotating in the horizontal from being full w-sideways at the front to pointing into the screen y-wards at the back; and red line rotating our way from pointing into the screen at front to being vertical at back.
The circle cross-section for each just fits itself to the orientation of the blue and red axes combined at each point.

Anyway this "tumbled" band is quit some interesting object.

Thanks Teragon. I feel we are getting closer now...

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!

Yes it is now that you mention it. The line cross sections turn with the ring and in the horizontal at the same time!

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.

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.

Also, I put together a general equation. It might be (one of four ways) to show all the basic Klein Band varieties. I want to test how the cross-section moves in some of the different angles before I can say.
But anyhow, this is what the cobbled together equation generates. It might or might not be the true equation. I said four ways (using placed negatives) but they all ultimately produce the same overall results anyway.


The parametric equations I cobbled together for this are:
w = (2 * (cos(t) * -sin(u) * sin(v / 2) + cos(u) * cos(v / 2)))
x = (6 * sin(v))
y = (6 * cos(v)) + (2 * (sin(t) * cos(u) * sin(v / 2) + sin(u) * cos(v / 2)))
z = 2 * cos(u + t) * sin(v / 2)
In the graphing program I added x and w together as: (6 * sin(v)) + (2 * (cos(t) * -sin(u) * sin(v / 2) + cos(u) * cos(v / 2)))
The v steps around the ring, u steps around each cross-section circumference, and t steps through the basic 4-twist Klein Band varieties...

[BTW: Noticed your post after I posted. Very cool! I'll have more look tomorrow.]

[gonegahgah]

Anyway this "tumbled" band is quit some interesting object.

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.

I just realised what you were meaning Teragon.
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?

[Edit...]
I realised this afternoon that the equation for the 3-twist Klein Band may be as simple as:
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))
This is simply the combination of my Mobius Band equations with a w-component. So for each visible line cross-section it is part of a circle cross-section that extends in the 4th dimension.
Here is the Mobius Band itself with no w-component. This is what we would see in our 3D slice of the 3twist Klein Band.

The base Mobius form leaves no direction untouched so when I graphed the Klein Band I first added the w-component to only the x-axis then only to the y-axis for comparison.
The rainbow effect would again perhaps be helpful if we had it...
The following images are (left) the 3twist Klein Band with the w-axis added to the x-axis, and (right) the same Klein Band with the w-axis added to the y-axis:

The PHI is set as 180 and PSI as 60.

Does this all look okay Teragon?

[gonegahgah]

I realised when doing some new pictures that 3-twist is not a good name for the simplest Klein Bands. That descriptor was probably just confusing things...
I'll try to think of a better descriptor.

Anyhow these new pictures just show off some simple Mobius Bands with different numbers of half twists. ie 1 half twist, 3 half twists and 5 half twists.

The parametric equations that draw these for the number of twists are:
x = (6 + 2 * (cos(u) * cos(v * n / 2))) * sin(v)
y = (6 + 2 * (cos(u) * cos(v * n / 2))) * cos(v)
z = 2 * (cos(u) * sin(n * 5 / 2))

The n should always be an odd number otherwise it is not a Mobius Band.
I used a PHI of 180 and a PSI of 75 to depict these Bands from a good perspective.

The formula for the w-direction would remain as "w = r * sin(u)" for each of these if we wanted to make them into '3-plane' Klein Bands...
The '3-plane' hopefully conveys that they only really change in the 3 planes rather than in all 4 planes...

[Teragon]

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?

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?

Almost. First of all, we need either a full 2-torus (without 'Moebius') twisted in 4D to get a Moebius 3-spheritorus or we need to cut a spheritorus. The cross section is a circular one, that's right, but the object is achiral in 4D (explained in previous posts).

I'm going to discuss the possibilities how to twist such a band in detail in some of my next posts. There are good ways to visualize them.
Firstly there is the number of twists that can vary.

Secondly there are 3 coordinates for a Moebius 3-spheritorus were the surface normal can go to, 2 perpendicular to the plane where the loop lies, one inside the plane. So the normal vector can lie perpendicular to the plane of the loop everywhere on the band (90°-band). Then there is the band, where the surface normal switches between completly in the loop-plane and completly outside of the loop-plane, just as the common Moebius strip in 3D (0°-band). There is a full 360° range of off-plane directions to go to for a 0°-band, but all these versions are identical - they're all rotated versions of the 0°-band. Finally the band can move between one direction perpendicular to the loop plane and any combination of the direction in plane and the second direction perpendicular to the loop plane, yielding an arbitrary minimum angle of the surface normal to the loop plane between 0° and 90°.



This is the 1-Moebius 90°2-torus. I'm going to visulize the 0°- and the 45° for you later.

Thirdly there is the possibility that the surface normal doesn't move straight between the two extreme directions, but shows an inclination to either side towards the third possible direction halfway. This is what you call a tumble. There are chiral tumbles and achiral tumbles, but it's an artifact of the deformation of the Moebius band to a tumble, not build in the band itself. If we allow for toroidal rotations the only parameter that's relevant for the Moebius 3-spheritorus is the number of twists.

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.

There are really just 90°, the rest are rotated versions of the 0-90°-bands. You know, there are only 2 basic toratopes in 3D: The 2-torus and the sphere. It is impossible to get a Moebius strip out of the sphere, even in higher dimensions. In 4D there are 5 basic toratopes: The spheritorus, the 3-torus, the tiger, the torisphere and the glome. These are the possible shapes to start from. That's all. Everything else would have to be some variation of these basic shapes. According to my level of knowledge each of these figure can produce cuts that are Moebius strips. If two or more of them produce the same Moebius strips, is another story. The spheritoric Moebius bands have just one loop, but an additional lateral dimension. That's why there are so many possibilities to twist them and they also occur in different shapes. Having two looped directions the rest of the Moebius strips in 4D is simpler in this regard, but harder to visualize.

[gonegahgah]

I'll go a few questions at a time so that I can understand it better... (I also have to sing early in the morning tomorrow and also have a concert to sing in in the afternoon!)

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?

I'm not quite sure to be honest. I need to look at the rotations a bit better; which I'll do here in this post...

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?

If you are happy that this is a shape and that it passes through our 3D-plane as a Mobius Band then the next question is: Is it a Klein Band?
Let's take a look...

It appears the red-line moves in a circle cross-section from facing into the screen (y-axis) at the front to being vertical at the back (w-axis) ie.
at v=0π(0°), u=0π(0°),1π(180°): (w,x,y,z) = (0,0,8,0),(0,0,4,0) ie. red-line is 2r in y-axis at front
at v=π/2(90°), u=0π(0°),1π(180°): (w,x,y,z) = (0,7.4,0,1.4),(0,4.5,0,-1.4) ie. red-line is 2r at 45° angle in x-z
at v=π(180°), u=0π(0°),1π(180°): (w,x,y,z) = (0,0,-6,2),(0,0,-6,-2) ie. red-line is 2r in z-axis at back
It appears the blue-axis doesn't change its orientation ie.
at v=π(0°), u=π/2(90°),3π/2(270°): (w,x,y,z) = (2,0,6,0),(-2,0,6,0) ie. blue-line is 2r in w-axis at front
at v=π/2(90°), u=π/2(90°),3π/2(270°): (w,x,y,z) = (2,6,0,0),(-2,6,0,0) ie. blue-line is 2r in w-axis at 1/4 around
at v=π(180°), u=π/2(90°),3π/2(270°): (w,x,y,z) = (2,0,-6,0),(-2,0,-6,0) ie. blue-line is 2r in w-axis at back

So the red-axis is changing orientation in our 3D-slice for each cross-section while the blue-axis remains locked.
In 4D a flat ground is perpendicular to the sky no matter what angle line we take in the ground 'cube'.
I extrapolate from that that even though the red lines are different orientations they are still all perpendicular to the rigid blue-axis.
And because they intersect the blue-axis midway to each they can all form legitimate circle cross-sections.

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]

I can start to see what you mean though Teragon.
If we build a path and put rectangle pavers down then it doesn't matter if they are left to right or right to left when laying them.
Equally if we put straight round pipes together it doesn't matter what there rotation is to each other.
So the 3-plain Klein band may just be a simplified version of one 4-plain Klein band... Is that correct?
[Edit: Although looking at the animation of the 4-plain Klein Band that twists to a cross-circle in the y&z-planes at the back; that doesn't appear to be the case. The 3-plain and 4-plain Klein bands seem to be distinct...]

[gonegahgah]

It would also seem that you could have matching 3-plain Klein Band varieties for each type of Mobius Band that we have. ie. 1/2 twist, 3/2 twist, 5/2 twist, etc.
Even more interesting it would seem that you can have 4-plain Klein Bands with different twists for the blue and red-lines.
You could have 1/2 red twist + 1/2 blue twist, 1 red twist + 1/2 blue twist, 3/2 red twist + 1/2 blue twist, 1 red twist + 3/2 blue twist, 1/2 red twist + 1 blue twist, etc...
Our Mobius Bands only have one direction of twist but a Klein Band has two directions of available twist. Does that sound okay Teragon?

[Teragon]

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?

I see, that would be a 0°-band, looking like an ordinairy Moebius band, just growing in width (r) and shrinking again (but only if it passes our 3-plane under the proper angle).
The form of the equations looks very promising, but you can't just go and plot 3 random coordinates out of the four, because there are only two parameters and the way the surface cuts the shown cross section doesn't always make sense (just try). As a basis for further calculations I suggest the following parameter form:

x(t,u,v) = sin(v)*(r(2u-1)*sin(v/2)*cos(t)-R)
y(t,u,v) = cos(v)*(r(1-2u)*sin(v/2)*cos(t)+R)
z(t,u,v) = r(2u-1)*sin(v/2)*cos(t)
w(t,u,v) = r*sin(t)

v: [0,2Pi]
u: [0,1]
t: [0, 2Pi]

If you want to plot a cross section in the x,y,z-plane that's moving in w I suggest this form (t is the time coordinate in this case):

x(t,u,v) = sin(v)*(r(2u-1)*sin(v/2)*sin(t)-R)
y(t,u,v) = cos(v)*(r(1-2u)*sin(v/2)*sin(t)+R)
z(t,u,v) = r(2u-1)*sin(v/2)*sin(t)

v: [0,2Pi]
u: [0,1]
t: [0, Pi]

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...

It's fine to make use of different models to get a better picture, but one should be aware of there limitations and how the get together as well. Your distinction is based on the concept of surface tangential vectors, which I think may be misleading at some points and is rather complicated. As I pointed more than once, defining vectors inside a rotationally symmetric surface is redundant, because their positions inside the surface are not defined by any feature and thus are meaningless, if taken by themselfs. More precisely there is one redundant degree of freedom in the description (in 3D, in 4D there are even two), the red and the blue bar can swap sites inside the surface, rotate by some angle or stay the same within one loop - the surface they describe is the same in any case. I could redefine the path of the blue and the red bar to describe the same object and get different results for what is a 3-plane and waht is a 4-plane. In fact you can define the red and blue line in a way that they show a tumble for any kind of twist, namely when one of them goes at 180° where the other one has started at 0°. Our cross section may be described by either one normal vector or a second rank tensor, but taking two vectors are dangerous, if we're not careful enough.

When I look at the surface normal vector instead, it traces out a 2-surface in any case. This surface lies in 3D in the case of the 0°-bands (and also for the versions between 0° and 90°), propably in 4D in case of the tumbled band, but only in 2D in case of the 90° band. So the outcome is similar, although its definition is a different one.

In any case it's a worthwhile observation that there is any number 1/2+n possible for both rotations. I don't understand this kind of tumbles yet, so I can't tell how to characterize them systematically. The problem is that there are many other possiblities to twist a band without a continous tumble. In this case the sense of rotation would simply change throughout the course of the loop. For example the surface normal could go from z at 0° to y at 90°, to w at 180°, to -y at 270° and to -z at 360°. Or imagine a symmetric tumble, where the red line rotates in a normal manner, but the blue line reverts its sense of rotation halfway! So I've got no clear classification for all types of possible twists, but as you'll see later, what I can offer is a way to depict a twist visually in a simple way!

From my charts you will (in theory) be able to see
- if two bands are identical (maby it's not that obvious in every case, I'll have to think about it)
- which minimum angle the surface normal confines to the plane of the loop
- if a band is simple or twisted in some more complicated way
- if the band is chiral or achiral
- the full range of possible twists that lead to Moebius strips
Some twists can get unclear on the other hand, as lines start to overlie, e.g. the 1 1/2-fold twist.

[gonegahgah]

I look forward to your category and comparison chart Teragon.
I'm getting ready for our convention in Melbourne so I'll be a bit scarce for a while but I'll keep checking in...

[Teragon]

Just check the forum once in a while. There are many things to do off the computer too, so I'm writing every now and then.

Moebius 3-spheritorus (gonegahgah's Moebius strip)

Full name: x°n-Moebius 3-torus
Family: Moebius 3-spheritorus
Cut of a full 3-spheritorus, twisted ½+n times
Circle extruded, twisted ½+n times and closed to a loop
Cross section: Circle
Open directions: 2
Closed directions: 1
Twisted directions: 1
Chiral (90°) / achiral (0°)

We are again cutting a torus, but now a 4D spheritorus. The first picture shows a cross section through the spheritorus.



Again the red plane is moving around the loop of the torus, but now it's actually a 3-plane and we can see only a 2D cross section of it. The red arrow is again normal to the red plane, always pointing in the direction of the loop. The black line inside the torus indicates how the torus may be cut at this location, but as we can only see 3 out of the 4 dimensions we cannot see all the possible orientations of the cut at this point.

So now that we know where we are on the loop we continue by replacing the direction of the red arrow by the 4th dimension and get this picture:



We see now that the red plane is actually 3D (for the sake of clarity only drawn inside the torus where it intersects). The axis the red plane revolves around in the first picture is actually a plane/2-line that shows up here as the grey plane between the two spheres. What is actually cutting the spheritorus at any position of the red plane is the dark red plane (the thick black line in the first picture was only a cross section of it). The surface normal, shown as an orange arrow, has two degrees of freedom - it can trace out any path on the red sphere within one revolution around the spheritorus as long as it ends 180° from where it originated.

If the orange arrow is pointing in the direction of the grey line it's lying inside the plane of the loop (0°-direction). If the orange arrow is pointing in any direction on the grey circle, it's perpendicular to the plane of the loop (90°-direction). We see now that in 4D the surface normal can do a twist fully outside the plane of the loop. Mind that the 0°-direction is dependent on the position on the torus in an outer coordinate system, while the 90°-directions are fixed.

We could depict the kind of the twist the strip does in a color diagram as we did for the Moebius 2-torus (I'm not going to do this) with the difference that we need a more complex color scheme to represent it. The lightness now shows the angle relative to the off-loop 2-plane, while hue shows the angle along the off-loop plane. (The figure is from the internet)



There's a far more clear way to do it. We can simply draw the path of the surface normal (orange arrow) on the red sphere while moving one time around the spheritorus.



The poles of the loop represent the 0°-directions pointing towards the center of the loop, while the equator represents the 90°-directions. The spheres on the top both represent 0°-bands. As you can see the minimum angle to the plane of the loop here is 0°. As the equator plane is outside the plane of the loop we can rotate the sphere around the axis going through the poles. The plane of the loop is not affected. Thus the two diagrams on the top represent the same Moebius strip and there is only one possible 0°-band!
Besides that's why the Moebius 3-spheritorus is an achiral object. Any band with a twist from z to x can be transferred into a band with the twist going from z to -x by a rotation of 180° inside the xw-plane.

The left sphere on the bottom represents a 90°-band, the right one represents a 45°-band. All these bands are represented by half great circles on a sphere. Mind that it doesn't matter where on the great circle the half circles start as we could define the starting point point of our diagram anywhere on the loop.

The figure below shows Moebius bands with different tumbles, meaning that the axis of rotation itself is rotated within the course of the loop. Some of them are 0°-bands, some of them descend from the 90°-band and have minimum angles between 0 and 90°. Intuitively we might say that these tumbles do have a chirality. You cannot rotate a lefthanded loop into a righthanded loop by spinning the sphere. It may be a bit more subtile, it's not clear enough to me yet.



Identical tumbles look different depending on the starting point you choose for the diagram. What distinguishes the tumbles on the top from the tumbles in the middle is only the starting points I've chosen for them. Of course the chirality and the inclination towards the plane of the loop also varies but it doesn't affect the shape of the curves.

There are only certain twists allowed on the sphere. If we do a second loop around the band the tumble has to continue smoothly and the surface normal has to point always in the opposite direction to where it pointed to in the round before! You can easily check this if you prolong the twist up to the point where it closes itself. (I can post some pictures later if there is general interest. This post is already long enough.) Moreover it turns out that for every orientation of the "untumbled" twist there are muliple possibilities to do a tumble - the rotational axis has to do an integer number of full rotations within one loop (one half loop on the sphere diagrams). These higher modes can be seen as an analog to the different possible numbers of twists within one loop, but the condition is n instead of n+1/2. While mulitple twists cannot be shown on the sphere diagrams, multiple tumbles look really nice (again maby later).

It turns out that the spheres on the top and in the middle show classical smooth tumbles while the ones at the bottom do not. They show a kind of tumble too, but because there is only one half tumble within one revolution around the band, the direction of the tumble changes its sense at the starting-/endpoint of the loop. There are also different modes for these tumbles, the rotational axis has to do n+1/2 rotations. In total we may characterize the mode of the tumble by just one number m = 2n with the even numbers representing the tumbles that keep their sense of rotation ("even tumbles") and the odd numbers representing the tumbles that change their sense of rotation ("odd tumbles").

Finally there are twists that are neither straight foreward nor tumbled. Every trace on the sphere is possible, as long as every point on it (indicating a direction) mirrored on the center of the sphere gives the point where we get to when we get one time around the strip.

I see now that it would be better to show the whole closed path on the sphere diagrams, i.e. two revolutions around the strip for more than one reason. The Identical curves with different starting points will look identical. The feature of every point on the curve beeing 180° to another point will become much clearer. (On the other hand the creator has to take care if this is actually the case.) It will become clearer whether the object is chiral or not, because you see if you can get the other version by rotating the sphere.

I quess that is a lot of information in a short text. Hope it's not too confusing. Enough for today, I'll do the rest of the Moebius 3-spheritorus another time.

[gonegahgah]

I find myself swinging back to my original equations now but with a better understood interpretation.
As a reminder they were:
x(u,v,t) = (R + r * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v / 2) + cos(u + t) * cos(v / 2))) * sin(v)
y(u,v,t) = (R + r * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v / 2) + cos(u + t) * cos(v / 2))) * cos(v)
z(u,v,t) = r * (sin(v / 2) * sin(u + t) * cos(t) * cos(v / 2) + cos(u + t) * sin(v / 2))
where 0 ≤ t ≤ 2π (though that doesn't really matter as the pattern repeats) which looked like:


My interpretation now is that these are just middle section 3D cuts of the the varieties of Klein Strip available in 4D.
You could call it a footprint if the Klein Strip were embedded in the ground up to its middle.

I'll try to explain this soon but a couple of new realisations have cemented this.

The only way for a 2Der to view a whole cross-section of a Mobius Strip is to stand it on its side.
If they then embed it into their 2D slice at midway the Mobius Strip will appear as a simple circle.
If the Mobius Strip moves our 3D left or right then it will appear to become an incomplete almost circular line.
Something similar too:

Which decreases at slower intervals to a dot and then disappears.

We figure a 4D Klein Strip should do a similar thing in our 3D world.
This time however we don't have to stand it on its side; we can lay it flat.
That aside, if the Klein Strip matches its Mobius Strip counterparts, we should get a plane figure in our 3D slice.
The cool thing is that if their orientations match then in our 3D middle slice of the Klein we will see a 3D Mobius Strip.
How cool is that! Which we get to see in the above animations (at what would be 90° and 270°).


Moving the Klein Strip sideways in 4D would see a similar effect as well.
The full Mobius Strip we see would separate at its flat point and shrink towards the opposite side.
One end would appear to move outwards slightly and the other end would appear to move inwards.
The opposite side would stay in place but grow narrower.
The whole path would grow narrower overall as well again to nothing by the opposite side.

But, what about the other angles of orientation into 4D?
A Mobius Strip can have only left and right twists but a Klein Strip can twist in any and all of the 360° of sideways available in 4D.

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.
I considered this seriously...
I then considered what this would look like if we moved those varieties sideways in 4D...
I've come to the conclusion that they would appear no different to the 4Der than a 3D-donut not matter what we do.
So to me this means that a flat ring path will not be a 3D slice of a Klein Strip.
I believe such transformations would appear to the 4Der as a Mobius strip morphing to a 3D-donut and back again.
So again, what about the other 360° of varieties of a Klein Strip. How would these appear to us?

Well, if we consider the 0° (180° & 360°) phase of my animation above...

If we were to take them as a 4D footprint (rather than our 3D slice) and you were to project them up vertically, what would you get?
The flat front, as a footprint to the 4D, is like us trying to walk on a line. To them it is an edge.
If we extend the flat front upwards (& downwards) it becomes a wall (a very thin wall but so is that back of a Mobius Strip).
The circular opposite side, as a footprint to the 4D, is like a path to them.
If we don't extend that path upwards it remains a path.
So here we then have an example of an object that is a path at one orientation and a wall at the opposite angle.
Sounds very like a Klein Strip. My conclusion is that it is.

As we know 4D footprints and 3D slices are very similar objects. They are just taking a different set of 3-axes.
So my conclusion is that such a 3D slice (ribbon on one side; cylinder on the other) can in fact be a 3D slice of two varieties of Klein Strip.

My discussions with Teragon had convinced me that I was wrong.
These further dissections of the matter in my mind however lead me to believe that the formulas were actually correct all along if just a little misconstrued on my part.
Instead of the animation being various rotated slices of the same Klein strip I now conclude that it is just middle slices of all the 360° of Klein varieties.
I should add that without my discussions with Teragon I may never have been able to get back around to what is hopefully a more correct interpretation of my original idea.

I would welcome to recommence these discussions, if I may, to determine whether that is now the correct conclusion...
It may take some time but hopefully I can submit supporting pictures for the above; when I can work them out.

[gonegahgah]

I should add the following comparisons too:


This one extends into the 4th dimension primarily from flat front progressively rotating around to extending only up/downwards (as can be seen in our 3D slice above).
The back section which is (and we see as) a 3D-cylinder forms the thin Klein wall (no extension into 4D sideways).
The front section (we see as a line) is part of the Klein foot path (a cylinder unseen off into 4D sideways).


This one extends into the 4th dimension from all points.
The back section is a thin 2D wall in our view and is a cylinder wall into the 4th dimension.
The front section (we see as a line) is part of the Klein foot path (a cylinder unseen off into 4D sideways).

Both of these varieties have a footpath at the front and a wall at the back! Cool.
However, they are not identical to each other. They are actually only two varieties of a unique circle (or semi-circle) of varieties...
If you paint one half side one colour and the other half side another colour then I guess you could consider it to be a full circle of varieties then.

[gonegahgah]

The following parametric equations maybe depict the sideways movement through the Klein Strips:

x(u,v,s,t) = (R + r * cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v/2) + cos(u + t) * cos(v / 2))) * sin(v * cos(s))
y(u,v,s,t) = (R + r *cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v/2) + cos(u + t) * cos(v / 2))) * cos(v * cos(s))
z(u,v,s,t) = r * cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * cos(v/2) + cos(u + t) * sin(v / 2))
where t is the variety of 4D Klein Ring and s steps us from the ana side of the object to the kata side of the object.
ie. 0 ≤ t ≤ 2π and -π/2 ≤ s ≤ π/2

The following is possibly an animation of the type 0 Klein Ring passing 4D sideways through our 3D space:
https://commons.wikimedia.org/wiki/File:Klein00A2K.ogg
The following is possible an animation of the type 90 Klein Ring passing 4D sideways through our 3D space:
https://commons.wikimedia.org/wiki/File:Klein90A2K.ogg

I'll have to look at this closer but its heading in the right direction I hope...

I realise, looking at this again, that there is definitely one error to correct so far...
I'll try changing the equations tomorrow to:
x(u,v,s,t) = (R + r * cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v/2) + cos(u + t) * cos(v / 2))) * sin(v * cos(s * cos(t)))
y(u,v,s,t) = (R + r *cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * -sin(v/2) + cos(u + t) * cos(v / 2))) * cos(v * cos(s * cos(t)))
z(u,v,s,t) = r * cos(s) * (sin(v / 2) * sin(u + t) * cos(t) * cos(v/2) + cos(u + t) * sin(v / 2))
I should explain the parts of the equations one day... Surprisingly it does make some sense...

The reason for the proposed change is because the Klein Ring version, that reveals itself as a Mobius Ring in our 3D space, should not retreat to the front; as it is presently doing.
The version, that reveals itself as the fattest middle version in our 3D space, should retreat as it is more in our middle 3D space already and less in the 4th dimension than the Mobius Ring looking version.
I suspect I also have to add an extra element that is similar to the Mobius Strip passing through a 2Der's plane space. I'll keep playing around with this...
The equation just keeps getting longer...

[gonegahgah]

Probably worthwhile to have an examination of the equations in their present forms:
x(u,v,s,t) = (R + r * cos(s) * (cos(u + t) * cos(v / 2) - sin(v / 2) * sin(u + t) * cos(t) * sin(v/2))) * sin(v * cos(s * cos(t)))
y(u,v,s,t) = (R + r * cos(s) * (cos(u + t) * cos(v / 2) - sin(v / 2) * sin(u + t) * cos(t) * sin(v/2))) * cos(v * cos(s * cos(t)))
z(u,v,s,t) = r * cos(s) * (cos(u + t) * sin(v / 2) + sin(v / 2) * sin(u + t) * cos(t) * cos(v/2))

You can notice they all basically share the following underlined parts:
x(u,v,s,t) = (R + r * cos(s) * (cos(u + t) * cos(v / 2) - sin(v / 2) * sin(u + t) * cos(t) * sin(v/2))) * sin(v * cos(s * cos(t)))
y(u,v,s,t) = (R + r * cos(s) * (cos(u + t) * cos(v / 2) - sin(v / 2) * sin(u + t) * cos(t) * sin(v/2))) * cos(v * cos(s * cos(t)))
z(u,v,s,t) = r * cos(s) * (cos(u + t) * sin(v / 2) + sin(v / 2) * sin(u + t) * cos(t) * cos(v/2))

First off comes:
1. (R ~ This is the radius of the Klein Ring. It appears only in the x and y equations as our ring only circles in the horizontal.

Then the first common part starts with the following (which also affects the second common part):
2. + r ~ This is the maximum path cross-section radius.
3. * cos(s) ~ This is not correct and I'm working on fixing it. It was meant to simulate moving the shape sideways in 4D.

The rest of the first common part creates the central mobius plane which is common to all the simple Klein Ring varieties:
4. + (cos(u ~ Creates the base mobius plane by stepping around its cross-section.
5. + t) ~ Rotates the shape around its cross-section just for effect (1>0>-1>0>1). Doesn't affect the shape.
The first common part is then multiplied by either:
6a. * cos(v/2) - ~ The horizontal plane component of the mobius ie. full flat cross section at front to add nothing at back (1>0>-1).
6b. * sin(v/2) + ~ The vertical plane component of the mobius ie. no vertical at front to fully vertical at back (0>1>0).

The second common part that appears is to create the axial component perpendicular to the mobius plane above:
7. * (sin(v/2) ~ This allows the varieties to never add anything to the front around to adding the full variety fatness at the back (0>1>0).
8. * sin(u ~ Let's us step around each cross-section to draw its outline.
9. + t) ~ Rotates the shape around its cross-section just for effect (1>0>-1>0>1). Doesn't affect the shape.
10. * cos(t) ~ This gives us our variety of Klein Rings by making the visible bulk squish and expand (1>0>-1>0>1).
This second common part is then multiplied by either:
11a. * sin(v/2))) ~ The horizontal plane component of the fatness ie. no fatness at front to full current fatness at back (0>1>0).
11b. * cos(v/2))) ~ The vertical plane component of the fatness. Adds, in a mobius oval fashion, nothing at front to current max at back (1>0>1).

Finally for the horizontal plane the resultant x and y are pushed to their correct orientation around the ring:
12a. * sin(v ~ for the x plane to get the x component of the cross section at the current place in the ring.
12b. * cos(v ~ for the y plane to get the y component of the cross section at the current place in the ring.
with the v modified by:
13. * cos(s ~ Replicates the receding that occurs if we move our 3D slice ana or kata-wards of the object (1>0>-1>0>1).
14. * cos(t))) ~ As we step towards the the two Mobius like varieties receding reduces to zero as it doesn't need to recede (1>0>-1>0>1).

As mentioned, I know that (3) is wrong but I'm fairly confident that 13 and 14 are correct.
The following animation shows how one Mobius axis is created and another axis is created that is perpendicular to that axis.
When you add these together you get the Klein Ring varieties:

[quickfur]

I'm late to the game, but just wanted to point out that the so-called Klein "bottle" is actually not a bottle at all, at least not in the sense a 4Der would think of a container that can hold liquid. I think you guys have probably figured this out, but just wanted to confirm.

Basically, the only reason it's called a "bottle" is because it consists of a closed 2D surface that wraps around in 4D back upon itself with a "twist", so that it becomes non-orientable. It's pretty much a direct analogue of the Mobius strip in 3D. In 3D, the Mobius strip is a line (expanded to a strip in order to make its ends have horizontal extent) that wraps around back upon itself, twisted so that what was originally the "top" surface connects with what was originally the "bottom" surface. Similarly, in 4D, the Klein "bottle" is made by taking a cylindrical tube (in the 3D sense), assigning, say, a clockwise orientation to one of its ends, and twisting it through 4D such that it connects back upon itself in counter-clockwise orientation. How is such a "twist" possible? It's because in 4D, a 3D object can be flipped onto its mirror image (just like in 3D, a 2D object can be flipped onto its mirror image, while in 2D that's impossible). The cylindrical tube, being a 3D object, can therefore be flipped onto its mirror image. Or, since we assume a flexible tube, one end of the tube can be twisted onto its 3D mirror image, so that it connects back to the other end in the "wrong" flipped-ness.

But since a 3D tube in 4D can't hold 4D water -- just like a Mobius strip in 3D can't hold any 3D water either -- the Klein "bottle", being a twisted 3D tube, is actually not a bottle at all. A 2D "surface" in 4D behaves more like a string than a surface; it doesn't divide space (i.e., separate one side of 4D space from the other, which is required in order for a container to hold fluid), and it can be knotted, etc.. 3D objects in 4D are "flat", so both the original 3D tube and the Klein "bottle" are flat objects in 4D, and cannot hold any water. The Klein "bottle" is just the 4D analogue of a Mobius strip, really.

[quickfur]

And interestingly enough, the Klein bottle construction in 4D actually doesn't require the original 3D tubing to be hollow at all. It can be a solid 3D cylinder, and as long as it's flexible, you can twist it in 4D such that the circular "lid" on one end is twisted onto its mirror image when it connects back to the other "lid". This produces a non-orientable 3D solid, basically the equivalent of a "filled" Klein "bottle". Except that it's not really filled in the sense of filling with liquid, it's just the solid version of the Mobius strip vs. just a pair of parallel edges without the middle of the strip between them.

Or another way to think about it, is that 3D objects embedded in 4D have two "sides", not in the 3D sense, but in the 4D sense of having a surface facing the 4D halfspace on one side of the hyperplane that the 3D object sits in, vs. the opposite side of that surface facing the other 4D halfspace on the other side of the hyperplane. You may say this is the ana side and the kata side of a 3D object. The Klein bottle twist is then just a matter of twisting the 3D hyperplane such that these two sides are interchanged, and then wrapping it around so that the two ends of the cylinder connects. So the ana side of one end connects to the kata side of the other end, and vice versa. Exactly like how a Mobius strip connects the "top" side of one end to the "bottom" side of the other end, and vice versa.

[gonegahgah]

I hadn't really thought about the bottle aspect so that is interesting QuickFur.
It is fortuitous that you speak about twisting Mobius Strips as well.
As you say, Mobius Strip can be made from a long rectangular piece of paper.
If you take this paper and give it a half twist along its length and joining the two ends, voila!

The principle is similar for a Klein Ring.
You start with a long cylinder which is just a strip to a 4Der and you join the ends.
A continuous twist still needs to be involved I believe.
I don't think you can just twist a donut into the 4th dimension at one point and call it a Klein Ring.
That would be like attaching the 2D paper above like a chain and adding an afterthought twist.
It wouldn't be a Mobius Strip; and the same goes for a Klein Strip.
So the approach would be the same as you mention QuickFur.

Looking at 'perfect' Klein Rings allows us to presume the 180° of varieties.
A Mobius Strip has only two varieties which are left and right varieties.
In 4D those two varieties become one as it is possible to rotate it 180° in 4D leaving it mirror image in our 3D.
The same goes for the 'perfect' Klein Rings.
There are only 180° of varieties because they can be flipped through 4D to get the other 180° of identical varieties.

I've rejigged the formulas a bit because I wanted to have the drawing begin from the fattest middle.
The formulas tended to lend themselves more to the drawing beginning from the flat section.

I wanted to do this so I could give the impression of rotating the part that we see the most bulk of.
The result is the following:

The left image shows the image starting with the fat middle cut through and twisted one side backwards and the other side forwards. We then twist this oppositely into 4D which does rotate the whole fat section.
So I wanted to simulate that behaviour.
I've included the image on the right to show how the two axes are created.
These are then added together to achieve the result on the left.

I'll work to add the modified formulas tomorrow...
I also want to show how these look when we move the Klein Rings sideways into 4D.
I'm have a picture in my head what this will look like in our 3D space.
I just have to work out the formulas to do this.
That's partially done as I mentioned with the receding.
I just have to work out how to depict the rapid reduction correctly...
I've got a picture in my mind of how this works...

[quickfur]

Well yes, it would have to be a continuous twist, not just at a single slice. And you can't start with something already attached, since it wouldn't have the right non-orientable topology. You have to start with unattached ends of a tube / cylinder, give it a twist, then glue the two ends together.

Anyway, in 3D there's a whole set of interesting and bizarre looping shapes that you can obtain by cutting a Mobius strip. Well, more precisely, dividing a Mobius strip, i.e., cut parallel to the edges, not to break the loop. Depending on whether you cut it exactly halfway in the middle, or 1/3 of the way, or 1/4 of the way, etc., you can obtain various new Mobius strips and sets of interlocking Mobius strips. A friend showed it to me once... it's pretty fascinating (and mind-boggling!). I wonder what kind of interlocking shapes we'd get if we did analogous cuttings on the Klein "bottle" (or Klein Ring -- I like that name ).

(And yes, now I'm just begging to know what happens if we cut a projective plane this way... )

[gonegahgah]

Thanks Quickfur. Sounds challenging. Hopefully do-able at some point in the future as well as the side stepping first suggested by ICN5D.

It just dawned on me this morning that I need to change the way I depict the rotation itself.
It occurred to me that the rotation I am showing does not give a good representation of the actual process.
I'm happy with the overall shape but somehow I need to taper/untaper, and not just transport, the drawing lines to show how the rotation more accurately occurs.
If a part of the path rotates from our space into the 4th space then it needs to appear to taper off while being replaced with path that's coming out of 4th space.
That shape itself shouldn't be the only thing that tapers it would appear. I have to think about how that will look?

I guess once again the 2Der observing a Mobius Ring standing on its edge with the ring in their plane is the best analogy.
All the 2Der sees is what we call an edge. So painting one side of the Mobius one colour and the other side another colour is useless as they won't see our two colours.
Instead what we have to do is paint each molecule in the Mobius Ring so that they align with the paper's orientation.
Each molecule has to be painted in a rainbow fashion around its 360° in line with the paper.
So the rainbow paint is impregnated into the paper itself.
By doing this the 2D slice the 2Der sees is a rainbow. This goes from the front upwards in say a clockwise fashion and down in an anti-clockwise fashion through the colours.
If we were to do this with just two colours the line would be one colour from the middle front to the top and bottom and the second colour around the back.
The 2Der basically only sees two points that are perpendicular to Mobius surface; That is the a point right at the front and a point opposite at the back if they get behind it.
All other points are rotated out of the surface perpendicular.

Our 3D view will do a similar thing for the 4D Klein Ring.
However there is not just one central line rotation but a plane of rotations.
So it is probably more worthwhile to use a rainbow of colours for the different untwisted angles in the cylinder path.
Then work out how these will look in our 3D slice when they are twisted through 4D in the different ways.

If we only had left and right rotations for the Klein Ring then we would only see an edge too; as in the Mobius appearing form of the Klein Ring.
But whereas we can only rotate clockwise and anti-clockwise, the 4Der can rotate in any of 360° of sideways ways.
This is why we get a greater variety of 3D views of a Klein Ring than would a 2Der of a Mobius Ring.
If a 2Der were to observe a Mobius Ring from a 4D space this would be altogether different and the number of varieties would be equal to Klein Rings.
Though the 2Der trapped to a line ring would probably see little difference once they aligned correctly.

If we were to just rotate a cylinder from our 3D space into the 4th Space along one axis we would see it ovalise along its length (if done perpendicular to the axis) until it becomes just a rectangle presence.
This is different to moving the object sideways into 4th Space where the cylinder would just appear to grow and shrink in size.
The rectangle would be the actual perpendicular surface to the 4Der. When looking at the full cylinder we are looking at what to them is the edge.
Same as a 2Der looking at a square. Their face is our edge.
If we treat the square as 0 thin (impossible and we wouldn't see it but for math purposes) then some interesting things arise.
If the square is rotated into our sideways the 2Der is left no longer looking at their considered square face (edge) or even at our considered square face but are left looking at an orientation of the square that we can not see.
Mathematically they are looking at a line that has greater depth than looking perpendicular to square as we would look at it, even though we are referring to the square as 0 thin.
If you give the square some sideway depth then this is easier to understand this. Looking at the face perpendicular to its plane is less deep than looking at it from an angle.
The following demonstrates:

The left 2Der looking at a square perpendicular from our perspective sees a line that has less bulk behind it than does the 2Der looking at it from an angle on the right.
But if the object is moved sideways into our 3D space perpendicular to their view it will run out faster to compensate for that.

Based upon a zero thin lower space model, in line with this, I assume this is why we see changing bulk sizes when we rotate objects into 4th space.
Also, the principle of angle of view is important I believe.
The left and right 2D viewers above don't see exactly the same line even though it is through the same cut point.
I am certain that their space sees a different orientation of that higher space line.
And the same goes for us I feel.
Unless we are looking from the 4Der's view directly perpendicular to the object we don't see exactly the same thing as them.
On any angle other than perpendicular we see a slice of that object in a way that they cannot see.
A close analogy is if we had red wood painted blue, they would always see blue, but we would see red except at the perpendicular (under a zero thin model).

So for the Klein Ring, the only part that we see the same as the 4Der are the flat areas as they are exactly as seen by the 4Der.
To them the whole thing is flat. Anything we see as not flat is our unique angle of view that is unavailable to the the 4Der.
They can see more at once but not from the angles that we get to see it from.

That's why I'm looking to use more than two colours to show up the different orientations we are seeing relative to how they are rotated into 4D.
Come to think of it, I believe Taragon was doing something similar.
The main problem is that I need to represent two axis of orientation as the cylinder rotates between y-z + w-y and y-w + w-z to create the various varieties of Klein Ring.
So rather than the use of a rainbow circumference I need to use a rainbow sphere somehow to leave a trail around the Klein path...
The main thing is that I think this is needed to evoke a sense of the rotation that is involved into 4D...

What does all the above jibba jabba mean? Essentially it is that it is for us as it is for a 2Der...
If they look at the midline of our Mobius Ring they will have very little idea of the actual rotation that is occurring along that line.
Each point along the line is at a different orientation to the them but they don't have diagonal arrows to show this rotation being only in a 2D plane themselves.
At this orientation only a single mid-point at the front and one at the back are orientated the same in both the 2Der's and our world.
They and we look at all the other points around the middle ring from a different possible exposure angle.
They see the points 'surface' from an angle that is inside the object whereas we see only the perpendicular surface of those points.
Doesn't seem like much of a difference but I think it is.
They could use a spread dots to show that the rotation is more clockwise left to right around to the top looking up or more clockwise right to left around to the bottom looking down (or both vice-versa).
These would taper towards being closer together at the middle point in front of them to show that that is the least twisted section of the Mobius Ring.
They could alternately use colour or shade too if that makes any sense to them.

In 3D we have a little more detail to play with...
However the spiral I was using is not enough to show how the path from the front point rotates around to either up or down while the hidden perpendicular point at the front rotates around to either forwards or back (fat form).
Nor its perpendicular counterpart where the path from the front point rotates around to hidden in the forth dimension while the hidden perpendicular point at the front rotates around to up or down (Mobius form).
Or one of the varieties in between of course.
It is these that I want to depict and provide easier recognition of that process. I will have to think how best to do this?

For the 2Der's Mobius they could use a continuous rainbow.
For our 3D slice of a Klein Ring we could add shade as well so around to top would be lighter and around to bottom would be darker. The rainbow itself would circle between the x, y and w axis.
I don't have the luxury of co-ordinating that much colour so I'll have to think up something simpler...

[Teragon]

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.

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 o fit. 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.

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.) 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. 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). To get a feel about it, here's just a common 3D Moebius strip, rotating through four dimensions:



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.

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.

It depends on the mentioned twist of the Moebius strip. Only a 0°-band will yield a slice that is identical to the 3D Moebius strip. But as you state anyway in the end, every frame of your animation is the same cut taken for a different Moebius strip.

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.

On the rest I do agree.

[Teragon]

Considering you second post: Exactly! These are the two extreme cases of what I like to call Moebius-Spheritori - the 90° and the 0° one.

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).

[gonegahgah]

Hi Teragon,
Nice to see you again

I'm getting there with my new depiction. The following is a hastily put together first frame.

The equations are getting longer and longer and hopefully I can tidy them down somewhat...
I've managed to get half of the new animation (one red and one blue section) mostly working now.
I think only two colours (total four sections) will be necessary but I'll see how that feels when done.
I'll keep plugging away and hopefully get it completed soon...

I've noticed already that the join at the fattest part is not as smoothly continuous as I would like.
I suspect I'll have to rejig some part of the formulas in someway.
The initial animation will at least give the idea hopefully once complete and then I can look to find what needs to be re-engineered.

[gonegahgah]

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.

Thank you Teragon. When I came back to this thread I was sad to see your images gone. Looking at them again I have a better feel for them now.

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.

Absolutely, I feel I'm more on track to add sideways movement as an animations. Hopefully rotation as well at some point.

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.

Me too, but I can't do that with what I have yet sadly.

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.)

Awesome Teragon . I'm pleased to hear that you are also making this for our solid objects in 4D! I look forward to seeing that .

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.

Maybe one day you can also help me to create a program to show 4D objects using my rotated projection model. That's something I hope to make one day.

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).

Like we see a square as having an interior (or two sides) but we don't think of a line in the same way; whereas a 2Der does see a line that way.

To get a feel about it, here's just a common 3D Moebius strip, rotating through four dimensions:

That also nicely shows how the Moebius strip can go from the left version to the right version by rotating it through 4th Space. [Which I notice you mention next...]

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.

These colour versions make clearer sense to me now, cool! I guess that is the process of my brain adapting to new models?

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.

I've had another conclusion (or realisation) as well. We discussed a lot about interiors previously somewhere in this thread...
We spoke about how the solid part of the Klein Ring representation contained an 'inside'.
I had, for a bit, thought that they must be hollow and you explained that they weren't; and you were absolutely correct.
My new realisation is that when we see a 'solid' of a 4D object in our 3D slice that it is either one of: 1) a 4D solid, or 2) alternately that we are looking at a flat 4D object edge on.
The distinction is that the solid seen is inside but not surface. It's all about the angle of viewing.

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).

I might need you to explain this a bit more please Teragon.