## Traversable Klein 'bottle' paths

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Traversable Klein 'bottle' paths

Just as a moebius strip is a path - though non-traversable in 3D - so can a Klein 'bottle' 'tunnel' actually be a path open to the sky in 4D; I'm guessing?
It was interesting to see discussions of Klein bottles which are new to me and only came to my attention when mentioned in the facebook page for the game Meigakure that is coming out some time.
I'm guessing that unlike a non-Klein circuit that a Klein circuit will require an actual bridge? Is that right?
gonegahgah
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### Re: Traversable Klein 'bottle' paths

I'm following is my figuring of what a 3D slice of a simple klein 'bottle' would look like.

Does this conform with what the knowledgeable people here would concur with?

If it does then I can see that there is no need for a bridge afterall.
I also realised that any sort of mobius like structure can't be traversed - at least without sticky feet like a gecko.
If you are on the up surface and you walk around a mobius strip then eventually you have to be on the opposite side to where you were.
The opposite of up is down so at some stage you must be upside-down.
So to answer my own question, it isn't possible to traverse a klein 'bottle' in 4D for us upright only creatures.

I'm keen to know if my depiction agrees with actuality.
gonegahgah
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### Re: Traversable Klein 'bottle' paths

I've been puzzling what the change of shape of this would be as we rotate it through the fourth dimension.
This has made me realise a few things...

Firstly that a slice can be vertical or it can be horizontal or some angle and orientation between.
The slices I'm mainly interested in for projection are vertical slices as this maintains at least a vertical orientation between each of the dimensions.

What this means for the object, I have depicted, is that if we rotate the given vertical slice, shown, around the horizontal; it will have an interesting visual effect.
What we would see is that the flat part shown at the front would stay fairly much as it is.
That is it forms a 4D cylinder path section with no vertical height and rotating horizontally just lets us see all 360° of its rotated sectional slices.

Now, like a mobius strip, part of the klein path twists so that its 'opposite sides' or paths are at some point standing on their 'edge' sideways; making them like back-to-back cliffs.
For the present orientation, in my depiction, this can be seen in the back vertical part of the path.
But what happens if we rotate our slice sideways into the 4th dimension?

Going back to our flat 'cylinder section' of the path, it is flat on the ground; so all of the its cylinder occupies the 4D ground space.
But with the 'cylinder section' that is on it's side; instead part of the cylinder is vertical and its perpendicular diameter is still in the horizontal space.

What this means for the object depicted is that as we rotate through the 4th dimension horizontally it will transform from the shape shown.
It will transform to one where the on-its-side section will balloon out to a circular (or cylindrical) cross section. The rest of the path will also transform to a morph between the flat front and this changing back.
So from flat at front to squashed ovals to wider ovals until a full circle is formed at the back.

An interesting thing is that you only need to turn 90° to go from the depicted flat to a cylinder. Going a further 180° returns you to a cylinder. A further 90° returns you to the current orientation.
Further interestingly those two cylinders are not identical. One is the overside of the Klein strip and the other is the underside.
This could be visually depicted, as is sometimes done for a mobius strip, by painting one half one colour and the other half another colour.
It is with further note that it might be useful to depict that those paths have a twist by showing a slow twist striation along the length.

In the present orientation us 3Ders couldn't walk around this Klein strip.
However, when rotated 90° we 3Ders could walk along the top of the depicted Klein strip; but a 4Der couldn't; just as we couldn't walk along a razors edge; as it would appear to them.

Reading about Klein bottles being discussed at the Meigakure facebook page also raised a question as to whether I am discussing Klein 'bottles' or not?
I assumed that a Klein bottle was representing a 4D path and not a 3D path in a 4D space. Could someone tell me the correct interpretation?
Although the bottle is hollow in 3D I had a assumed it was not in 4D and formed a 4D path. Is it instead a 3D path in a 4D space?
gonegahgah
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### Re: Traversable Klein 'bottle' paths

The construction of the klein bottle does not actually require a bridge, if you resolve the surface into two dimensions.

Take a rectangle, and divide the left and edges into 5 parts, labeled A-F. Across the top, label the five parts 1 to 5. This would divide the rectangle into 25 game-squares.

You first make a cylinder by joining the sides, so the letters match. So going from A5 you walk not into A6, but A1. This is because the 1 column follows the 5 column. From the top, the numbers run 1-5 clockwise, at top and bottom.

The second part is to join top to bottom. The section through C shows one circle, and the AF join can be either inside, outside or separate from the C circle.

I: The separate circle is made by joining top to bottom by joining the ends of a long tube or hose-pipe. Because A and F are inverted from their original place, the circles run anticlockwise.

II The circle around the C-circle is formed in the same way you might take a sock off: rolling the hem towards the toe, outside the sock. The orientation of this circle does not change, the hem still remains clockwise from the top.

III The circle inside the C-circle is done the same way, but the hem is tucked into the sock as it goes down. It's still clockwise.

There are six ways of connecting A to F. So if A and F are both I or both II or both III, the resulting figure is a torus.

If one connects A as II (outside) and F as III, (inside), it's a torus, still, but formed by rotating an 8 around a vertical, ie 8 8 .

The connection of A as I and F as II or III gives rise to a klein bottle. Both of these are the sections before and after the handle crosses inside.

When the connection is made inside, one has to get the outside circle of I inside the circle so it is co-axial with the other circles. This means that the connection of I to III represents the interior of the bottle, where the bottom end is now part of the internal stork.

When the connection is made to the outside, A must yet to swallow the C circle, which means that we are looking at a connection made in the 'handle' of the bottle.

For a game based on squares, all one really needs is a rule that going south from F,n means that you end up in A,5-n. No handle required. As it is in 4D.
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### Re: Traversable Klein 'bottle' paths

(Please read from the bottom up, because I accidentally pasted the quotes in reverse choronological order)
gonegahgah wrote:I've been puzzling what the change of shape of this would be as we rotate it through the fourth dimension.
This has made me realise a few things...

Firstly that a slice can be vertical or it can be horizontal or some angle and orientation between.
The slices I'm mainly interested in for projection are vertical slices as this maintains at least a vertical orientation between each of the dimensions.

What this means for the object, I have depicted, is that if we rotate the given vertical slice, shown, around the horizontal; it will have an interesting visual effect.
What we would see is that the flat part shown at the front would stay fairly much as it is.
That is it forms a 4D cylinder path section with no vertical height and rotating horizontally just lets us see all 360° of its rotated sectional slices.

Now, like a mobius strip, part of the klein path twists so that its 'opposite sides' or paths are at some point standing on their 'edge' sideways; making them like back-to-back cliffs.
For the present orientation, in my depiction, this can be seen in the back vertical part of the path.
But what happens if we rotate our slice sideways into the 4th dimension?

Going back to our flat 'cylinder section' of the path, it is flat on the ground; so all of the its cylinder occupies the 4D ground space.
But with the 'cylinder section' that is on it's side; instead part of the cylinder is vertical and its perpendicular diameter is still in the horizontal space.

What this means for the object depicted is that as we rotate through the 4th dimension horizontally it will transform from the shape shown.
It will transform to one where the on-its-side section will balloon out to a circular (or cylindrical) cross section. The rest of the path will also transform to a morph between the flat front and this changing back.
So from flat at front to squashed ovals to wider ovals until a full circle is formed at the back.

An interesting thing is that you only need to turn 90° to go from the depicted flat to a cylinder. Going a further 180° returns you to a cylinder. A further 90° returns you to the current orientation.
Further interestingly those two cylinders are not identical. One is the overside of the Klein strip and the other is the underside.
This could be visually depicted, as is sometimes done for a mobius strip, by painting one half one colour and the other half another colour.
It is with further note that it might be useful to depict that those paths have a twist by showing a slow twist striation along the length.

In the present orientation us 3Ders couldn't walk around this Klein strip.
However, when rotated 90° we 3Ders could walk along the top of the depicted Klein strip; but a 4Der couldn't; just as we couldn't walk along a razors edge; as it would appear to them.

I am sorry I cannot help you on this one as I am still not good at slicing an object that has a continuous twist in the 4th dimension.

However I am guessing that if gravity always points down regardless of where you are, then after passing through the twist, the 3D person within the cylindrical hose that made up the klein bottle, should fell to the floor. Which means there would be no inversion possible along the vertical direction

(Sketch showing near the twisted region viewed as a parallel projection, not as a slice obtained from intersecting an angled hyperplane. Bottom is the analogous 2D version that occurs for a moebius strip standnign upright with a 2Der walking in the middle, area region. Grey is what the person should be like if gravity is not always downwards, but get twisted by the twist also. Black is what is predicted will happen after the person left the twist as he/she fall to the floor, get up again and continue to walk)

http://i.imgur.com/x12t6jQ.png?1

However, if the twist (which in the 3D projection you show, the line region) is parallel to the direction of gravity, then it is still possible for the person to become left right inverted after passing through it, without any ill effects discovered (since 3Ders embedded in a 3D manifold in 4D cannot normally perceive the bend along the extra dimension, w)

gonegahgah wrote:Reading about Klein bottles being discussed at the Meigakure facebook page also raised a question as to whether I am discussing Klein 'bottles' or not?
I assumed that a Klein bottle was representing a 4D path and not a 3D path in a 4D space. Could someone tell me the correct interpretation?
Although the bottle is hollow in 3D I had a assumed it was not in 4D and formed a 4D path. Is it instead a 3D path in a 4D space?

Yes this is a Klein bottle, it is analogous to the moebius stripe made by twisting the paper 180 deg and then glue the ends together.

A Klein bottle is basically a moebius strip rolled up into a cylinder (since you need two moebius strips glued at the circular boundary to make one)

A moebius strip is a non orientable 2D manifold (i.e. a non orientable surface or hedrix) with a circular boundary. If we make it "hollow", then all you left with is the circular boundary, which can be deformed continuously into a circle (i.e. homeomorphic to a circle).
A moebius stripe need at least 3 dimensions to be made (otherwise you will end up having intersections)

(Following I am not very sure...)
To define something walking on top of a n dimensional manifold, you need to use normal vectors (vectors that are orthogonal to an infintesimal regions on the surface) at all points of the manifold. Therefore the normal vectors (hence the entity that walks on it) need to be at least n+1 dimensional. In an orientable surface, you can cover the manifold with normal vectors all pointing either inwards or outwards, thus you can talk about the "inside" or "outside" of the manifold. This is not possible on non orientable manifolds as the twists in the geometry of the manifold causes the normal vectors to flip, thus the entity can end up walking upside down as you have noticed when it passes through the region where the twist took place, and the surface end up having only one side
(end not sure)

A Klein bottle is a non orientable 2D manifold with no boundary. It is hollow and you need at least 4 dimensions to make it (otherwise you have self intersections).

The most common way to make a klein bottle is to start with a cylinder hose (i.e. only the curved surface if the cylinder), twist one end in 4D and then gluing the ends of the hose together, as shown in wikipedia, and here
Another way to make a klein bottle is to glue two moebius strips together

For the "klein bottle" you are referring to that 4Ders can theoretically walked on top, analogue to the area of the moebius strip that an ant can walk on, you end up with a related object called the solid klein bottle

It is basically a klein bottle with the volumetric region in the middle filled in and it is a non orientable 3 dimensional manifold
https://en.wikipedia.org/wiki/Solid_Klein_bottle

Or in other words, an ordinary moebius strip except that the "paper" you use to fold it is cylinderical

gonegahgah wrote:I'm following is my figuring of what a 3D slice of a simple klein 'bottle' would look like.

Does this conform with what the knowledgeable people here would concur with?

This is actually not a 3D slice of the Klein bottle. It is actually one of the many 3D projections of it

More details here: https://en.wikipedia.org/wiki/Klein_bottle#3D/4D%20pinched%20torus

If you try to draw a vector pointing from the bottom face to the top face, then after passing through that line (which is a circle oriented in the e.g. zw place, but the w axis is "outside") you will notice the relative positions of the faces (hence the arrows) flipped vertically.
Nothing happens for vectors parallel to the line, though.

Thus only directions along one of the axis is flipped (two if you include the normal vector pointing in and out in the w direction, which corresponds to whether a 4Der is standing "w side up" or "w side down")

gonegahgah wrote:Just as a moebius strip is a path - though non-traversable in 3D - so can a Klein 'bottle' 'tunnel' actually be a path open to the sky in 4D; I'm guessing?
...
I'm guessing that unlike a non-Klein circuit that a Klein circuit will require an actual bridge? Is that right?

1. I have heard that both Klein bottles and moebius strips are non orientable, but I have no idea what do you mean when you say they are non transversable?

ANS:
gonegahgah wrote:If it does then I can see that there is no need for a bridge afterall.

1. I also realised that any sort of mobius like structure can't be traversed - at least without sticky feet like a gecko.
If you are on the up surface and you walk around a mobius strip then eventually you have to be on the opposite side to where you were.
The opposite of up is down so at some stage you must be upside-down.
So to answer my own question, it isn't possible to traverse a klein 'bottle' in 4D for us upright only creatures.

I'm keen to know if my depiction agrees with actuality.
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### Re: Traversable Klein 'bottle' paths

Hi Secret,

I am sorry I cannot help you on this one as I am still not good at slicing an object that has a continuous twist in the 4th dimension.

This can be your initiation as it has been mine
Thank you for finding that term 'solid Klein bottle'. Now I know the correct term to use.

I like how you have added arrows to the picture.
Here's my addition of arrows just to show the actual 'surface' direction of the path at each point - that can be visibly seen (or an approximation of as I'm doing it by eye).
You can see more clearly in this depiction how the surface (over and under) twist around to create the mobius like nature...

If we go anticlockwise from underneath then this twists towards the outside until we are on the outside surface at the back.
The path then continues to twist around the back and up to the top surface at the front.
It then continues to twist towards the inside until we are at the foward facing surface at the back.
Finally this twists downwards until the we are back at the beginning under the path on the downward facing surface, and so on continuously.

I was remembering that our own square pavers have an up-side and an under-side just as a 4Der's cylindrical/prism combo paver also has an up-side and an under-side.
Just as our paver up-side and under-side are squares so are 4Der's pavers have an up-side and an under-side both of which are cylinders (with variations of course).
The 'face' is the flat surface that we would consider walking on. Just as for the mobius strip this 'face' twists so that it faces sideways and then under making it impossible for us to traverse all the way.

Something I realised since is that for a cylindrical path that my depiction should really show ovals rather than rectangles which morph from a line at the front.
These then morph out to ovals and then back to a vertical line at the back with the up-side now facing forward and the under-side now facing back.
You're reply has inspired me to have a go at depicting that, Secret, to be more beneficially accurate. It will involve a lot more lines I suspect.

I noticed that one of the links you gave me also shows an example I was describing:

This is what I was describing as being the same object but with our vertical 3D slice rotated 90° into the 4th dimension.
The interesting thing with this view is that you can only show the surface up and down on the pinched line for this one. Any other arrows would need to point out into the 4th dimension for the opposite surfaces.
In the 3D slice I depicted we can show all the surface direction arrows as these all twist into our 3D slice. This is a nice feature of a uniform twist where the surface goes from flat up/down to flat forward/back in our 3D slice.
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### Re: Traversable Klein 'bottle' paths

Thanks Wendy too,

When the connection is made to the outside, A must yet to swallow the C circle, which means that we are looking at a connection made in the 'handle' of the bottle.

I'm just wondering? A hollow Klein bottle doesn't have to connect to the outside in 4D does it?
I was thinking that despite a Klein bottle appearing to have an inside and outside in our 3D world that both surfaces would actually be inside in a 4D world?

I was thinking about how to make a hollow Klein bottle in 4D.
Instead of a cylindrical surface you would have cylindrical void surrounded by a 4D shell.
Instead of walking through the cylindrical volume you would walk 'through/on' the inside of the 4D shell.

You could do a similar thing with a mobius strip and just have solid volume edges (one overall edge of course but one side would point in the opposite direction of the other (aka towards each other).
We could then effectively walk inside the mobius strip 'bottle' if we had sticky feet and still walk twice the length each lap.
A 2Der depicting a mobius strip would tend to draw and think of it as a mobius bottle; whereas we don't.
Would a 4Der likewise think of our hollow Klein bottle as a Klein strip?
Also the edges would have to be very thick or the 4Ders would feel like they were walking on a tightrope; wouldn't they?
Fine for us to walk a hollow Klein bottle in 4D (with sticky feet) but I suspect 4Ders would prefer the solid Klein bottle variety (still with sticky feet); would they not?

That also leads me to another question...
If we encased the hollow Klein bottle with a 4D wall; wouldn't we need to provide a hole to get through that wall into the hollow Klein bottle in 4D for us 3Ders?
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### Re: Traversable Klein 'bottle' paths

Some interesting dimensional perspectives come out of this it would seem perhaps.
We 3Ders talk about mobius strips, Klein bottles and solid Klein bottles...
But, what do the other dimensional beings talk about?

We talk about Klein bottles and solid Klein bottles. We also talk about mobius strips.
But a 2Der doesn't have our left-right sidedness.
If they were to envision our mobius strip I think they would think of it as having two forms: a mobius bottle and solid mobius bottle.
A mobius bottle would be just what we consider to be a mobius strip but without any inside (as they see 2D shapes as usually having).
A solid mobius bottle would be just what we consider to be a mobius strip.

The 4Der, like us with the mobius strip, might not even consider to give the Klein shape a name for when it does not have any insides.
Like us with our mobius strip it may be that they only consider the Klein shape when it is filled.
For them they may simply call it a Klein strip as to them it is a strip and not a bottle at all.

Does that sound correct?
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### Re: Traversable Klein 'bottle' paths

In 4D, a 2d sheet is not a surface, and can be indeed knotted. There is a thing called the criss-cap which represents a square cut out of a hedrix, and pasted back with one of the ends reversed.

But a surface supports an out vector, which means a string can bound a region in 2D, but not in higher dimensions, and likewise a hedrix can bound only in three dimensions, and higher one can completely avoid it, or turn it into knots or whatever.
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### Re: Traversable Klein 'bottle' paths

I've also realised some of my errors in my depiction.
I was working along creating the shape I envisaged using Sketchup. Lot's of lines as I said.
But it occurred to me that the centre line I was adding should run through the middle but then how would it be visible?

As always seems useful, we can go back to our ever helpful 2Der and determine what they see first with our lesser models.

For a mobius strip the 2Der will see only an entrance line that rises up and let's say they have a little gap between the entry line and the mobius strip itself; like a train platform onto a train.
They jump this little gap to get onto the mobius strip. They then turn their 2D space 90° through our space so that they are facing along the current section of the mobius strip.

What they see is a short (and line thin) section of the mobius strip; as most of it curves off into our 3D space further along and is out of their line of sight. If we have a yellow stripe along the centre of the mobius strip, and they are standing on it, then they will see this go for a certain distance and then cut out giving way to grey path for much further which then gives out to empty space. Depending on their exact angle the yellow strip may appear to travel along evenly, or dip, or rise very slightly continuing into the grey path which then terminates in their point-of-view.

The 2Der would walk along this path and, every now and again, turn their 2D space through our 3D space one way or another (to us left or right) and see which offers a longer path (matching the flow of the curve). They could even follow the yellow strip much of the way but eventually they would have to leave this and follow an upward path so that they can walk over and along the edge of the path as it stands on its side in our 3D space where it begins to twist over. Otherwise they eventually run into a vertical wall impeding their path at the opposite reach. Fortunately the path pavers having some thickness which offer a means for the 2Der to walk on top of the mobius strip's highest point via the path's edges. This is we 3Ders would find difficult to do; except maybe as tight-rope walkers.

If it were that the paver's could have no 3D thickness it would be more difficult for the 2Der to traverse our mobius strip as the continual combined twist/turn would make any in-line straight distances very awkward to be of any use to the 2Der.

Once they reach the extreme opposite highest point, they then continue around in a similar fashion on a slightly descending path, rotating their 2Der space into ours as they go, until they are once again back where they started. Unfortunately they have defeated the purpose of the mobius strip and turned the path into a single full loop. To traverse the mobius strip correctly they would need sticky feet and the ability to twist their 2D space around both the horizontal and vertical simultaneously in order to follow the twisting path it describes.

Now for us 3Ders on a Klein strip...
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### Re: Traversable Klein 'bottle' paths

I came across some of the same stuff on Klein bottles and the mobius strip, about a month ago in this video. I just now grasped the torus with a crimp in it. Is this surface the same as the Klein bottle, but 'unfolded' in some way? The two of them seem like different angles of the same surface. Or, is the klein bottle a projection, and the crimped torus a slice?

Also, can we get one of these to pass through a 3-plane? I think that would look awesome.

Does anyone know what the implicit equations for these surfaces be?

There also seems to be a clifford torus based surface, as well! What's the deal with this thing? Wouldn't that just the the klein bottle, as projected in 3D?
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### Re: Traversable Klein 'bottle' paths

Hi ICN5D,

Having explored these a little now, and needing to do more admittedly, I'm inclined to think of these things as I think a 4Der would think of them.
We wouldn't refer to a mobius strip as a solid mobius bottle but a 2Der might be inclined to discover and think of it this way.
In a similar fashion a 4Der is unlikely to think of the Klein shape as being a solid bottle. For them it would be considered only as a strip.

The depiction you mention of the Klein strip as a torus with a crimp in it is sort of a projection and a slice together. It ignores the vertical and projects only all the sideways directions.
Also we have to remember that this projection depicts only the crimp part correctly as a slice (because it has a top and a bottom) but is a little flawed in depicting the rest of the shape; as occurs for 3D projection of 4D shapes.
The 'volume' shown depicts only one side of the projection. The other side occupies the space outside the object including where we are; because of the strange twisting we have to do to project 4D objects in 3D.

You should be able to walk around to the back of the object and see the other side as, due to twisting, it is now at the back. The crimp depiction shows us only all of the one side at once (though mostly hidden inside).
What I've realised is that a true 3D slice of a Klein strip, where we share a common vertical, will rotate from a mobius strip (0°) through to a flat ring strip (90°) through to a mobius strip (which to us the back) (180°) through to a flat ring strip (270°) back to a mobius strip (which to us is the front). You can also view various partial slices depending on how you align with the Klein strip.

So if you coloured half the Klein strip length one colour and the other continuation a second colour you will see the 180° view above as being a different colour than the 0° view.
It's quite cute to think of. The reason we get to see a flat ring at 90° and 270° is the same reason a 2Der gets to see a line ring if they we able to lay their 2D slice of our 3D space on its side looking at a mobius strip.
It too would appear as a ring.

Correct me if I've made any false assumptions again...

PS. Sorry, I have to correct that slightly. Got to rush so I'll do so later... Just the part about rotating and seeing the back...
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### Re: Traversable Klein 'bottle' paths

The correction seems to be that the edge will always be uppermost at the back of the Klein strip. So no matter how you rotate through the extra 360° of sideways direction the Klein strip will always seem to transition from one side to the other along the line that bisects the back; or oppositely to however the the strip goes from one colour to the other.

So as per the following image on the left of a mobius strip, painted two colours to highlight the orientation switching, I think a 3D slice of a Klein strip would show a similar look even in its two flat ring orientations at 90° and 270°.
We would see it from the front as two halves painted different colours. If we then wandered round the back, and we couldn't see underneath or over the top, we would see only one colour no matter how we rotated through the 360°.

This also points up something I've mentioned previously about two useful rotation planes for us 3Der's in a 4D world (whereas a 2Der only really has one useful rotation axis of their 2D space into ours).
If we 3Ders look at a sphere we can either scan a vertical line across from left to right; or we can look at the sphere and rotate around a dot to where we are upside-down looking at the sphere at 180° upside-down and then back to upright at 360°. Not very useful unless we want to walk around on our hands but both those rotations can be achieved in 4D while remaining upright.

So as I mentioned for the Klein strip in the last few posts you can either choose the rotation to scan through the segments of the Klein strip from one point on the circumference through to one diametre back to the opposite point on the circumference (where the flat path that resolves from our 3D slice keeps the same orientation as we rotate) or you can choose rotation that simply rotates the ring in place so that it goes from the mobius strip appearance at 0° and 180° to a flat ring at 90° and 270°.

It is probably also safe to infer that the mobius strip, where our 3D slice is at 0° and 180° turned into the 4th dimension, would look to us to have opposite twists to each other.

Can someone knowledgeable here concur whether that and the rest above is correct or not please?
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### Re: Traversable Klein 'bottle' paths

There's one more thing I should add that I thought of last time and forgot to include.
I imagine that as we rotate our 3D slice into the 4th dimension (in the rotate around rather than across form of rotation) that we would see another cool effect - I'm guessing...

That is I feel we would see the mobius strip rotating through to the 90° flat ring° to the 180° mobius strip to 270° flat ring back to the start as appearing to have a counter-rotation.
That is the left side would appear to rotate top to front and the right side would appear to rotate bottom to front giving the nice transition from flat ring to mobius orientations 1 and 2 of the one side of the Klein strip.
Rotating our 3D slice the counter way would see these rotate the corresponding opposite directions.

Does that sound right or wrong? I'm just trying to imagine what the juncture would look like...
Certainly sounds like something I would love to animate...
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### Re: Traversable Klein 'bottle' paths

Though that would have been interesting for the two halves from the flat front of the Möbius to appear to rotate counter to each other I don't think that is what actually happens; as we rotate our 3D space into their 4D space.
Instead what appears to happen is that the the Klein strip appears to twist to a point at the 90° mark (or at the opposite cycle at their 270° mark).
I say appear because in reality it maintains it length during the rotation process but it just appears to us that it shrinks to a point.
It's like when the 2Der wonders around our Möbius strip and gets to the back which appears to them to shrink to a point where you have no left right maneuverability.
We know that it still is full width even though there is only a dot passage for the 2Der. It is this which makes it more difficult for them to appreciate a Möbius path.
I believe it simply comes down to a similar difficulty of representing 4D depictions in 3D.

It further highlights that we can walk on top of their Klein strip just as a 2Der can walk on top of our Möbius.
In both cases we and they only complete one loop in the process instead to the two total loop lengths provided by either strip.
A 4Der could not walk along a Klein strip in any way just as we can not walk along a Möbius strip in any way.
Lower dimensional creatures think they can but only achieve half the distance.

It is interesting to note that Wikipedia (not the end-all source I know) says that it may have been called a Kleinsche Fläche ("Klein surface") and then misinterpreted as Kleinsche Flasche ("Klein bottle").
As I say, I prefer to call it a Klein strip as a nod to our 4Der friends.
I'll add some further notes soon in relation to these. I want to clarify between scanning around into the 4th dimension and rotating perspective in the 4th dimension with some diagrams.
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### Re: Traversable Klein 'bottle' paths

Final depiction hopefully. I don't think you would get a flat ring at any stage. As for the 2Der on a Möbius strip we can only stand at the back point by standing on top of the edge when we have the view just right or right enough.

A few points:
- Whereas a Möbius strip forms a line edge to a 2Der when turned just right a Klein strip forms a circle edge to us 3Ders when turned the right way.
- Just as for a 2Der, where the two dot edges form a line between them being a cross-section of the path, the Klein strip edges are opposing lines (that form one circumference) with a circle between as our cross-section.
- The twist will not change no matter how we rotate our 3D space into the 4th dimension.
- The twist just appears to morph squash as it does the opposite of taper through to the back around the edges as the back goes from circle to line to circle to line back to first circle.
- When we rotate our 3D space in the 4D space using rotation (not scanning) the Klein strip changes from a Klein strip, to a pinched torus back to a Klein strip (now orientated 180° but still on the front side) to another pinched torus.
- If we continue this rotation finishing the full 360° we end up back at our original view being a Klein strip.
- At no time from the front view can we see the back view; we have to wander around to the back to see this.
- Same at the front we can either look at it from the top path side; or we can crouch under and look at the underneath side.
- We can't rotate into the 4th dimension when using upright rotation to achieve these views; we have to actually move in our 3D space.
- So a simple rotating animation will show a Klein path with a yellow stripe along the middle but the yellow stripe will disappear as soon as we rotate into the 4th dimension because it is hidden by the circular edge.

Hopefully that is finally right! I'll try to animate what I mean one day but that's a lot of lines to draw. A software depiction would be nicer. If there already is a video I'd love to know...

I should add that scanning into the 4th dimension can also be used to increase the surface you can walk on.
The 2Der can walk along all the cross-sections of our Möbius strip while remaining on top (except the very back which is like a wall).
In a similar fashion we can use scanning to move around the cross section of the Klein strip.
When we scan, instead of rotate, the Klein strip appears to grow or shrink (until it disappears; though the ring circumference remains the same central diametre) just giving us a narrower perch to walk along.
In this way we keep a full circular cross-section we can wander around (just as the 2Der has a full line cross-section). Although it is not as directly easy to visualise it as being this way it is so.

I should also add that what we walk on is all that is available to the 4Der as well though we would not see same slope they see just as the 2Der does not see our Möbius slope except as being forward up or down.
Just as the 2Der can turn scan their 2D space around in our 3D space to follow our Möbius 'around'; we also can scan our 3D space around the 4D space (and also rotate which 2Der's couldn't do; or end upside-down).
When our 3D space is rotated to showing a pinched torus section of the Klein strip the torus itself is not solid; it's surface only is the cross-section of one part of the Klein strip; whether that be perpendicular or some angled cross-section.
So whether the orientation looks like a Möbius strip or a pinched torus; the 4Ders feet will be no more inside either of those than would our feet.
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### Re: Traversable Klein 'bottle' paths

Many many lines later...

Here is a depiction of a 3D slice of a 4D Klein strip in classic pose:

Here it is directly from front on; so we can see where the pinch is (the place where the Klein strip is fully 'in' the 4D ground space):

Here it is looking at it from front but from above:

Hope these agree with what you would expect. The pinched torus mentioned in a previous post doesn't quite correctly depict a Klein strip. I believe these are more accurate.
These are the 3D slice images when we are at 90° and 270° into the 4th dimension. At 0° and 180° it will just look like an ordinary Möbius strip.
Last edited by gonegahgah on Mon Aug 03, 2015 8:58 am, edited 1 time in total.
gonegahgah
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### Re: Traversable Klein 'bottle' paths

Are there any programs available that allow you to put equations in:
x(ϴ,ζ,Ϟ) = ...
y(ϴ,ζ,Ϟ) = ...
z(ϴ,ζ,Ϟ) = ...
and then spit out the 3D plot of the diagram?
gonegahgah
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### Re: Traversable Klein 'bottle' paths

Nice renders, by the way. It has taken the form of a mobius torus, by the looks of it. I'm not sure what program has that kind of parametric input. Is that a different coordinate system, the symbols ϴ, ζ, Ϟ ? Toroidal, maybe?

If you're able to re-write ϴ, ζ, Ϟ as the adjustable parameters a, b, c (which seems like you can), then CalcPlot3D is an excellent program. That's what I use for my renders and animations.

What are the parametric functions you are using for the Klein strip? I can help you write the explore function to turn and slide in 4D.
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### Re: Traversable Klein 'bottle' paths

Thanks ICN5D. Yes a,b should be all that's needed. A t (time) will be useful for animating the rotation.

I haven't quite got the parametric functions yet. I found some for a torus, a pinched torus and a mobius.
I have made a twistor (twisted torus) and a pinched twistor:

The later one looks close but ultimately it is still just a pinched torus - which is not what we want - as can be seen from the following perspective:

I haven't made a mobius pinched torus yet but am working on it. I think it would be possible to also model it with paper which is cool. I'll do that and take a picture...

I'll have to model some math shapes and crunch the numbers to work out where I want to get and where I am...
Cheers
Last edited by gonegahgah on Fri Aug 07, 2015 12:13 am, edited 4 times in total.
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### Re: Traversable Klein 'bottle' paths

Almost there I think...
Here's what I have created so far...

However, it should go to a full cylinder cross-section at the back...
Unfortunately so far it goes to a pinch at the back so I need to do something by half...

I'll look at my formulas again tomorrow to look at making it correct.
Last edited by gonegahgah on Fri Aug 07, 2015 12:15 am, edited 1 time in total.
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### Re: Traversable Klein 'bottle' paths

Here you go ICN5D, the final form of the Klein strip when our 3D slice is rotated into the 4th space at 90° and 270°.

Here are views of it from the front and back:

Here is a view from above:

Here is the more classic view as per what we usually see for a Mobius strip (actually it might be back to front but that is okay):

The final parametic equations:
x(u,v) = (R + r * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v / 2))) * sin(v)
y(u,v) = (R + r * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v / 2))) * cos(v)
z(u,v) = r * (sin(v / 2) * sin(u) * sin(v / 2 + π / 2) + cos(u) * sin(v / 2))

Where R is the radius of the donut and r is the radius of the cross section at the back...
... and both u and v step from 0 to 2π.
For my model I have stepped u 30 times and v 90 times with R = 6 and r = 2.

I'll make an animation soon showing how the Klein strip changes as we rotate our 3D slice around in the 4th dimension...
Enjoy...
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### Re: Traversable Klein 'bottle' paths

Here's an animation to show what the Klein Strip looks like as (left) we rotate around in the 4th dimension and (right) as we side step into the 4th dimension keeping our 3D slice at 90/270°:

You can see how for the rotation on the left it morphs from the full shape to a Möbius Strip and back again.
Originally I animated it for the full 360° and it did as I expected and basically repeated the same sequence twice.
So I only uploaded half the animation as it is no different than the full animation.
Kind of looks like a heart beating

The right image is what would appear when we side step into the 4th dimension. When doing this the Klein view just grows larger then smaller.
In real life we don't side step to see things but instead turn our head and scan around and in a game I would expect to use scanning more than side stepping.
However, as in real life a scan covers a lot more area than does a side stepping view so the side step provides for closer examination for our animations here.

These do help to depict the different forms of rotation. A scan will have similar results to a side step. A rotation produces different results.
You can use a combination of scanning and rotation to see the Klein Strip (or any 4D object) in different aspects.
By using both you can see the full extant of the surface and how it slopes even if you can't see it all at once...

Rotation offers a means to examine a 4D object more closely whereas scanning allows you to view the terrain much faster.

The parametic equations for both of these, if you are also interested in them, are:
1. Rotational viewing:
x(u,v,t) = (R + r * (sin(v / 2) * sin(u) * cos(t) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * sin(v)
y(u,v,t) = (R + r * (sin(v / 2) * sin(u) * cos(t) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * cos(v)
z(u,v,t) = r * (sin(v / 2) * sin(u) * cos(t) * sin(v / 2 + π / 2) + cos(u) * sin(v / 2))
where 0 ≤ t ≤ 2π
2. Side step viewing:
x(u,v,t) = (R + r * cos(t) * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * sin(v)
y(u,v,t) = (R + r * cos(t) * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * cos(v)
z(u,v,t) = r * cos(t) * (sin(v / 2) * sin(u) * sin(v / 2 + π / 2) + cos(u) * sin(v / 2))
where -π / 2 ≤ t ≤ π / 2
3. Scan viewing (will be closer too):
x(u,v,t) = (R + r * cos(t) * cos(t) * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * sin(v)
y(u,v,t) = (R + r * cos(t) * cos(t) * (sin(v / 2) * sin(u) * cos(v / 2 + π / 2) + cos(u) * cos(v/2))) * cos(v)
z(u,v,t) = r * cos(t) * cos(t) * (sin(v / 2) * sin(u) * sin(v / 2 + π / 2) + cos(u) * sin(v / 2))
where -π / 2 ≤ t ≤ π / 2
gonegahgah
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### Re: Traversable Klein 'bottle' paths

Well, check that out! Awesome!

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.
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### Re: Traversable Klein 'bottle' paths

Just for fun here is an animation showing how the shape is formed:

You can see that the section being drawn rotates in a clockwise direction. This agrees with how both a Mobius strip and a Klein strip rotate around along their length with a half twist.
You could also have it rotating anti-clockwise and it would be the left-hand form of the Klein strip.
It is easier to see how the section is rotating when it is more oval shaped. When it gets more circle like the rotation still exists but we don't see it...
[Except that I've now modified the anim above and added a full traversal (two loops) of the Klein Strip and also divided it into two colours.]
[Because of this you can now more easily see that the line across the centre of the stepping circle rotates all the way; showing that the circle rotates all the way.]
[You can also see that the cyan and magenta exchange sides as the traversing circle swaps from one loop side of the Klein strip to the opposite loop side.]
[Just think of it as a vacuum cleaner cleaning both sides of the Klein strip; even though to our brains it appears to only travel along one surface...]
[It's also interesting to note that the diametre line between the cyan and magenta halves describes a Mobius path in the animation for the total length of the single side a Mobius strip ie. two loops.]

As you can see, as I mentioned previously, we can only walk along the Klein strip over the top completing just a single loop; just as a 2Der can only walk over the top of our Mobius strip completing only one loop too.
The 4Der can see the whole Klein strip at once allowing them to appreciate the fullness of it; just as we can appreciate a Mobius strip in its fullness which is difficult for a 2Der to fully appreciate.
Last edited by gonegahgah on Sat Aug 08, 2015 12:40 pm, edited 7 times in total.
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### Re: Traversable Klein 'bottle' paths

Hi ICN5D,

Thanks
I could do that. Basically it would be side steps of the fuller form through to side steps of the Mobius form where they all just appear to grow then shrink in the radius (r) of the tube for each side step...
I haven't thought too much about rotation of the object itself; just the rotation of our perspective. Of course the Klein strip could be standing on its end (like a Mobius standing on its side).
For a 2Der when they see a Mobius strip they only see a straight line section of it; unless we tip it on its side then they can see it as a full circle from the edge.
Same goes for us when we see a Kelin strip we only see a ring but if it is tipped on its side we see it as a sphere. I haven't thought too much about that at this moment...

It would be cool to have a Klein pet wheel to walk around; like a mobius wheel for pets... Though you would need interesting mechanics to keep it turning such that you were always upright.
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### Re: Traversable Klein 'bottle' paths

Here's a render showing how the Klein strip (3D slice of) just looks like two Mobius strips stuck together that balloon away from each other but which cross over at the pinch point:
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### Re: Traversable Klein 'bottle' paths

I'm not getting these (beautiful) depictions of the 4D Klein surface. The surface seems to self-intersecting, when going from the "outer" to the "inner" part, as in 3D, which is redundant in 4D.

I don't thing the 2D-Klein surface in 4D may be considered the analog of the Moebius strip in 3D. A Moebius strip in 3D is a non-orientable, open surface bounded by a 1D line. With another Moebius string attached edge on edge, one can create a Klein bottle, the surface is now closed, but self-intersecting.

In 4D the same Klein surface is still closed, in the same manner as a circle is closed in 3D, but it corresponds to an 1D objekt in 3D. It's nothing you can walk on in 4D anymore. You'd have to give it a certain thickness or put it on an open 3D volume, like you have to do with a circle in 3D if you want to walk around it. Still you could easily cross from one side to another at any position on the Klein surface without turning upside down and you could go anywhere "on" the Klein surface without turning headfirst. The Klein surface doesn't possess a normal vector. It's clearly not analog to the Moebuis strip in 3D, where the normal vector is mirrored when you cross the edge. If you're walking along the 2D border line of the Klein bottle, you could get to the other side at one place without crossing it or jumping across it.

There is some object that is more analog to the Moebius band in 3D. Along one dimension the object is closed to a circle, in two dimensions it is open, another one is the direction where the normal vector points to. Going around the strip (you have to be an ant), the normal vector is rotating through one of the dimensions where the surface is open. In this configuration the remaining dimension is just a redundant degree of freedom, it's a trivial extension of the moebius band to 4D, with the difference that the normal vector may always point in a horizontal direction.

Now comes the beautiful thing: It should be possible to close the Moebius surface in the remaining direction to a torus. You can change sides either by going around the small loop or by going around the big loop. There are another two possibilities to go in a circle and return to the same place on a torus: Going around in both directions at once. In this case on one circle you would switch sides at the opposite location at the torus and reach the point where you started from with the orignal orientation, on the other circle, although the normal vector turns because of the curvature of the object, you would always stay on the same side. The bonding surface of this object is a 2D torus.

There is also a variation where one loop of the torus is a moebius band, the other one is just a normal band. Maby two of these "Moebius tori" may form an intersecting 3D version of the Klein surface in 3D, but my imagination doesn't reach that far yet.

What is deep in our world is superficial in higher dimensions.
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### Re: Traversable Klein 'bottle' paths

Hi Teragon. Thanks for that. What you have discussed seems to be okay. I'll try to have a more thorough read tomorrow.

I would like to distinguish between 'surface', 'bottle' and 'strip'.
From what I've read 'surface' and 'bottle' seem to be used in reference to 'Klein Bottles'.
My exploration here has brought me to something more solid (in a 4D world) that I refer to as a 'Klein Strip'.
I refer to it this way as the 4Der would see their Mobius equivalent as a strip; and not a bottle.
It's only us 3Ders who would come up with the idea of making a bottle from two Mobius strips as that is more logical to us.

As you mentioned, things need to have a thickness - whether in 3D or 4D. If we talk about walking then paver stones need to have thickness or they would crumple under out feet.
However, usually a Mobius strip is depicted without that thickness and my Klein strips are likewise depicted without this thickness. Of course it needs to be there.

As I say I'll re-read your post tomorrow. You did trigger a realisation for me which I will go and fix...
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### Re: Traversable Klein 'bottle' paths

Agree with your definitions. So a "strip" in 4D is a 3D object?

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.

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.

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. In another sense it's derived from the 2d Klein surface (closed in two directions). Is this the "Klein strip" you're talking about? This object might be identical with the Moebius strip I described in the last paragraph of my post above. In one direction on the strip you get to the other side with one loop, in another direction you just do a loop without anything happening (analog to the Klein surface in 3D) and yet another direction makes up the width of the strip (analog to the Moebius strip in 3D).

Hope that was somewhat comprehensible.
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