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.
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?
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?
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?
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.
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.
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.
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