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:
![Image](http://i64.tinypic.com/28jgroh.png)
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...