A Mobius curve is a single closed curve. It has only one plane. If you draw a line around the the curve you will discover that you travel on both the "outside" and "inside" yet meet your starting place. The Mobius curve where place and time turns back on itself again. Here in a painting by M. C. Escher we see ants traveling on a Mobius curve:
An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface.
In making a Mobius curve, you take a long thin rectangular piece of paper, give one of the narrow ends a 180 degree twist and glue or tape the ends together. What happens when you cut a mobius band all around its length?
This is so much fun that you *must* try this yourself. But the result is...you get just one band of paper, not two, amazingly enough.
Location: Wivenhoe, not too far from the Clacton VOR
And when you have damaged your brain sufficiently, try a Klein Bottle which is, of course, a single sided bottle with no inside or outside, has zero volume and yet can contain a liquid. Unfortunately it can only truly exist in a 4-dimensional universe. However, a 3-D representation (that is the equivalent of a 2-D photo of a 3-D object) can be made and bought. I bought my scientist father one as a prezzy.
The mobius strip is made by turning one end through 180 degrees and joining it to the other. A Klein Bottle results from stitching two mobius strips together, effectively joining one end twisted through 180 and the other directly. In mathematical terms, by multiplying the y-axis of a radian function by -1 and the x-axis by +1.
There is no technical objection to multiplying both axes of a radian function by -1. This would create an extremely interesting model, which although I have yet to see an acceptable 2D illustration of this 5 dimensional construction, can be imagined as long as you try not to look at the whole model in one view. (The model may appear impossible, but only when you consider the time element of the equation (time = 't') as being constant throughout the body of the model. )
For those who are interested in following the Blacksheep on an exploration of topology, this is an excellent starting point. Yes goudie, it may make your head hurt at first, but it is more mind expanding than any of the drugs people experimented with in the sixties.
Once you master the spatial possibilities, advance to the possibilities of the fourth dimension being different at all points in the structure. Once you get this picture, mere weirdness becomes normality and quantum mechanics becomes questionable. Perhaps I'm slightly mad?
If you want to fly the smoke trail Been Accounting, a figure of eight knot would make a visualisation of a trail on the surface of the simple five dimensional structure that I mentioned earlier.
Location: Dublin, Ireland. (No, I just live here.)
You sometime hear people attributing magical qualities to the Möbius Curve or Klein Bottle, as if they do something physically impossible. If you bend a surface in the 3rd dimension, it's no longer a purely 2-dimensional surface, is it? So you haven't really gained anything "new" in the process.
To bend a surface (i.e. two dimensions) in the third dimension, you might try manipulating the fourth dimension, bnt. As you say, you gain nothing for of course, a klein bottle encloses no volume. But once you accept that the fourth dimension is not fixed, the concept of volume is no longer necessary or relevant. Being three-dimensional creatures we find that difficult to visualise, but there's really no need to invoke magical qualities.