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Paper Klein Bottle Net

I was talking with a maths-teaching friend recently, we touched on Klein bottles, and I remarked there was a paper net for making one. I was shocked to find that the only paper Klein models on the bit of the net that Google knows about are either rather pathetic things that are too mathsy to be visually satisfying or extremely complicated things that would take far too long for children to make in a maths lesson (but which look jolly good, and would be worth having for display purposes, if you don't fork out for a glass one).

I worked this cubist Klein bottle net out quickly - it's not very good because it doesn't have proper tabs to stick everything to, but tabs date back to before they had stickytape, eh? :) I made one of these at school several years ago. I think at that time, I just put tabs on every edge available, and trimmed the ones I didn't need.

I initially made a version of this net where the handle was all one piece, but that was fiddly. I believe it's easier to fold/stick the lower piece (inside of the handle and the part of the tube inside the cuboid) before the upper piece. I also think that splitting the handle into two parts makes the form marginally easier to understand, even if the model's a little less robust. It's not particularly difficult to rearrange the handle back into one piece if you prefer.

One advantage of the models I dismiss as 'pathetic' is that they're very quick to make, and hence good for demonstrating what happens when you cut a klein bottle - the classic cut is the vertical slice to make two equal Möbius bands, but apparently you can perform a horizonal cut to make two unequal bands, so here's a description: Make a tube, flatten it in the middle and fold in half, cut a slit in the side of the tube, near an end, insert the other end through the slit, and sticky-tape the edges together.

Since the net above is almost symmetric, you could make two half cubist Klein bottle pieces to make the Möbius strips evident.

I'd rather like to make another net based on a triangular mesh; I think that would be visually more pleasing while remaining relatively simple, but it'll take quite a lot of thought and experimentation.

Sheila Morgan discovered that one could weave a Klein bottle from a single wire, and found that spoke to her metaphysically.


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This page last changed Wed Feb 15 10:24:20 2006