Moebius Mind Meltdown
erco
Posts: 20,259
Speaking of enigmas, try this sometime: http://www.exo.net/~pauld/activities/mobius/mobiusdissection.html
Splitting a paper moebius strip down the middle twice yields a surprise:
A mathematician confided
That a Mobius band is one-sided,
And you'll get quite a laugh,
If you cut one in half,
For it stays in one piece when divided.
Splitting a paper moebius strip down the middle twice yields a surprise:
A mathematician confided
That a Mobius band is one-sided,
And you'll get quite a laugh,
If you cut one in half,
For it stays in one piece when divided.
Comments
What you mentioned is fairly well-known. But there is a much more exotic split. This may be a bit more of a mind meltdown.
According to Martin Gardner of Scientific American, if you take a Klien bottle and split it down the middle -- you get two Moebius Strips of opposite twists. This is a bit more profound as you go from a 3-D object to a plane surface loop.
There is a proof for this in "The Colossal Book of Mathematics" that is done by creating a paper model of a Klien bottle and then bisecting it.
http://www.amazon.com/Colossal-Book-Mathematics-Paradoxes-Problems/dp/0393020231
Hmmm. Other than still owning Martin Gardner's book "Science Puzzlers" from way back in 1966, I'm beginning to suspect that I've lived a sheltered life.
Nope, it is just that Parallax Forums has a lot of people that actually think and read.
http://mathforum.org/kb/message.jspa?messageID=7756465
http://www.youtube.com/watch?v=BVVfs4zKrgk
Gets kind of hairy at the 1 and a half minute point.
Part two is here:
http://www.youtube.com/watch?v=x7d13SgqUXg
It took me a long while to convince myself that you could turn a bicycle inner tube inside out through it's plug hole without tearing anything if the rubber was stretchy enough.
Seems topologists are quire happy to turn spheres and toruses inside out even without any holes in them. They just have some odd rules like "self intersection is OK", "causing a sharp crease in the surface is not OK"
It was actually only a week or two ago that came across the fact that Klein bottle can make Moebius strips. So the relationship between the two and topology have been on my mind of late. Something has shifted in my thinking of what surfaces are, and possibly a relationship to knots.
No sure where this all leads, but there is a dusty corner in my mind with a lot of rather obscure geometric and mathematical observations.
Hyper-cubes are likely to turn the mind to mush.
There still are thinks that can be discovered. For instance, I have a perfectly good and useful model of the creative and destructive cycles of the five elements in Chinese medicine modeled with an icosohedron. I've never seen it used in any text on Chinese medicine, but it works perfectly to decrible the whole, inclusive of the two poles being the major division yin and the yang.
I keep getting new insights, so I keep studying. Also recently spent a lot of time with exploring how the Tower of Hanoi puzzle might apply to Forth two stack engines in terms of keeping processes in order. So far, no conclusions.... but interesting.
I used to entertain kids with that and the Mobius strip trick.
And so, folding is a very valid mathematical topic, but in the case of the Moebius strip, it is a twist that does all that is wonderful.
Okay, more food for though for everyone.
Moebius Strips actually come in two varieties -- plain and knotted.
Not sure where this might lead, but it is a fact -- both twisted and knotted together.
http://www.knotplot.com/moebius/
Gauss Braid theory, of course.
We also seem to have Moebius strips existing in molecular forms and even on crystaline form. I am not sure if this qualifies as a third category for crystals as there are now classic crystals and quasi-crystals.
http://www.youtube.com/watch?v=4PO7LqmZk5M