Tuesday, November 17, 2009

Möbius (Moebius) Strip: One Side, One Edge; Topology's Fun

"Nov. 17, 1790: A Rather One-Sided Affair"
This Day in Tech (November 17, 2009)

"1790: Mathematician, astronomer and physicist August Ferdinand Möbius is born in Schulpforta, Saxony (in modern-day Germany).

"Möbius has name recognition today because of the Möbius strip, which is a clever topological surface with only one side and only one edge.

"Speaking of name recognition, Möbius probably pronounced the name something like MER-bee-oos (first syllable rhymes with her, but with the r barely spoken). Webster's New World College Dictionary recommends pronouncing it MAY-bee-us or MOE-bee-us. But you often hear MEE-bee-us, and you sometimes see the alternate spelling Moebius.

"As for the Möbius strip (or Möbius band), many budding young topologists have discovered that it's possible to create one of these seemingly impossible objects with nothing more than..."

The Möbius strip may actually be the Listing strip. Another topologist, Johann Benedict Listing. This Wired article has a truly painful play on words, involving the lesser-known topologist's name. See if you can spot it.

The article itself is mostly about August Ferdinand Möbius, who was born on this day in 1790. And, of course, his famous one-sided band.

Which really does have one side and one edge. Putting one together, following the (really simple) instructions in the Wired article, you can try drawing a line down the center of just one side of the paper strip. The results are a bit counter-intuitive.

(Then, try cutting Möbius strips in two, lengthwise: first, by cutting it down the middle of the strip; then, with a fresh strip, cutting it lengthwise, about a third of the way in from one side.)

Moving along:

Somewhere in the sixties or seventies, I ran into a bit of verse that went something like this:

A mathematician named Klein
Thought the Moebius strip was divine.
Said he: "If you glue
The edges of two
You'll get a weird bottle like mine!"
(John Baez[!], This Week's Finds in Mathematical Physics (Week 155), Department of Mathematics, University of California, Riverside (August 16, 2000))

Several places sell what they call Klein bottles - or Klein-somethings: Actually, those products are models of Klein bottles (properly, Klein surfaces), projected onto the three spatial dimensions we're familiar with. Just like that picture of a Möbius strip isn't a Möbius strip, but a projection of one onto the two spatial dimensions of the picture.

This post is getting dangerously close to being 'educational.'

Here's a video (it's silent):

"The Klein Bottle"

bothmer, YouTube (November 8, 2006)
video, 1:23 (no sound)

"The Klein Bottle is a surface on which you can move from outside to inside without crossing an edge. This shows that inside and outside are not universal concepts.

"In this movie Klein's Bottle is constructed by gluing an rectangle along the edges. Then the bottle is cut up again to yield a Moebius-strip.

"This Video was produces for a topology seminar at the Leibniz Universitaet Hannover." (http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1)

This is one of the best visual presentations of a Klein surface I've seen. What's missing is an explanation of what's happening, when the cylinder seems to go through itself.

The "Klein bottles" we have are projections into three-dimensional space of an object that needs four spatial dimensions. The Klein surface doesn't got through itself, any more than the Möbius strip does - although it seems to, on the two dimensions of that picture.

Here's another video of a rotating (projection of a) Klein bottle: in Christmas colors, yet. It's 'squishier' than most. And, again, silent.

"Klein Bottle 1"

wh8191, YouTube (September 27, 2008)
video, 1:21 (no sound)

"Rotating Klein bottle, rendered in POV-Ray"

A reasonable question is "are Möbius strips and Klein bottles real?" The answer is yes and no.

Yes, we can make models of them. It's quite easy in the case of the Möbius strip - and fairly straightforward for Klein bottles, as long as we accept the fact that we live in three spatial dimensions.

No, Möbius strips and Klein bottles aren't "real" in the sense that there's a perfect Möbius strip and an ideal Klein bottle somewhere. Unless Plato was right about that sort of thing. In the case of the Klein bottle (properly, Klein surface), even if there was a "real" one, we could only see a projection of it onto the three-dimensional space we live in.

Both the Möbius strip and Klein bottle are mathematical models.

"Four spatial dimensions?!" It's possible to make mathematical models of objects in four dimensions, just like we do for three- and two-dimensional objects. A sort of Topology 101 explanation is this:

Take a point. Move it. You've got a line that has one spatial dimension. a point on the line can move either one way or the other, in one dimension.

Take the line. Move it at right angles to the one dimension of the line. You've got a surface. A point on the surface can move in either - or both - of the two dimensions.

Take the surface. Move it in a direction that's at right angles to both of the surface's dimensions. You've got a volume. (Or, a surface three-dimensions?!) A point in this volume can move in any - or all - of the three dimensions.

Take the volume. Move it in a direction that's at right angles to all of the three dimensions (yes, it's possible - for a mathematical model). You've got a (volume with four dimensions). A point in this (volume with four dimensions) can move in any - or all - of the four dimensions.

This fourth spatial dimension may or may not reflect something in the physical world. It's not the same thing as time - although quite a number of entertaining stories have been written which assume that to be so - but time can be considered as a "fourth dimension" - as Einstein did, about a century back now.

Oops. There I go, getting "educational" again.

Time to stop writing.

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