Loopy logic: A Möbius strip made of a material that changes colour with bending pressure. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density.
Credit: Nature Materials/Starostin & van der Heijden
PARIS: Scientists have cracked a 75-year-old riddle involving the Möbius strip, a mathematical phenomenon that has also become an art icon.
Popularised by the Dutch artist M.C. Escher, a Möbius (or Moebius) strip entails taking a strip of paper or some other flexible material. You take one end of the strip, twist it through 180 degrees, and then tape it to the other end.
This creates a loop that has an intriguing quality – dazzlingly exploited by Escher – in that it only has one side.
See Escher's Möbius Strip II (1963) from the M.C. Escher Foundation web site.
Mathematical conundrum
Since 1930, the Möbius strip has been a classic poser for experts in mechanics. The teaser is to resolve the strip algebraically – to explain its unusual shape in the form of an equation.
Now, in a study published in the journal Nature Materials that lyrically praises the strip for its "mathematical beauty," Gert van der Heijden and Eugene Starostin of University College London, in England, present the solution.
What determines the strip's shape is its differing areas of "energy density," say the experts in non-linear dynamics.
"Energy density" means the stored, elastic energy that is contained in the strip as a result of the folding. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density.
If the width of the strip increases in proportion to its length, the zones of energy density also shift, which in term alters the shape, according to their equations. A wider strip, for instance, leads to nearly flat, "triangular" regions in the strip, a phenomenon that also happens when paper is crumpled.
Not just esoteric
The research may seem esoteric, but van der Heijden and Starostin believe it also has practical applications.
It could help predict points of tearing in fabrics and also be useful for pharmaceutical engineers who model the structure of new drugs.
"One of the classic problems in mechanics is to find the shape assumed by a Möbius strip – the famous band that is closed with a half-twist and which has the intriguing topological property that it only has one side," said mathematician John H. Maddocks in an accompanying commentary that also appeared in Nature Materials.
"This abstract mathematical question, dating back to at least 1930, is also of practical scientific interest as single crystals in the form of a Möbius band have now been grown," said Maddocks, of the Swiss Federal Institute of Technology in Lausanne, who was not involved in the study.
The Möbius strip was named after a German mathematician, August Ferdinand Möbius, who discovered it in 1858. Another German, Johann Benedict Listing, separately discovered it in the same year.
Learn first
A moebius strip is a strip closed on to itself with a half twist. If you take a pencil and start drawing a line along the surface, you will end up where you started, except the line you just drew will have spanned the entire surface! That's the beauty of it. Try doing that to a rubber band and see what happens. Just look it up, try it and convince yourself what a fool you were to post your comment.
What are you talking about?
If the mobious strip only had one side then how would you twist it?It would have to have AT LEAST 2 sides and thats not counting the edges.So you are nowhere near knowing what you are talking about.
Whats wrong with you?
How would you POSSIBLY think that a strip of anything would have just one side? Let me put it like you would understand. Think about a shoelace... If you twist it does it not still have 2 edges? Just think about it.
Watch and learn
The Möbius strip
The only one with one side,
The only one with one side, sure, if you ignore spheres, ellipses, and random blobs with no edges.
One side
A sphere has a volume, or in the case of a ball an inside and an outside that arn't connected. A mobious strip could be made without volume (in theory). Once you create a volume you create an inside and an outside.
Shayne McKinney
Mobius solids
Take a equilateral rectangular prism, twist one end 90 degrees, reattach the ends and a mobius solid is produced.
The mobius solid is comprised of a single surface and a single edge.
Twist the end 180 degrees, reattach the ends, a 3 dimensional intersection of 2 mobius solids is produced.
Increase the number of sides, twist an appropriate amount (360/# of sides), a solid is created made up of multiple intersecting mobius solids.
Increase the number of sides to approach infinity, the result approaches a torus. I hope it's jelly filled. ;-)
Half of you dont know what your talking about!
If you was anywhere near as smart as you thought you was, then you would know how you sounded. I've left a couple of comments for some of you just to let you know what I could see you was figuring out to be wrong. But dont get me wrong i'm no mathematician by any means. But really some of the stuff i've seen on here was just common sense overlooked. You can leave me all the hateful messages you want to, because i'm going to be checking this on a regular basis now.
ignorance
If you were anywhere as smart as you thought you were, you would know how to properly use the verb "to be", something a person of intelligence can usually do correctly by, oh, age 7 or 8. Also you don't know what "you're" talking about. Moron.