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Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:

"The unorientable surfaces are never discussed in the literature since the primary interest of mathematicians in surfaces is in the study of one complex variable, number theory, algebraic geometry etc. where all of the surfaces are oriented."

(I won't give context, other than a page number: 446.) This got me thinking, perhaps non-orientability is purely an invention of topologists. Surely non-orientable manifolds play an important role in other areas of mathematics? Are there "real world" examples of non-orientability phenomenon in the natural sciences?

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My understanding is that we still don't know whether space-time is orientable. In fact, as far as I understand, we don't even know whether space is orientable. No particles have been observed that have changed orientation after going on a journey, but that might simply mean that no one has tried sending them along the "right" path. –  Alex B. Nov 12 '10 at 15:31
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@Alex, the fact that we observe CP violation (i.e., weak interactions have a preferred orientation) suggests that any such path would have to pass through a region where the laws of physics are qualitatively different from the universe we see. –  S. Carnahan Nov 12 '10 at 16:10
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@Mark- I thought you were asking for examples of non-orientability throughout mathematics (hence the quoted "real world"), but all the answers seem to be about the real real world. Which did you mean? –  Dylan Wilson Nov 12 '10 at 16:31
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@Alex: The fine structure constant describes the effective coupling strength of electromagnetic force, and once physicists had some grasp of renormalization, they understood that it would vary with energy scale. That news is decades old. I think the important principle (which seems to have held up rather well to date) is that if you take a sufficiently small, freely floating isolated laboratory and move it around, doing the same experiment here or there, you should not be able to detect the change in location. That principle would be necessarily violated in the case of nonorientability. –  S. Carnahan Nov 12 '10 at 17:55
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It seems that you are also interested in examples of non-orientable surfaces in math. I thought of the following. You can study algebraic curves in $\mathbb{R}^2$ as curves in $\mathbb{R}P^2$. If you blow up a point of $\mathbb{R}P^2$, a basic operation in algebraic geometry, you get a Klein Bottle (more generally, blowing up points of $\mathbb{R}P^2$ gives you all non-orientable surfaces). Unfortunately, I know nothing about real algebraic geometry. So, I can't say if people working in this field use these spaces in an essential way. –  James O Nov 14 '10 at 9:21
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15 Answers

Some industrial conveyor belts are hooked up like a Möbius strip, so I've heard, in order to wear evenly on "both" sides.

Of course nonorientabilty has got to show up in more fundamental physical ways. It's too natural to not be utilized by physical theory in some way. Configuration spaces are not necessarily orientable, for example. I'm sure someone can come up with a natural example.

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Another practical application: the hammer loop on a pair of painter pants usually has a half twist in it. –  Richard Kent Nov 12 '10 at 15:51
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People might be surprised at just how many patents there are that use the strip: google.com/patents?q=mobius+strip –  J. M. Nov 12 '10 at 16:16
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Here is a (poor) photo of B.F.Goodrich's Möbius conveyer belt: mathforum.org/mathimages/index.php/Image:Conveyor.jpg . –  Joseph O'Rourke Nov 12 '10 at 16:23
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There is a humorous short story based on this idea by W. H. Upson. See kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf636. –  Richard Stanley Nov 12 '10 at 19:45
    
Well, this example is more about non co-orientability. If we were to leave in a non-orientable space, we might have orientable conveyer belts which would be one-sided. But of course, my remark is far-fetched. –  Benoît Kloeckner Feb 14 '12 at 1:05
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Nice question! This may not qualify as a definitive answer, but there is a pattern of molecules that goes under the name Möbius aromaticity, which is at least close:
         Mobius
And this Technology Review article describes a prediction of "a new form of crystalline carbon made entirely of Möbius-molecules of graphene." Möbius strips have also been formed from DNA, but artificially, as described in this Science Daily article.

And moving beyond the molecular world, there is this:-)
                  Recycle

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Möbius-aromatic compounds don't really occur in nature, though there is this Nature article on a compound with this property: dx.doi.org/10.1038/nature02224 . It is interesting that the concept was proposed in 1964, and only practically realized with the synthesis of a compound with this property nearly forty years later. (Yes, I'm a chemist. :) ) –  J. M. Nov 12 '10 at 16:07
    
@J.M.: Thanks for the expert clarification! –  Joseph O'Rourke Nov 12 '10 at 16:16
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Why don't they occur in nature? –  Mariano Suárez-Alvarez Nov 12 '10 at 16:24
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@Mariano: In naturally occuring reactions, (due to random collisions and the like) the products with lower energy will generically be preferred. The strain energy in these Möbius rings is high (due to the twisting; the $p_z$ orbitals prefer to be parallel (as that's what the lower energy π-bonding eigenfunctions look like)) compared to nontwisted rings. –  j.c. Nov 12 '10 at 20:01
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Yes, there is a fair bit of, to use chemical terminology, steric and torsional strain in these. When Hückel aromaticity is good enough for Nature's purposes, there's really no need for Möbius. –  J. M. Nov 12 '10 at 23:23
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Chiral molecules.

See the last paragraph here.

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The real projective plane is the space of orientations for "nematic liquid crystals": these are materials (often found in your TV or computer screen!) composed of molecules shaped roughly like rods, which can point in any direction in 3D. However, they have no head or tail, so two antipodal orientations are identified. We can model nematic liquid crystals thus by a map from $U\subset \mathbb{R}^3$ to $\mathbb{RP}^2$.

The topology of the real projective plane thus comes into play when one thinks about "topological defects" in these materials. A topological defect is a sort of singularity, where in some tubular neighborhood of this defect the material is continuous, but at the points of the defect, there is a discontinuity. Furthermore, this defect is topological, in that it cannot be homotoped away locally.

With a bit of oversimplifying, $\pi_1(\mathbb{RP}^2)=\mathbb{Z}_2$ means that there is one nontrivial type of line defect (since $S^1$ surrounds a line) and $\pi_2(\mathbb{RP}^2)=\mathbb{Z}$ means that there are an infinite number of types of point defects in 3 dimensional nematic liquid crystals.

Here's a schematic image of a cross section of a line defect and a corresponding path on $\mathbb{RP}^2$ corresponding to a circuit around it. These are both from Jim Sethna's page:

line defect path on RP2

Here's a photograph of droplets of nematic liquid crystal under crossed-polarizers from the lab of David Weitz. I won't say too much about the colors, but they correspond roughly to the orientation of the molecule. The sharp points at the center of each droplet are one or more point defects, discontinuities in orientation. The dark brush-like structures coming out of each point are the regions where molecules are oriented in directions parallel to the polarizers - thus it's kind of like the inverse image of two different points on $\mathbb{RP}^2$.

droplets

Roughly speaking, a homotopy class of a map from a 1- or 2-sphere to the projective plane being nontrivial, means that the defect cannot be smoothed away (otherwise there would be a homotopy to a constant).

This is part of a much bigger picture of course; and there are other nonorientable spaces that describe the order of materials. I've been vague above because all of this is explained quite beautifully in the article by N.D. Mermin, The topological theory of defects in ordered media Rev. Mod. Phys. 51, 591–648 (1979). For a quicker introduction, this online essay "Order Parameters, Broken Symmetry, and Topology" by Jim Sethna covers the basics.

I love this stuff, so let me know if you have any questions and we can correspond further.

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Also, projective spaces are very important in algebraic geometry, and finite projective spaces are a big topic in coding theory and design theory. Finite spaces are of course not manifolds, but this is an example of the application of the development of certain non-orientable spaces. –  Kimball Nov 12 '10 at 22:00
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@Kimball, as you no doubt know, (2n+1)-dimensional real projective spaces are orientable, as are all complex projective spaces. I don't know how to think about orientability for finite projective spaces. –  j.c. Nov 12 '10 at 22:21
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There was a study where they took thousands of digital pictures of "natural images", rendered them in grayscale, and looked at all the 3x3 pixel squares which arose in such pictures. Using topological data analysis they found that (after some normalizations) their data points actually clustered around a Klein bottle embedded in the 7 sphere! Here's a paper that talks about it, and tells you where to look for the Klein bottle:

http://www.math.uiuc.edu/~ghrist/preprints/barcodes.pdf

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I have not read the paper closely yet, but this has to be one of the cooler things I've seen in a long time. To me it seems so implausible! –  R Hahn Nov 13 '10 at 5:30
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Well, these microscopic examples (molecules), and small-scale examples (pulley-belts), and far-out, conjectured cosmological examples are all well and good, but as a Bostonian born and bred I am disappointed that you are all apparently too young to remember the amazing example of the famous Moebius subway line that was part of Boston's MTA (earning it the nickname "Moebius Transit Authority"). The details can be found in A. J. Deutsch's remarkable story, "A Subway Named Moebius":

http://www.iblist.com/book12352.htm

And it was made into a famous prize-winning movie called strangely enough "Moebius":

http://www.dm.unibo.it/bologna2000/mosquera.html

And there is also a famous ballad written about it. (See : http://www.stationreporter.net/boston.htm

Well, let me tell you of the story of a man named Charley on a tragic and fateful day. He put ten cents in his pocket, kissed his wife and family, went to ride on the M.T.A.

Chorus: Well, did he ever return? No, he never returned and his fate is still unknown. (What a pity! Poor ole Charlie. Shame and scandal. He may ride forever. Just like Paul Revere.) He may ride forever 'neath the streets of Boston. He's the man who never returned.

(See : http://www.stationreporter.net/boston.htm for the full lyrics)

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+1 For the song! –  Andrew Stacey Nov 12 '10 at 18:11
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If you consider every guitar store on a saturday afternoon part of the real world

http://music.princeton.edu/%7Edmitri/deeppurple.mov

More generally, an explanation of why music is essentially a sequence of points in a non-orientable orbifold (well, sort of)

http://www.sciencemag.org/cgi/content/full/320/5874/346

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It seems to me that real projective spaces are immensely important throughout topology, especially in the study of certain cohomology theories and as classifying spaces... These are only orientable half the time.

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It's true that real Grassmanians occur "naturally" as classifying spaces of real vector bundles. Are these important in other branches of Mathematics? –  Mark Grant Nov 13 '10 at 8:51
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It seems that nature "is" a Klein bottle in the following sense. There is a growing field in applied topology (yes, I said that) which goes by "topological data analysis" or sometimes "persistent homology". As I understand it, it works like this: take a manifold embedded in Euclidean space and suppose you have a way to extract a huge number of sample points from the manifold. With enough sample points it is in principle possible to build a simplicial approximation to the manifold which is refined enough to capture its local structure, and one ought to be able to recover actual topological invariants of the manifold by computing them for these approximations and letting the number of sample points approach infinity. This might be how the human eye assembles intuition for the geometry of an object from just looking at it, a process which collects a very large but finite number of sample points.

Now here's a crazy idea. Let's crank things up a notch and imagine the following experiment: go out in the world, take a huge number of high resolution photographs of images of clouds and trees and other natural objects, break them down into patches of 3 pixels by 3 pixels, and treat each patch as a point in $\mathbb{R}^9$. In other words, we are sampling the subspace of $\mathbb{R}^9$ of "natural images". Question: what would the topological structure of this space be? Crazy, right?

Well, this experiment has actually been done and it turns out that the space of natural images is our good old non-orientable friend, the Klein Bottle:

alt text

Who would've thought?

References: Informal, Formal

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How does this vary if you choose a different "sample size"? What if you pick 2x2 "pixels"? Or 4x4 ones? Or 2x3, or generally n x m ones? This is certainly fascinating, and it makes me wonder how generally it holds... –  Simon Rose Nov 13 '10 at 16:17
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@Paul: this was already mentioned in an answer by Lucas Culler. –  Qiaochu Yuan Nov 14 '10 at 12:34
    
This is AWESOME! –  tetrapharmakon Feb 13 '12 at 11:43
    
@Simon: 2x2 probably is too small to see things happening; but I know people have looked at 5x5 and found the klein bottle, as well as some enrichment of the structure there. –  Mikael Vejdemo-Johansson Feb 13 '12 at 23:07
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This is the sort of question that people consider when teaching intro topology courses. For example at the Ross Mathematics Camp this summer, Jim Fowler gave this example (on slide 3) of a Moebius band appearing in biology:

A yeast cell has some sort of life cycle (grow, sleep, reproduce, or something); the life cycle can be thought of as the circle $S^1$. Apparently, a colony of yeast cells tends to synchronize its life cycles. If you have two yeast colonies $A$ and $B$, then they will be separately synchronized, but the two colonies will be out of phase in general. If you combine the two colonies into one big colony, then they will gradually re-synchronize to be in phase. The question is, how is this new phase related to the old ones?

Suppose that the resulting phase depends only on the two input phases (this is particularly reasonable if the colonies $A$ and $B$ are the same size, etc.). That is to say, it is determined by a function $f: S^1 \times S^1 \rightarrow S^1$ of the input phases. If $A$ and $B$ happen to be in phase to start, then of course we'd expect the phase to remain the same: $f(x,x) = x$. We're not allowing anything to distinguish $A$ from $B$, so $f$ ought to be symmetric, i.e. $f(a,b) = f(b,a)$, so it actually defines an $S^1$-valued function on the Moebius band $M$. Reasonably, $f$ should be continuous.

So what we're asking for is a retract from $M$ to its bounding circle. Of course, this is absurd. So there is no such function.

I'm not sure what the resolution is, and unfortunately don't have a reference to the original work. Adding in parameters, making things probabilistic... the obvious approaches don't easily resolve the underlying topological issue!

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Probably the assumption that has to go is continuity. If the phases are not perfectly opposite, then they tend to pull together while if they are perfectly opposite, then maybe they're in an unstable equilibrium. –  Tim Campion Nov 24 '10 at 19:04
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In this paper, the energy landscape of the cyclo-octane molecule (foreign words to me!) is shown to be a Klein bottle and a 2-sphere fused along two circles. Using persistent homology techniques discussed in the Ghrist paper mentioned by Lucas, the homology has been experimentally recovered by Mikael Vejdemo-Johansson (I learned the above from this talk).

I'm under the impression that these questions are directly related to protein folding, which is a major concern of modern biology. The basic idea is that the space of possible ways the protein can be folded forms a manifold, with some function on the manifold governing which possible folds take place.

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Old automotive cassette players used Möbius magtape to avoid the need for rewinding. That was used also in some "last N hours" data recorders to avoid losing data during rewind.

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Why is Mobius better than just a simple closed loop? –  Willie Wong Jan 6 at 11:39
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