Are there examples of non-orientable manifolds in nature?

Question

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?

Answer 1

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 article "Order Parameters, Broken Symmetry, and Topology":

Here's an old photograph of droplets of nematic liquid crystal between crossed polarizers from the paper P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Novel Colloidal Interactions in Anisotropic Fluids. J. Science (1997) vol. 275 page 1770.. 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 either of the polarizers - thus it's kind of like the inverse image of two different points on $\mathbb{RP}^2$.

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, the paper cited above "Order Parameters, Broken Symmetry, and Topology" by Jim Sethna (published in 1991 Lectures in Complex Systems, Eds. L. Nagel and D. Stein, Santa Fe Institute Studies in the Sciences of Complexity, Proc. Vol. XV, Addison-Wesley, 1992) covers the basics.

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

Answer 2

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.upenn.edu/~ghrist/preprints/barcodes.pdf

Answer 3

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:

Who would've thought?

References: Informal, Formal

Answer 4

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.

Answer 5

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:
         
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:-)
                  

Answer 6

Lanyards, strings used for holding name cards, often has the shape of a Mobius band.

A picture of lanyard: Many more can be found on Google Image.

The purpose of using this shape is probably to help the card stay flat on the chest.

This is certainly a real world application of the Mobius band. It is a bit amusing to see so many people wearing an unorientable, one-sided surface! Showing a lanyard and pointed out its shape can be a convenient way to introduce the Mobius band to people.

(I learned this from someone else. A search on the web shows that somebody has noticed this earlier: [https://www.reddit.com/r/AskReddit/comments/l890m/why_the_hell_are_keybadge_lanyards_made_into_a/])

Answer 7

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 1950 story, A Subway Named Moebius, available in full at https://www.fadedpage.com/books/20210556/html.php

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, available (with full lyrics) at YouTube: The Kingston Trio — M.T.A.

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! // etc)
He may ride forever 'neath the streets of Boston.
He's the man who never returned.

Answer 8

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.

Answer 9

Before 2009, the daily route of London's Circle Line was non-orientable. Mostly, it took the yellow route, but see how it used the pink and green arcs on the east side of this map:

"As trains are constantly running in the same direction around the line, the wear on the wheels becomes uneven. To combat this, once each day the train journeys from Tower Hill to Liverpool Street via Aldgate East instead of directly via Aldgate. This effectively reverses the train, and puts the wear on the opposite set of wheels."

So effectively it had a Möbius route, until they trivialized its topology instead.

Answer 10

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.

Answer 11

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

Answer 12

The two-dimensional real projective space $\mathbb{R}P^2$ also crops up in quantum mechanics in the theory of identical particles, and explains why there are only two possible types of particles in three dimensional space: bosons and fermions.

Reference: Leinaas, Jon Magne; Myrheim, Jan (11 January 1977). "On the theory of identical particles". Il Nuovo Cimento B 37 (1): 1–23

Answer 13

See this image from Beautiful Proteins

Answer 14

Musically, a crab canon, such as the one in Bach’s Musical Offering, can be conceived as notated on a Möbius strip: http://www.youtube.com/watch?v=xUHQ2ybTejU

Answer 15

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!

Answer 16

As mentioned by Jim Conant, Möbius conveyor belts were patented. Here is a 1949 patent to Owen Harris, for an abrasive belt, as described by Clifford Pickover:

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          ^{Image from Pickover, p.44.}

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Pickover, Clifford A. The Möbius Strip: Dr. August Mobius’s marvelous band in mathematics, games, literature, art, technology, and cosmology. Thunder’s Mouth Press, New York (2006).

Answer 17

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.

Answer 18

a mobius strip roller coaster.

http://en.wikipedia.org/wiki/M%C3%B6bius_Loop_roller_coaster

Answer 19

Not exactly nature, but not designed like that intentionally. Or so I hope.

Answer 20

Chiral molecules.

See the last paragraph here.