Mobius strips, are generally held to have only one side - if one marks a place on the strip and on what looks like the other side then a pencil can draw a line between these two points without ever leaving the surface of the strip.

Technically this property is called non-orientability; and what we've done in the above demonstration is show that both 'sides' are path-connected.

So our intuition is shown to be limited and a little foolish.

However, one can as why then does it look as though it has two sides; a little thought shows that it is because we examine locally; in the same way that the earth looks flat, but in fact is round, and we resolve this tension by saying that the earth is locally flat; similarly we can say that the mobius strip is locally two-sided but not so globally.

(Now if the default adjective applied to the concept is local rather than global we could just simply say that the mobius strip is two-sided).

So, can we say that the Mobius strip has a front and back, despite all the write-ups it has had, by adding 'locally' as a qualifier?

  • 4
    As you know, every manifold is locally orientable since each is locally homeomorphic to R^n. So any point on any surface is contained in a neighborhood with a front and back. Oct 29 '14 at 4:53
  • It was your answer to the question why objects have a front & back that inspired this question ;) Oct 29 '14 at 5:20
  • I agree completely with the reasoning in the question. Mobius strips do not have a global front and back but have a local front and back.
    – AndrewC
    Oct 29 '14 at 20:57

If you take a point on a mobius strip, and some small enough open neighborhood around that point, the neighborhood is "equivalent" to some open neighborhood of zero in the regular euclidean plane (or, if the point is on the edge, equivalent to a neighborhood in the half-plane). This means that the mobius strip is indeed "locally" orientable, in that locally you can consistently define orientation on the strip.

However, I want to point out that when you say the strip "looks" as though it has two sides, there is some question of whether you actually mean the same thing as mathematicians mean by "locally orientable". Restricting our attention to 2-manifolds, a mathematician means by "orientable" that there is some consistent choice of "clockwise" and "counterclockwise" loops on that surface, i.e. that if I draw a counterclockwise loop on the surface and then move it around, I can't arrive at a clockwise loop.

But is this really what you mean by saying it "looks" like it has two sides? It seems to me that you are rather thinking of it like a paper, where you draw on one side or the other; yet mathematically, the strip is merely a set of points, and unlike a paper or a physical mobius strip it need not be embedded in 3-space. Worth considering.

  • This all reminds me of a conversation in a common room many years ago: A:"Wales is like a big park." B:"No, parks are like a small bit of Wales" A:"Well I meant that Wales is locally parklike." B:"That's like saying that the plane is locally like a 2-manifold."
    – AndrewC
    Oct 31 '14 at 19:29
  • Good point about the needlessness of embedding in 3-space. +1
    – AndrewC
    Oct 31 '14 at 19:31
  • @AndrewC Thanks for correcting the errors. That is a fantastic conversation.
    – 6005
    Nov 1 '14 at 22:49

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