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The earth’s orbit around the sun exhibits a torus topology with the revolution as one axis and rotaion around earths axis as the other. But I cannot think of any examples where the topology is a projective plane. Does anyone have a good real world example?

Michael Hong
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    What do you mean by "real world phenomena"? The projective plane is nonorientable, so it can't be embedded into 3-d space (see https://math.stackexchange.com/questions/2745135/how-to-visualize-the-real-projective-plane-mathbb-rp2-in-three-dimensions-i). On the other hand, if you think of objects doing something in Euclidian space, the objects you tend to think of (like a planet orbiting in space) tend to be implicitly embedded. Probably the projective plane can appear as embedded in spacetime as a 4d manifold, but that won't be visualisable anyway. – rosecabbage Oct 29 '22 at 09:29
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    What I mean by real world phenomena is some phenomena that is easily relatable to layman that exhibits a configuration space that is a projective plane. It doesn’t have to be physical objects embedded in 3D space. The earth’s rotation around the sun is not actually a torus but its configuration space has a torus topology. – Michael Hong Oct 29 '22 at 09:57
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    Would you count the set of straight lines through a given point in space as a “real world phenomenon”? That's often taken as the definition of the projective plane. – Hans Lundmark Oct 29 '22 at 10:02
  • That is exactly what I wasnt looking for. Haha Abstract and geometric examples are common but remain elusive to the layman… – Michael Hong Oct 29 '22 at 12:59
  • You need to define layman – Taladris Oct 29 '22 at 17:42
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    I really don't get it when people want someone to precisely define something that is not mathematical or purely logic. You kill CREATIVITY, because CREATIVITY does not have a precise mathematical definition!!! – André Caldas Oct 29 '22 at 18:51
  • What I mean by layman is normal people who are not math or physics majors. – Michael Hong Oct 29 '22 at 21:30

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A version of this question came up awhile ago on MathOverflow and there was a great answer given there: the configuration space of a nematic liquid crystal is $\mathbb{RP}^2$, because a nematic liquid crystal is basically a rod. This is relevant to classifying what are called topological defects of materials made out of nematic liquid crystals. You can check out the link for more details and some nice pictures.

Qiaochu Yuan
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