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The 𝐏𝐚𝐮𝐥𝐢 𝐈𝐝𝐞𝐧𝐭𝐢𝐭𝐲 tells us how to rotate a 2 component 𝒗𝒆𝒄𝒕𝒐𝒓 of complex numbers, called a "spinor", around an axis θ̂.
for spinors, a rotation by 2π yields -1
only a rotation by 4π yields identity
..sound familiar?pic.twitter.com/ECGtLshlH2
An object that existed in a Möbius-strip-shaped universe would be indistinguishable from its own mirror image
This crab's larger claw switches between left to right with every circulation
(GIF not mine)pic.twitter.com/n3JHJdy0jJ
*Locally* it’s still distinguishable from its own mirror image. So if spacetime turned out to be nonorientable, and you took a trip around the universe along an orientation-reversing path, that would make for a veeery confusing return home!
Is there a sort of potential field that can possibly result in paths that are globally non-holonomic with respect to a symmetry of Euclidean space? Iirc things like Aharonov Bohm and a spin-field coupled Hamiltonian for fermions only have non-holonomy on SU(2).
@InertialObservr this hurts my brain in a very deep, satisfying way. I love your Möbius animation, too 
As you say, so much for geometric intuition
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