720 degrees.

Objects are made of bosons (integer-spin particles) and fermions (half-odd-integer spin particles), and the wave function of a fermion changes sign upon being rotated by 360 degrees. To get it back to its original state you must rotate by another 360 degrees, for a total of 720 degrees. This fact is the basis of Fermi-Dirac statistics, the Pauli Exclusion Principle, electron orbits, chemistry, and life.

Mathematically, this is due to the continuous double cover of SO(2) by SO(3), where SO(2) is the internal symmetry group of fermions and SO(3) is the group of rotations in three dimensional space.

A fermion can be modeled by a sphere with strings attached. It is possible to see that a 360 degree rotation will entangle the strings, which another 360 degree rotation will disentangle. You can also demonstrate this with a tray, which you hold in your right hand with the arm lowered, then rotate twice as you raise your arm and end up with the tray above your head, rotated twice about its vertical axis, but without having twisted your arm.

Hospitals have machines which take out your blood, centrifuge it to take out certain parts, then return it to your veins. Because of AIDS they must never let your blood touch the inside of the machine which has touched others' blood. So the inside is lined with a single piece of disposable branched plastic tubing. This tube must rotate rapidly in the centrifuge where several branches come out. Thus the tube should twist and tangle up the branches. But the machine untwists the branches as in the above discussion. At several hundred rounds per minute!

References

R. Penrose and W. Rindler Spinors and Space-time, vol. 1, p. 43 Cambridge University Press, 1984

R. Feynman and S. Weinberg Elementary Particles and the Laws of Physics, p. 29 Cambridge University Press, 1987

M. Gardner The New Ambidextrous Universe, Revised (Third) Edition, pp. 329-332 W. H. Freeman, 1990

M. Gardner Riddles of the Sphinx, pp. 46-48 Mathematical Association of America, Washington DC, 1987

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