This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
What is the smallest rotation that returns an object to its original state?
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!
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
The New Ambidextrous Universe, Revised (Third) Edition, pp. 329-332
W. H. Freeman, 1990