This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

What is the smallest rotation that returns an object to its original state?

geometry/rotation.s

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

Continue to: