16 pages for non-brainiacs on the Hopf fibration by David Lyons

  • mapping from S³→S²
  • ƒ(a,b,c,d) = (a²+b²−c²−d²,,)
  • in general a linear transformation in 3-D requires 9 parameters (3×3 matrix — see general linear group)
  • but a rotation only requires ≤4 parameter
  • mapping from S³→S²
  • ƒ[a,b,c,d] = [a²+b²−c²−d²,2(ad+bc),2(bd−ac)]
  • but a rotation only requires 4 parameters
  • understanding maps from high-dimensional spheres to low-dimensional spheres is a Hard Problem
  • Gimbal lock, composition of rotation maps
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    to get the general three-angle rotation group
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    but this is ugly and wrong—not because there are too many trig words, but because if you play around with it enough you’ll see that—just like the North Pole and South Pole have redundant longitude coordinates—various combinations of [phi;,theta;,psi;] can overlap each other or even get caught in a Gimbal Lock.
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