π₁(SO(3)) = ℤ/2
A 360° rotation in SO(3) is NOT contractible — you need 720° to return to the identity!
SU(2) is the double cover:
SU(2) ≅ S³ (simply connected)
SO(3) ≅ ℝP³
Spinor ψ ∈ ℂ²:
R(2π): ψ → −ψ (sign flip!)
R(4π): ψ → ψ (back to start)
Current θ: 0.00 rad
Phase: 1
Belt twisted: 0×
Red belt = twisted; green = untwisted
A 360° rotation in SO(3) is NOT contractible — you need 720° to return to the identity!
SU(2) is the double cover:
SU(2) ≅ S³ (simply connected)
SO(3) ≅ ℝP³
Spinor ψ ∈ ℂ²:
R(2π): ψ → −ψ (sign flip!)
R(4π): ψ → ψ (back to start)
Current θ: 0.00 rad
Phase: 1
Belt twisted: 0×
Red belt = twisted; green = untwisted