Parameters
θ = 0.618 (golden ratio − 1)
Rational? No — C*-algebra is simple
UV = e^{2πiθ} · VU
K₀(A_θ) ≅ ℤ²: generators [1],[p_θ]
τ([p_θ]) = θ (trace): 0.618
Type: Simple, non-nuclear for θ∉ℚ
Noncommutative torus A_θ: C*-algebra generated by unitaries U,V with UV=e^{2πiθ}VU.
θ∈ℚ: A_θ ≅ sections of matrix bundle over T² (Morita to C(T²))
θ∉ℚ: simple C*-algebra — no nontrivial closed ideals.
Rieffel projection p_θ: explicit self-adjoint idempotent with trace θ. Classifies K₀(A_θ)≅ℤ+θℤ.
Connes: A_θ = C(T²) deformed in direction of irrational foliation. Noncommutative geometry: metric, Dirac operator, spectral triple all generalize.