We derive the inverse fine structure constant lpha^-1 = 137 from Fibonacci numbers that appear naturally in the geometry of biological microtubules: lpha^-1 = F₇ F₆ + ₊=₁^7 Fₖ = 13 imes 8 + 33 = 137, with a correction lpha^-1 = 137 + arphi/45 = 137. 0360 matching the experimental value 137. 036 to < 10^-6 relative error.
Jason W. (Sat,) studied this question.