Recursive Integer Sequences, Detected in Solar-Cycle Periodicities Measured in Numbers of Rigid Rotations of the Sun

Abstract

Consecutive integers in the recursive number sequences, the Fibonacci sequence (Fn) and the Lucas sequence (Ln), are detected in the lengths of solar-activity variations from ≈ 1 yr to ≈ 12 yr, measured in rigid rotations of the Sun at the helioseismologically determined frequency in the radiative zone, $433 \pm 3$ nHz. One rotation is denoted by the symbol $\Omega $ . Firstly, in the new international sunspot-number record (Ri) the most frequent (modal) sunspot-cycle length, which is also the period defined by autocorrelation for the recurrence of sunspot cycles, has been $144 \pm \approx 2~\Omega $ ( $\mbox{F}_{12} = 144$ ). The most frequent length for a descending leg of the cycle has been 89 ± 2 $\Omega $ (F $_{11} = 89$ ), and for an ascending leg 55 ± 1 $\Omega $ (F $_{10} = 55$ ). Secondly, there is some observational evidence of Ri spectral peaks at the consecutive Ln numbers of $\Omega $ : 18 $\Omega $ (≈ 1.3 yr), 29 $\Omega $ (≈ 2.1 yr), 47 $\Omega $ (≈ 3.4 yr), and 76 $\Omega $ (≈ 5.6 yr), which are harmonics of the 144 $\Omega $ period divided by the first four F $_{{n}} > 1$ : 2, 3, 5, and 8. The numbers of $\Omega $ : 144, 89, and 55 may be kinematical thresholds in the dynamo process starting at sunspot maximum, when the poles change polarity and the process is re-set. The ratio of two consecutive Fn or Ln converges to $\frac{1+ \sqrt{5}}{2}$ , hence it is suggested that this proportion plays a role in solar behavior over time, described numerically. The length ratio $\frac{1+ \sqrt{5}}{2}$ also is characteristic of fivefold symmetry in space. Since the icosahedral group is the link between numerical and spatial expressions of fivefold symmetry, it is proposed that the presence of icosahedral symmetry in the large-scale geometry of the Sun could also be tested.