ANGULAR ANOMALIES IN PARAMETRIC ELLIPSES

The paper presents the results of a computational experiment studying the dependence of angles θ₁ and θ₂ on the parameters of length (L) and height (h). The experiment was performed for values of λ = L/h in the range 6 – 21.64. The data were obtained using the ATAN2 functions in Excel, taking into account trigonometric transformations. The results demonstrate the nonlinear dynamics of the angles with a change in λ, which indicates the critical influence of the ratio of geometric parameters.It all started with a routine job in Microsoft Excel. I built ellipses using classical formulas, checking geometric models.But when I entered the data for the parameter L (half-axis length) and angle θ, the calculation column returned something inexplicable:At L=3.5, the angle was stable at 340.5°. At L=4.14, it was abruptly divided into 149° and 2 11°. And at L = 10.82, it dropped to 6° altogether."Excel bugs?" "Excel bugs?" I thought. But the anomalies were real.A pattern that cannot be ignored.Despite the lack of mathematical justification, I have recorded clear patterns (Table 1, graph 1, graph 2).Conclusion: This is either an Excel error, or: Ovchinnikov's laws, at λ = L/0.5: 1. At high L/h, the spiral tends to a straight line (θ = 180°). At λ ≤ 7.0 → θ ≡ 340.5° Angle: θ ≡ 340.5° The system ignores changes in the L-energy barrier ??? The system is "frozen" in its initial state. There is not enough energy to overcome the potential barrier. 2. Bifurcation (λ = 8.28) Critical value: λ = 8.28 At 7.0 ≤ λ ≤ 8.28 Angles: Two stable states - 149° (lower) and 211° (upper) Bifurcation of the angle - quantum tunneling??? Assembly disaster. The slightest disturbance picks a branch. The bifurcation point (λ=8.28, θ∈ (149°, 211°) 3. Stabilization (λ > 8.28), at λ > 8.28 → θ → 180° ± δ(λ) At λ ∈ [8.0, 20] Angles: 170° - 188° Formula (1) Exponential tendency to 180° with fluctuations. Angles in physical systems are quantized by discrete numbers (340.5°, 149°, 211°, etc.). 4. Decay anomaly (λ = 21.64, θ = 6) При λ > 20 Formula (2) "Value 6°" new state of matter ??? Singularity at λ < 5, Bifurcation at λ = 8.28; Decay at λ>20."I am not a physicist or a mathematician. I just saw a pattern, and I'm asking you to help me decipher it."Conclusion/ Invitation to a dialogue. Fact: Anomalies exist and are reproducible. Mystery: Their nature is unknown. Perhaps it is:1. An artifact of the model (but why is it correct for 90% of the points?).2. The manifestation of quantum effects in the macrocosm, or... a new physics section, born in Excel."If you have found out these patterns, please contact me. If not, repeat the experiment:Set h = 0.5, L = [3.0, 10.82]. Find your 340.5°, 149° and 6°. Mathematics is the language of the universe. Perhaps we have heard her new word"