New regularities of segment division according to the golden ratio

The paper investigates four problems on the dividing a unit segment by the "golden" proportion. Namely, the general model of the unit segment "golden" division, the decomposition of a square trinomial, the "golden" division of a unit segment by a point with coordinate $ x<\frac{1}{2} $ the "golden" division of a unit segment with loss of "memory".

In this article, the concept of decomposition is used as elevation to the degree of a quadratic trinomial. The binary division of a unit segment into two unequal parts with the properties of the "golden" proportion is realized at an arbitrary point in the phase plane $0 p q$ , and the decomposition of a square trinomial leads to the formation of recurrent sequences with Fibonacci properties. It can be noted that the well-known "golden" ratio between the parts of the binary division is most likely a partial imitation of the theorems of Viet and Poincaré. The rules of the "golden" division for the case $x>\frac{1}{2}$ are well studied. Therefore, the regularities for the case $ x<\frac{1}{2} $ were researched. Despite the fact that the numbers $\psi, \varphi$ are expressed through each other, from the point of view of the "golden" division, both realizations with quantitative characteristics $\left.Y_{\varphi}\right|_{L=1}=\varphi$ and $\left.X_\psi\right|_{L=1}=\psi$ are independent and equal, although their quantitative characteristics can be related to each other with the appropriate formulas. Geometric progressions were constructed for numbers $\varphi$ and $\psi$ for whole positive values $n \geq 0$ of the exponent to confirm the independence and equality of both models. Quantitative characteristics of the "golden" division of a unit segment by two points with coordinates in intervals $x>\frac{1}{2}$ and $ x<\frac{1}{2} $ interconnected by a nonlinear relation of parabolic type $\psi=\varphi^2$. In the classical "golden" section theory, it is assumed that after distribution, the parts of the segment do not change their spatial directions, and they coincide with the direction of the original segment, i.e. $\alpha=0$ . In this article the case $\alpha \neq0$ was studied when, after the distribution, the spatial orientation of the distribution elements changes. The angular dependence of the "golden" division of a unit segment with the loss of "memory" of its parts on the spatial orientation after division, shows a known angle $\left.\alpha\right|_{p\to1}\to\frac{\pi}{3}$ of inclination on the lateral surface of the Hyops.