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In this work, we generalize and describe the golden ratio in multi-dimensional vector spaces. We also introduce the concept of the law of similarity for multidimensional vectors. Initially, the law of similarity was derived for one-dimensional vectors. Although it operated with the values of the ratio of the parts of the whole, it created linear dimensions (a line is one-dimensional). The presented concept of the general golden ratio (GGR) for the vectors in a multidimensional space is described in detail with equations. It is shown that the GGR is a function of one or more angles, which is the solution to the golden equation described in this work. The main properties of the GGR are described, with illustrative examples. We introduce and discuss the concept of the golden pair of vectors, as well as the concept of a set of similarities for a given vector. We present our vision on the theory of the golden ratio for triangles and describe similarity triangles in detail and with illustrative examples. Full article

People have had an interest in harmony issues for thousands of years; however, there is still no elaborated system of views on these questions. Ancient Greeks understood harmony as an agreement of opposites. A surge of interest in the study of the harmonic aspects of being occurred in the twentieth century due to the development of systems science, particularly regarding synergetic system effects. At the same time, there are still relatively few applications of synergetics because of the absence of an accurate methodology for the identification of system harmonicity. The aim of this research is to develop the methodology for the quantitative assessment of system harmonicity by considering a practical example: the quantitative assessment of the harmonicity of the road safety provision system (RSS) and its dynamics during the last 15 years (2006–2020). In addition, the impact of the COVID restrictions on population mobility in Russia in 2020, on the change in the harmonicity of the road safety provision system, is considered. During the research it was established that the quality factor g of the Russian road safety provision system changed from g₂₀₀₆ = 1.9565 to g₂₀₂₀ = 2.4646, which promoted the decline of the relative entropy of the Russian road safety provision system from H_{n RSS2006} = 0.8623 to H_{n RSS2020} = 0.7553. The deep reason for that change was the modification of relation between “weights” or the significance of the contribution of different elements of the cause-and-effect chain in the formation of the factual level of the road accident rate in Russia in the last 15 years. The main conclusion of this research is that the harmonicity of the Russian road safety provision system, assessed by the normalized functional general utility GU_n, has been increased, and it has already exceeded the level of harmonious reference systems GU_n = 0.618. In fact, the normalized functional general utility GU_n of the Russian road safety provision system increased from GU_nRSS2006 = 0.615 to GU_nRSS2020 = 0.652 (by 6.0%), from 2006 to 2020. Simultaneously, the share of the normalized used resource X_n declined, allowing a conclusion to be drawn about a significant improvement in the balance “efficiency-quality” of the Russian road safety provision system. The COVID lockdown played a positive role in this process. Harmonicity of the Russian road safety provision system, assessed by the normalized general utility GU_nRSS, increased by 0.46% from 2019 to 2020. Full article