Golden Ratio Function: Similarity Fields in The Vector Space

In this work, we generalize and describe the Golden ratio in the multi-dimensional vec-tor space. We also introduce the concept the law of similarity for multidimensional vectors. Ini-tially, the law of similarity was derived for one-dimensional vectors. Although it operated with such values of the ratio of parts of the whole, it meant linear dimensions (a line is one-dimensionality). The presented concept of the general golden ratio (GGR) for the vectors in the multidimensional space is described in detail with equations and solutions. It shown that the GGR is a function of one or a few angles, which is the solution of equations, or the golden equa-tion, described in this work. Main properties of the GGR are given with illustrative examples. We introduce and discuss the concept of the golden pair of vectors and the set of similarities for a given vector. Also, we present our vision on the theory of the golden ratio for triangles and de-scribe in detail the similarity triangles with illustrative examples.