Golden Ratio Function: Similarity Fields in the Vector Space

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.