Notes on the Golden Ratio: The Golden Rule of Vector Similarities in Space

TL;DR

This paper generalizes the concept of the golden ratio to multidimensional vectors, exploring its properties, dependence on angles, and applications to similar vectors and triangles in space.

Contribution

It introduces the concept of a general golden ratio (GGR) for vectors in multidimensional space, extending the classical ratio to higher dimensions and angles.

Findings

GGR depends on the angles between vectors

Properties of GGR are illustrated with examples

Theory of similar triangles related to GGR is discussed

Abstract

In this work, we have abstractly generalized the similarity law for multidimensional vectors. Initially, 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 concept of the general golden ratio (GGR) for the vectors in the multidimensional space is presented and described in detail with equations and solutions. It shown that the GGR depends on the angles. Main properties of the GGR are given with illustrative examples. We introduce and discuss the concept of the similar 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 describe the similar triangles with examples.