![The vector a=[3,2]′ and its four similarity vectors.](https://www.researchgate.net/publication/389229979/figure/fig5/AS:11431281318361406@1742524826542/The-vector-a3-2-and-its-four-similarity-vectors_Q320.jpg)
February 2025
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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.
![Figure 7 shows the graph of the positive roots Φ 1 (í µí»¼) calculated in the interval of interval [0,2í µí¼]. We call the function Φ(α) = Φ 1 (í µí»¼) with this graph the general golden ratio function, or the GGR function. For this function, the minimum is 0.6180 at the angle-point í µí¼, and the maximum is 1.6180 at 0 and 2í µí¼. The Golden ratio equals to 1 at angles 2/3í µí¼ and 4/3í µí¼ (as shown in Fig. 1) . The mean of the Golden ratios in this interval equals to 1.192880, approximately at angles 1.7385 and 4.5447, or 99.6075 ° and 260.3925 ° (or 180 ° ∓ 80.3925 ° ).](https://www.researchgate.net/publication/368305160/figure/fig5/AS:11431281118371813@1675740564714/shows-the-graph-of-the-positive-roots-PH-1-i-i14-calculated-in-the-interval-of-interval_Q320.jpg)
