Beauty in Mathematics: Symmetry and Fractality

The most important concepts underlying beauty are the concepts of symmetry and fractality, but the relationship of these concepts has not yet remained clear. For centuries, beauty was understood only as a stable order and symmetry. Synergetic worldview allows us to give a new assessment: beauty can be seen as an attractor, the result of self-organization of nature, or the flight of human thought. On the one hand, fractality can be considered one of the manifestations of symmetry in an expansive sense. On the other hand, symmetry can be considered a manifestation of fractality with a finite number of iterations. Thus, the concepts of symmetry and fractality are closely interrelated. Symmetry reveals in beauty a stable order, and fractality reflects in beauty the result of the self-organization of the chaos of nature or the freedom of human thought. Symmetry and fractality are two opposites, mutually complementing each other, aesthetically and mathematically mutually passing into each other. Thus, symmetry and fractals are the most important concepts for the disclosure of the beauty of the universe, which determine their importance for learning. The concept of self-similarity can serve as a basis for acquaintance with fractals.