FEATURES OF THE GEOMETRY OF SPIRALS AND THEIR NATURAL AND PRACTICAL APPLICATION

This work is devoted to the topical issue of geometric modeling of some types of spirals and their practical application in nature and various areas of human activity. A geometric and practical study of Archimedes' spiral and its inversion with respect to the pole - a hyperbolic spiral, as well as Fermat's spiral and its inversion on a circle of unit radius - lituus, was carried out. A mathematical description of spirals is presented and their geometric models are developed: definitions and equations in polar coordinates are provided, graphic formation schemes and algorithms for their construction are created. A comparative analysis of the researched types of spirals in relation to the oldest spiral of Archimedes, which was found for the first time in the III century BC, was carried out, and the geometric properties that distinguish the types of researched spirals from each other were determined. Special attention is paid to the study of spiral forms in the natural environment and in practical application. Various examples of natural spiral phenomena and forms are selected in the article, which proves the relevance of this geometric study of various types of spirals. It is shown that due to their geometric features, spirals are widely used in many areas of modern human activity: technology, physics, mathematics, astronomy, art, everyday life, etc. Products with spiral surfaces are used in various industries: oil, chemical, food, etc. Special attention is paid to highlighting spirals in architectural forms. The most vivid and characteristic examples of the use of various types of spirals are selected, and photographic images are given. It was concluded that with the development of science, there are new areas of use of the studied types of spirals, therefore, geometric research is a necessary condition for the possibility of using spirals in newly created objects of various industries and expanding the range of their application.