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1 | Ornament is one of the oldest methods of decoration, with the help of which people often tried to express their emotions, knowledge, beliefs, hopes, desires, and requests. Ornamental art can undoubtedly be considered as a completely independent type of applied arts, visualizing the conceptual thinking of its creator at a given moment in time. Therefore, the development of ornament is closely related to the history of the formation and development of the nations of the world. This article considers such a formation on the basis of traditional Kazakh felt carpets of different types, regardless of their manufacture method and purpose. The traditional shape of the carpet is most often a rectangle with a central symmetrical pattern edged with an ornamental curb. In the study of ornaments of traditional Kazakh felt carpets, attention is focused on the symbolism and conceptual meaning of Kazakh ornaments, borders and rosettes in particular. The work is based on a study of the national Kazakh ornament due to its dominant position in the visual arts of this country until the end of the 18th – early 19th centuries, as a result of the semi-nomadic way of life with its inherent unity with the outside world, as well as due to the gradual penetration of Islam with its ban on images of humans and animals. The study is also based on the laws of symmetry, on Curie’s principle in particular. As it was established, the most common methods of constructing curbs of Kazakh felt carpets are: the use of the sliding reflection operation and various combinations of the transfer axis. Also, when creating an ornament, attention is paid to the structure of the substrate - an invisible flat grid (lattice). The ornaments were explored as a single unit rather than its individual elements. Special attention was also paid to the fact that, in the Kazakh folk ornament, along with the shape, its color scheme plays a huge role. Although the value of color is noted in many works, a comprehensive analysis of motifs, background, and substrate has not been carried out. Physical symmetry (that of color) in the Kazakh ornament is no less important than the motif. In this work, the value of color in the life of Turkic peoples, which had been formed in the conditions of nomadic life in the endless steppes, was invested in the consideration of color design. Color for the nomad became not only a hallmark, it could also mean the directions of the world or characterized the quality of the object. As a result, the location of the types of elements of the ornament was determined with special points on the grid that have specific properties compared to the entire ornamental field. The role of an invisible substrate (grid, lattice) for building ornaments was confirmed. Such a mathematical analysis opens up great opportunities in the applied use of the results of this study. Keywords: ornament, Kazakh, symmetry, Curie’s principle, grid, motif, border of ornament | 342 |