The article developed the mathematical model for the synthesis of ornamental images and implemented the software editor of ornamental images, based on symmetry theory. The paper shows the fundamental role of symmetry. It is analyzed that the symmetry theory methods are used in physics, chemistry, biology, and engineering. It was found that symmetry is based on transformation and storage. In addition, the symmetrical system is based on a set of invariants that are built according to certain rules. It is shown that the symmetry of borders and the symmetry of mesh ornaments are used in ornaments. The synthesis of ornamental images is considered on the example of Ukrainian folk embroidery. The contribution of foreign and domestic scientists to the development of the symmetry theory and synthesis of images is analyzed. It is indicated that Ukrainian folk embroidery is the valuable property of the cultural and material heritage of people and an important source of research. It is analyzed that there are more than 100 types of different embroidery techniques. The role of famous Ukrainian artists in the popularization and organization of Ukrainian folk embroidery museums is presented. It is investigated that embroidery is built from separate motives or from ornaments. Ornaments consist of sub-ornaments. A sub-ornament is a pattern consisting of rhythmically ordered identical elements (built on one group transformation). Subornaments are divided into reports. The report is called the minimum for the area of the area that can cover the sub-ornament, using only transfers. The report, in turn, is divided into even smaller particles: motive or elementary picture. It is found that in embroidery ornaments there are 7 groups of stripe and 12 - plan. Mathematical models of images-ornaments synthesis for groups of a strip and plan groups are developed. Mathematical models are given for ideal ornaments. If offsets of axes or centers of symmetries, it is necessary to adjust the coefficients of transformation matrices displacement. Samples of embroidered ornaments of the corresponding plane and stripe groups are provided. Editor of image-ornaments has been developed, which allows the synthesis of complex ornamental images based on analytical formulas of elementary picture, sub-ornament, and ornament. Examples of real and synthesized samples of Ukrainian folk embroidery are provided. The scientific novelty of the work lies in the development of mathematical models of ornaments on the basis of symmetry groups on the strip and the plane. The practical value of the work lies in the development of an image editor-ornaments.
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