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Alavi M. (2021). Investigation of Two-Dimensional Geometric Patterns of Knot Arangements and other Cryptographic Methods on the Surfaces of Ancient Tiles (Kashis) and its Adaptation to Three-Dimensional Arrangement of Atoms in Crystalline Solids. JRA. 7(2), 197-208. doi:10.52547/jra.7.2.197
URL: http://jra-tabriziau.ir/article-1-314-en.html

The art and techniques of colorful tiling in domes, inscriptions, minarets of mosques, buildings and historical monuments from antiquity, is a valuable and important document about the decorations of ancient Iranian architecture [1].Ancient mathematicians, designers, and practitioners used the geometric methods of knot arrangement, Yazdi-bandi, Muqarnas,(Ahu pai) and other methods, with the help of geometric arrangement of squares, diagonals, rhombuses, and techniques such as "square-to-circle correlation" with the term "mandala. The identification and production of the primary units of geometrical "motifs" began centuries ago and includes geometric units in two dimensions. These motifs have been developed in different historical periods by geometers and designers. It is clear that mathematical-geometric laws have remained fixed in motifs throughout the past. This means that the laws of symmetry have a main origin, in a way that they have been fixed in different historical periods. On the other hand, alignment, balance and symmetry are necessary for each other. Geometry has been used since ancient times to measure the surfaces of buildings and agricultural lands. Points, line segments, angles, circles, squares and triangles have been used in the production of two-dimensional patterns in the order of regular patterns.