The application of a geometrical theorem of Khayyam in the drawing of Chahartoranj motif

One of the interesting and controversial arrays in mathematics is geometry, which is closely related to the geometry of motifs in Iranian art. The synchronicity of the use of geometry in the drawing of the widely used geometry motifs in Iranian architectural arrays is such that it seems that the foundation of traditional Iranian arts was based on it. Among the most important theorists in this field is Hakim Omar Khayyam Neyshaburi, a prominent philosopher and mathematician of the Seljuk period. One of his most important works can be considered a treatise on solving cubic equations and his studies on Euclid's fifth principle in the history of science.This research aims to solve the third degree equation obtained from Khayyam's geometrical theorem and by introducing the ruler (ruler), how to draw Chahartoranj motif. Toranj is one of the ancient Islamic and Iranian motifs used to decorate works. This pattern is usually rhombus or almond-shaped, and sometimes it is square or oval, and it is often placed in the middle of the background, and the inside is filled with leaves, flowers, slime designs, or images of animals and humans. When this pattern is repeated four times side by side, it becomes a Chahartoranj. For this purpose, the aim of the current research is to use the analytical descriptive method to study the use of Khayyam's calligraphy in drawing Chahartoranj and the importance of this motif in the history of visual and applied arts of Iran as well as the use of Chahartoranj images in architectural arrays, especially in Explain tiling.