Islamic design: a mathematical approach (Q1692409)

The book under review is a cultural history of the Islamic world, a history that started with the Revelations on Prophet Mohammed and lasted for several centuries, and a world that extended from the Atlantic to China. It focuses on Islamic art whose main manifestations took the form of tessellations by geometric figures, but which also has many other interesting components such as architecture, calligraphy, book illumination, etc. All of this is covered in Part I which consumes a little more than one fourth of the book. This part covers the history of the Islamic states describing the different styles of art that existed in the countries that hosted those states, such as Syria, Iraq, Egypt, Spain, Morocco, Turkey, Iran, India, and Central Asia. It also surveys the materials that were used, such as stone, brick, ceramic, and wood. It also talks about the art of book illumination that was motivated by devising attractive ways for the illumination of the Holy Quran. This part also has chapters on how Islamic designs were influenced by Greek geometry, and also by the Islamic values. Part II of the book is a detailed analysis of geometric tessellations that are often thought of as synonymous with Islamic art. A tessellation is a covering of the infinite plane by identical tiles in such a way that no overlaps are allowed and no gaps are left. The simplest tiles that can be used as tessellations are the squares, and extremely intricate animal-based tiles can be found in the art of M.~C.~Escher. Islamic art used tiles having the form of geometric figures that consist of stars, polygons, etc. Such tiles can be found in great abundance on the walls and floors of mosques and palaces and in street pavements throughout the Islamic world. To every tessellation, geometric or not, there is a mathematical structure called a group, actually a group that is usually called a \textit{2D crystallographic group} because of certain special mathematical properties that it has. The manner in which this group is associated to a tessellation is explained fully in a 21-page appendix at the very end of the book. Mathematicians can theoretically prove, without examining any tessellation, that there are exactly 17 2D crystallographic groups in the world of mathematics, and these 17 groups are all realized in the tessellations that exist in al-Hambra in Spain. This is amazing, and can be thought of as kind of perfection in Islamic tessellations. Part II contains hundreds of beautiful Islamic patterns, and thorough mathematical analysis, based on the material in the appendix, of many of them. These include patterns from Egypt, Morocco, Iran, Turkey, and al-Hambra. Some of these patterns are accompanied by computer-generated plates that replicate them in a polished and more accurate and colorful manner. Another mathematical aspect of the geometric tiles used in Islamic tessellations has to do with the regular polygons used therein. Some of the polygons used are triangles, squares, pentagons, octagons, and decagons, and all of them can be constructed using ruler and compass. However, some of the tiles contain regular heptagons and nonagons, which is a bit curious since these cannot be constructed using that tools. The book is so rich in material that makes it difficult to write a fair review of.