What are the mathematics behind Islamic geometric decorations? What is the essence that makes it so recognizable? One possible characterization is by pointing to the symmetry, hence group theory is what is needed to describe it. However, that may catch some of the local symmetry, which of course is part of the beauty of these designs, but it does not completely explain the overall structure as well as the finer geometric aspects of doubling and interweaving lines that define the patterns. Thus the description of the 17 wallpaper groups is not the end of the story.
Jay Bonner, who is a creative designer of these patterns gives here a detailed description of the underlying polygonal techniques that can be combined to form a myriad of possible designs. He comes to his conclusion by comparing the many designs that were used throughout the Islamic cultural history and by distilling from these the techniques that were possibly used. Some of the designs, and hence the assumed underlying techniques, were more popular than others or were particular for certain regions or periods. The possibilities of the more complicated ones were not always fully explored and they give rise to new original designs. After the decline of the craftsmanship of these Islamic designs, some renewed interest in the subject arose in the second half of the twentieth century. Some books were written on the mathematics of the symmetry groups used, and it became a popular subject for documentaries and picture books, but Bonner now supersedes the latter less mathematical approaches with this monumental encyclopedia. It is not only a nice picture book with over a hundred photographs of decorative art on monuments (in chapter 1), but there are also the 540 other illustrations, many of which consist of several parts that illustrate the construction and the results of the designs.
The first chapter starts with a quick survey of design techniques with pointers to many illustrations in subsequent chapters where a more technical discussion is given. The main objective of the chapter however is to illustrate by a chronological summary how the different techniques were used throughout the centuries of Islamic culture from the Umayyad Caliphate (7-8th century) till the Mamluk Sultanate in Egypt (13-16th century), how they evolved in Eastern Islamic countries as well as in North Africa and the Western Al-Andalus, and how the techniques were adopted in non-Muslim cultures.
The second chapter is a bit more technical and summarizes different classification methods. One can for example look at an underlying regular tessellation (isometric, triangular, rectangular, hexagonal), or the known plane symmetry groups can be used to classify the designs, but the method proposed by Bonner is by design methodology, and he gives arguments why the polygonal technique is probably the one that was historically most commonly used, and hence the proper way to classify. Other authors have proposed that historically different methodologies were used but there is less evidence for those proposals or they are only useful for simpler designs. The polygonal technique starts from a polygonal tessellation of the plane. Pattern lines in these polygons will define the eventual design. These pattern lines emerge at points on the edge under particular incidence angles and intersect the pattern lines from the other edges. Once the polygons are put together to form a tessellation of the plane, the global design will protrude and the underlying polygonal stratagem can be forgotten.
The incidence angle of the pattern lines at the midpoints of the edges can be acute median or obtuse, and there is a fourth possibility in which pattern lines start from two symmetric points on the edges. Depending on the incidence angles and the underlying polygonal pattern rotational symmetry will occur. The most common are fourfold, (with squares and 8-pointed stars), sixfold (with 3-,6-,12-, and even 24-pointed stars), or fivefold (5- and 10-pointed stars), but occasionally also sevenfold symmetry was used, and in the more complex designs we also find 11, 13-pointed stars. Usually the stars appear at the vertices of some regular polygonal grid and/or its dual.
The longest chapter by far is chapter three which is a thorough discussion of the polygonal technique. One possibility is to start from a tessellation of the plane that consists of one or several types of regular polygons (triangles, squares, hexagons, octagons). Sometimes one needs the systematic inclusion of an irregular polygon, which is then called a semi-regular grid. The pattern lines can be narrow or invisible like when they just delimit coloured mosaic tiles, or they can be widened or doubled. Moreover they usually do not just intersect but they form an ingeniously interweaving pattern.
But regular or semi-regular tilings are relatively simple and soon Bonner moves to tessellations composed of regular and irregular polygons decorated with suitable pattern lines that fit nicely together obtained by one of the four design possibilities (acute, median, obtuse, 2-point). Bonner systematically discusses the different possible symmetries that can be obtained in this way. There are two variants of the fourfold symmetry. The A version has a large and a smaller octagon and seven other polygons to tessellate. The B version has only one octagon and five other polygons, but still that leaves many possible tessellations. The fivefold system obviously involves decagons and pentagons but can also include many other convex and concave polygons. This fivefold system was very popular and Bonner discusses several variations depending on the shapes of repeat units, that are rhombi, rectangles, or hexagons, These repeat units will fill up the plane by translation. It's not a coincidence that the golden ratio appears in these designs. Sevenfold symmetry occurs is more complicated to deal with and therefore probably less frequently used. The starting point is a tetradecagon and a heptagon and pattern lines can be constructed by connecting the midpoints of edges that are k = 1,...,6 positions apart.
A second group of design methods are called non-systematic patterns by Bonner. This technique allows the construction of more enigmatic stars with 9,11,13, or 15 points. While in the previous group, a tessellation was formed using a limited set of polygons, in this group, just one characteristic polygon is used (rhombus, triangle, square, rectangle, hexagon) that tessellates the plane. The generation goes as follows. Take one of the polygons and generate at each of its vertices, equispaced radii are generated such that the incident edges of the polygon are two of them. The intersection points of the radii are used to generate a design pattern consisting of smaller polygons, and the whole design is then translated to cover the plane. Bonner describes many examples using this kind of technique, some are historical, but there are also possibilities for original designs.
The most complex design technique is called dual-level design. Basically one starts from a coarse level that generates a set of lines that are widened. These wide strips are decorated with a fine gain design, which is then extended to the whole plane. This gives highly complex structures of which historical examples exist. Although there are only two levels used, it has the characteristics of self-similarity and it creates possibilities for new multilevel designs. In a short final section, some ideas are given about how to apply such techniques to decorate a dome or a sphere.
I do realize that my previous attempt to capture the main points of the design methodologies is totally inadequate since one needs the graphics to understand them properly. You may want to look up the author's Facebook page or the website of his company, but none will match the abundance and clarity of pictures in this book.
In a short chapter 4 Craig Kaplan describes the software building blocks that will be needed to generate the pictures on a computer: tilings, fitting polygons together, generating patterns lines, producing rosettes, how to join widened lines or generate the weaving effects etc. There is very little mathematics here and it remains a high level description so that it will need additional computer and mathematical skills to actually produce the graphics, but it gives at least some useful guidelines.
The book is very carefully edited, especially the graphics are extremely nice and very informative. The only strange typo I could spot was that σ is called "delta" on page 361. The book is not written by a mathematician, nor is it written for mathematicians. It is an artistic designers (hand)book for Islamic(-like) geometric patterns. There is very little mathematics, but I am sure all mathematicians will love the beauty of the designs non the less. While reading the text, it takes a while to get used to the terminology. There is a glossary with a set of terms that are briefly explained at the end of the book, but these are necessarily short and their meaning will become only gradually more clear. When chapter one starts with a brief survey of the techniques, one is pointed to pictures in later chapters to get an idea of what is meant, but the proper explanation comes only in chapters 2 and 3, and if you are really interested how the graphics can be produced on a computed, one has to read chapter 4. Mathematicians may be used to books that are arranged in the opposite order: start with the definitions and tools and end with the applications. The (often forward) references to pictures in this book are however carefully and consistently done, so that with a lot of paging back and forth one becomes gradually familiar with the content and the ideas proposed. The book has the looks of a coffee table book, but it requires more than just casual reading to understand the design methods.