Islamic Design: a Mathematical Approach
This book is similar to Jay Bonner's Islamic Geometric Patterns that was recently reviewed here, although this one does not have the aplomb of a coffee table book. It is accompanying the database www.tilingsearch.org of the first author. Brian Wichmann also authored The World of Patterns (2001) which is a catalogue of 4000 plates of different tiling patterns, accompanied with a searchable cdrom. He has a mathematics degree from Oxford but he worked as a software engineer at NPL. The tilingsearch website is Wichmann's retirement project that conveniently replaces the cdrom. David Wade has written several books on Islamic art and he also has his own website patterninislamicart.com describing patterns illustrated with some 4000 related photographs. Of course many of the photographs and drawings in the book are also fount on these two websites. Several other related websites can be found by a simple web-search.
The books, just like Bonner's book, has two parts: the first is about the historical and cultural context, and the second about the mathematical analysis of the patterns. The first part has some mathematical interest as well because it sketches where this fascination of symmetry and patterns comes form. Its origin is the geometry of Pythagonas and Plato's philosophy. The monotheism of Islam created a sense of unity and this resulted in a successful jihad with a quick expansion of the young Khalifat in the period 650-700 CE. By conquering the existing ruling powers like Byzantium and Persia the Arabs assimilated also all their knowledge and cultural heritage. Hence neoplatonism came naturally into Islamic culture via encyclopedic efforts to translate the Greek philosophers.
There is also a religious component of course. While the Quran is like the Christian Bible, the Hadith (revelations) is like the Jewish Torah, a book of law where it is written that human and animal representation is not allowed (a rule that has a Jewish origin). Moreover the iconoclasm was also a way of submitting the wealthier cultures they had conquered. On the other hand, the abstraction of a design gave a sense of transcendence, and the repetitive tiling character can symbolize eternity.
Soon the Islamic territory became too large to be ruled by one central government and fights among different subregions based on religious, political, and cultural differences divided the superpower. Different styles were grafted on the basic ideas.
All of these ideas are clearly worked out in separate chapters of this book. The timeline at the end of the book is thereby very helpful. It spans the period from the fall of Rome in 410 CE and the consecutive rise and disintegration of the Islamic superpower till the invasion of the Europeans into the Arab lands in 1500, Also the glossary of Arabic words and of special terms used to describe the designs are useful.
The mathematical part has diverse topics to discuss. In the introductory chapter the key concept of a rosette is defined. A rosette has rotational symmetry with at its centre an n-pointed star. Most common are rosettes with n equal to 5, 6, 8, 10, 12, 16, 24, or, 32. This star is surrounded by an alternating sequence of kites and petals. Kites are rhombi with two long and two short sides having d2 (mirror) symmetry. The short sides fit into the inward pointing wedges of the star. The (usually larger) petals are 6-sided polygons, also with d2 symmetry and that in standard design have 2 radially oriented parallel sides. Think of a rectangle whose short edges are replaced by an obtuse and a sharp outward pointing arrowhead respectively. The petals fit in the wedges created by the kites and their sharp points touch the points of the star. (To properly understand this, just check one of the websited mentioned above.)
The symmetry of the design is denoted both in Orbifold and Hermann–Mauguin notation (a survey is included as an appendix of the book). Once the symmetry is fixed by the central pointed star, it requires a detailed analysis to fix the lengths of the edges and the angles to produce all the tiles that are needed to generate the overall design that will have several identical or different rosettes. Once the n is chosen and the length of the smallest edge, little freedom is left to fill up the whole figure. A first example is worked out for a 16-pointed star, surrounded by eight 6-pointed stars. The central star has a vertex angle of 45°, and all angles involved will be simple fractions of 45°. If the central star edge is chosen as unity, then the length of all other edges can be computed to capture the whole design with mathematical precision.
While in the mathematical design, the tiles fit tightly together, in a practical realization, the (mathematical) boundaries of the tiles can be replaced by lines with a certain width. Sometimes these lines are wide enough so that they can also be realized by tiles. These lines are like treads that run over the pattern. They can be painted in white or have different colours. If one follows one of these lines, then it will intersect with itself or with other lines. At intersections they can be strictly interlacing, meaning that they will interrupt the intersecting line or will be interrupted by it in a strictly alternating pattern. This give the visual impression that the line goes over or under the lines it crosses. If these lines are wider bands then they can have a quadrilateral tile at their intersection. A separate chapter is devoted to all these issues.
Like the analysis of the example of the 16-pointed star, other configurations are discussed in subsequent chapters.
The kathem is an 8-pointed star with vertex angles of 90° that is very common and which allows for a lot of variation. Only 17 different tiles are used with some variation in edge length to form all octagonal designs. Some variations are possible that have a central star that has 16, 24 or 32 points.
Similarly decagonal patterns are analysed that leave the ratio of two edges as a degree of freedom to bring variations into the pattern. Here the central 10-pointed star can be surrounded by ten 5-pointed stars or the central star can be 20-pointed surrounded by 10-pointed ones.
Designs with six-fold symmetry are called 6-fold delights in the book. Clearly here angles are multiples or simple fractions of 60°. The 6- or 12-pointed stars can be surrounded by stars with 5,6, or sometimes 9 points.
Sometimes there is a symmetry to be discovered at different scales. There is the micro symmetry, as described above, but if the size of the pattern allows to look at it on a macro scale, some other patterns may occur.
The above construction based on trigonometry does not always work for some patterns as is illustrated in the penultimate chapter. A different construction is then required which is based on the use of ruler and compass. Starting from one rosette and the centres and radii or neighbouring ones, all the rosettes and the intermediate pattern can be constructed using only these two instruments. This is explained in detail for an example of a design with 18 and 12 pointed stars with two (irregular) heptagons in between (a design from a door of a mosque in Cairo). A goniometric analysis is made for the pattern on a pulpit of another mosque in Cairo. Not all the authors agree on the mathematics that were historically used to make the designs. Because some boundary lines are wide, precise measurement of the pattern is sometimes impossible, or the design has been damaged or it has to be analysed using an unclear or noisy photograph. All this can leave some room for interpretation. The mathematical techniques (mainly trigonometry) used here should have been known at the time the patterns were made.
For all these patterns, existing examples are given showing that some patterns are typical for certain regions. There is for example a typical Moroccan style and a Byzantine style and designs typical for India etc. With few exceptions, the mathematical analysis of these patterns is not in depth and it would have been nice if there had been more details about the software used to generate all these patterns. Often photographs of existing artwork are used as the starting point for the mathematically generated pattern. Because measures can be imprecise, sometimes it is not clear that there are small flaws in the design that will only come to the foreground when implemented on a computer. So there is a chapter that illustrates some of these small errors either in the design or in some published analyses of designs. There are also some small typographical errors in this book. For example, on page 94 the authors refer twice to triangle X, but these are different triangles and there is only one X on figure 10.10; page 129 refers to an interlace discontinuity in Figure 10.2, while I think the idea is to refer to the kink in Fig.10.5; page 143 refers to angle Y in Fig. 14.9, but there is no Y in that figure.
The nice thing about this book is that it does explain many of the constructions, but it also shows that not all existing artwork is perfect and that different methods may have been used to generate the patterns. All these examples being generated over many centuries and in geographically very different regions explain the richness and diversity, and yet the underlying uniformity of these geometrical patterns. Note that just like Bonner's book, the analysis of this book is also considering only strictly geometric patterns.