Symmetry Through the Eyes of the Old Masters . By Emil Makovicky. De Gruyter, 2016. Hardback, Pp. 240. Price EUR 99.95, USD 140.00, GBP 74.99. ISBN 978‐3‐11‐041705‐0.

Humans have long been fascinated by symmetry. The oldest known works of pure art – designs carved into soft rocks up to 100 000 years ago – are repeating patterns of high symmetry (Henshilwood et al., 2009), and both the archeological record and museums have numerous examples. Explanations for this fascination range from prehistoric encounters with crystals (Waterhouse, 1972) to our (possibly) greater ability to process our perceptions of symmetric patterns (Gombrich, 1979) to kaleidoscopic images observed by Stone Age people during sensory deprivation, exhaustion or neurological disorders, or psychedelic intoxication (Spivey, 2005). It is not surprising that symmetry has been a major part of the human enterprise. Yet while symmetry may pervade art, it is less visible in art history books. Janson & Janson’s (1962) magisterial ‘History of Art’ has almost no symmetry at all [although there is a brief and almost dismissive mention of ‘ . . . a taste for symmetrical abstract patterns characteristic of Moslem art’ (Janson & Janson, 1962) in the largely architectural chapter on Islamic art]. Although aware of this lacking, Emil Makovicky presents his ‘little book’ not to address a deficiency but as a personal odyssey intended for ‘entertainment and inspiration’. The author, armed with a camera, traveled the world and brings to fellow crystallographers accounts of and images of interesting and suggestive works of art – along with diagrams to assist their analysis. The book explores zero-, oneand two-dimensional symmetries in art from a crystallographic point of view, i.e. symmetry under group actions. (There are two chapters on aperiodic and fractal works towards the end of the book, but most of the book consists of planar geometric crystallography applied to art.) It is organized by crystallographic structure, beginning with representations of unequivocally finite structures, and proceeding to increasingly complex works. The book starts with finite objects and friezes. There is no description of the point groups, so the discussion of finite structures is ad hoc. For example, in Fig. 1(a), a pillar of the Grand Mosque of Bursa has some Arabic calligraphy painted on it, with one side of the inscription being the mirror image of the other. Later in the text, there are a few more complex examples, such as the eightfold dihedral symmetry in Fig. 1(b). Friezes are handled more formally, starting with diagrams of the seven frieze groups. Then it’s on to Gothic vaults (which can be regarded as friezes with patterns consisting of the fluting), friezes spiraling up columns, friezes used as borders, and friezes wrapped around ceramics as in Fig. 2. The reader can compare the book’s diagram for the frieze group p211 (or p2, although the text prefers p211, and we will stick to the notation in the text) to the design on the vase, assuming that the pattern is solid black/hatched versus white. Here, the book slips in color symmetries without much introduction. As artists colored periodic patterns in symmetry-breaking arrangements, color symmetries are necessary for a crystallographic analysis of their art. Very roughly, one has a crystallographic group for the underlying pattern, and then one permutes the colors. Space precludes further discussion of color symmetries here: for a technical description, see, e.g., Jaswon & Rose (1983). Anyway, in Fig. 2, the point of the twofold rotation directly facing the viewer (or ISSN 2053-2733