Steps Towards the Analysis of Geometric Decorative Motifs Using Shape-matching Techniques

The classification of geometric decorative motifs has tended to emphasise the geometric by comparing the decoration to mathematically perfect constructions. Focussing on spirals as exemplars, this paper proposes that additional information may be gleaned from historic decoration on the assumption that such decoration often does not conform exactly to mathematically correct constructions. Shape-matching analysis methods combined with the analysis of stylistic variables are put forward as a means of comparing cultures; in particular, of looking for evidence of cultural diffusion and studying the manner and rate of evolution of geometric motifs. Introduction Spiral decoration is geographically widespread and long-established with the earliest known occurrence dating back to the Palaeolithic period [1: 10]. Formal geometric approximation techniques for spirals possibly date back to Theodorus in the fifth century BCE [2] and Vitruvius’ manual of the first century CE presented geometric constructions for spirals to be used in architectural decoration. Consideration of decorative spirals as a tool for studying cultural diffusion dates back to the late-nineteenth century; cultural contact routes across Europe have been proposed on the basis of similarities in spiral decoration [3: 141-2]. These early comparison systems did not tend to employ formal typologies, instead working from subjective identification of the co-occurrence of similar motifs to infer links. That these proposed links have sometimes been shown by improved dating techniques, to be false [4], supports the case for more precise and formalised typologies as a basis for cultural comparison. Classification of decoration using geometric constructs The possibility of comparing decorative schema using geometric motifs was suggested by established methods of cross-cultural comparisons which identify differences between cultural groups in the presence and frequency of occurrence of types of symmetry transformation [5]. Motifs with an underlying form which closely approximates to a geometrically definable construct may be evaluated according to a constant set of geometric properties. These properties offer the potential for a generic comparison method, applicable across a range of cultures, to be developed. Existing anthropological classification systems analysing a single geometric motif have tended to emphasise the mathematical basis of the motif and assign classes based on resemblance to perfect mathematical constructs [6, 7]. These type groups are applied on the tacit assumption that the producer aspired to create a regular construct – for spirals, a curve in which the radius is a continuous monotonic function of the angle [8: 357]. In some cases as, for example, with Gil et al.’s analysis of Islamic tilings [9], the assumption of a geometric basis to the designs can be readily supported so there is good reason to look for an underlying ideal structure in geometric terms. Decorative spirals, however, often differ sufficiently from mathematical ones to prevent secure identification of them as a defined mathematical type [10]. The sine wave plots of Figure 1 show considerable deviation of architectural spirals from mathematical forms even when the superficial similarity appears strong. This degree of variation is likely to be present in other geometric motifs, if more than one construction method is available, and renders classes based on the pure mathematical constructs inappropriate to the decorative motifs of many cultures. Bridges 2012: Mathematics, Music, Art, Architecture, Culture