Abstract
Degtyareva, Tatiana 
Great Britain/ Russia 
D.553  Wednesday, 18 July 2007, 17:00  17:30, Room D2 
Towards a topological theory of pictorial order: A study of compositional phenomena in Western art using geometrical models as analogues 
In this paper I will report on my attempt to establish a plausible explanation for some compositional phenomena that are well known in the literature of western art history such as the ‘rule of thirds’, contrapposto, and Sline. I seek to define the compositional phenomena in terms of their place and function within formcontent image structure and in relationship to the concepts of plastic metaphor and aesthetic qualities. As a background to my research, situated in art theory, I consider artwork from two perspectives, namely a holistic view of composition (as a complex whole) and semiotic models of structural analysis of visual language. In addition I make use of experimental data and scientific multidisciplinary research. The analysis of research literature reveals the lack of a unifying theory of pictorial order. The overall methodology for my partly theoretical and partly experimental study is built around the concepts of structure, modeling, and analogy and under the umbrella of synergetic philosophy.
Based on binarystructural and bonding features, a classification of compositional phenomena, designated into three stable constituents of order in artistic image, is introduced: (a) a container (or unit) of visual information within the image frame, which stands for the whole, or pictureworld; (b) proportional relationship between the parts of represented pictorial content within the container, which stands for interconnectedness and correlation; (c) and a hidden path (revealed in composition mostly through what Delacroix called ‘principal lines’) between the most distant points of the container, whose function is also to connect and unite. I posit that the constant characteristics ascribed to these configurations in composition coincide with the major problems studied in the discipline of topology (region of geometry) namely compactness, interconnectedness, and a path. Extrapolation of the properties of topological space onto the constituent elements of structure of artistic image generates the possibility of applying a ‘scientific method’ to the study of artistic image structures previously regarded as nonquantifiable. Finally knowledge derived from geometrical explanations of natural phenomena may enhance the existing ways of understanding art.
My assumption is that a general principle of optimization (or minimax) plays an important role in both coding and decoding visual information. I assume that in design, modeling or constructing the image an artist continually and mainly unconsciously makes a choice while seeking out a ‘perfect’ composition formula, or ‘exact’ configuration of form to convey the message. I hypothesize that these plastic formulae ‘crystallized’ in compositional structures, approximate the parameters (and configurations) of certain mathematical models that represent particular scientific concepts. The reference (by resemblance) of both compositional and mathematical models to their physical analogues causes the use of verbal metaphors in artists’ reflections on their creative practices (such as Titian’s ‘bunch of grapes’). To say more, I hypothesize that if pictorial order (in terms of compositional phenomena) has common morphology with certain mathematical models, in particular drawn from problems on optimization in topology, it is possible to describe and estimate the topological similarity between them. I view the mathematical models as referents, or constants which define the extreme parameters of the dynamic formcontent structure of artistic image (provisionally defined as a binarylevel system comprised from compositional phenomena and plastic metaphors), which is treated as a kind of soft model. To test this hypothesis and trace the affinity of compositional phenomena to patterns governed by general rules of organization, such as ‘minimax’ I have set up some experiments, and developed a method called double modeling. The technique of generating the database of processed samples of images, (though departed from harmonic/ geometric analysis by Hambidge, Bouleau and other), is developed to formalize and reduce the steps of structural analysis keeping in mind to minimize bias and to a certain extent achieve the replicability and validity of experimental results.
Following principles of analogical reasoning I searched for models appropriate for experimental designs both in scientific research and recreational mathematics, and finally chose three topological problems that visualize the optimization principle at work, namely a doublebubble conjecture, a quasicrystal structure model, and a shortest path problem. The affinity between compositional structures and geometric models is established using software processing as a tool (e.g. Photoshop). For Test 1 the method consists of comparison through superimposition of pairs of templates (namely processed reproductions of artworks which constitute the random sample database, and geometric schemata) and measurement with a ruler is applied as an additional instrument to judge the quality of approximation. For Test 2 a 3D animated model of the Sline is generated. The parameters of the model are subject to change while adapting the schema to templates of 3D images of artworks, processed in a certain way. Each image is inscribed into the geometric shape of a model and mechanically adjusted until the principal lines in the composition approximate the Spath oblique view. Results of experiments are categorized and tabulated. Analysis of obtained data reveals (in terms of qualitative estimation) the level of approximation of compositional structures to geometrical models.
If there is an analogy, it means there are numerical limits for (a) an optimal container of visual information, (b) for maximum density within a proportional relationship between the representational elements, and (c) for a shortest way of visual connection (through principal lines) between distant parts within an image. Thus certain numerical parameters in the organization of composition are responsible for economy of perception and by this make the imagemessage clear to the viewer. These constraints, as I surmise, underpin the nature of aesthetic qualities of art composition.
To sum up, I argue that topological order (or more precisely mathematical optimization) has the potential to provide a unified origin to certain compositional phenomena in western art and to enhance our understanding of what is called ‘secrets of old masters’ and the mechanisms of recognizing a seminal artwork. With the prospect of widening the scope of these experiments in future to different kinds of art databases, the developing theory aims to contribute to research exploring aesthetic qualities. I am interested in generating discussion with art educators about the possible application of the comprehension gained from these experiments in the facilitating knowledge in composition studies and in teaching art appreciation. 

