Visual arts and geometry
Knowledge and culture subproject 3: "Visual arts and geometry" of Leiden University Centre for Linguistics
The project examines to what extent geometric knowledge is innate by establishing a dialogue with anthropology and art history. These disciplines have a long research tradition into the universality of design patterns and the universality of art as a feature of cultures. Two classical problems will be examined: universal decorative patterns and basic proportional patterns.
The Munduruku live in isolated villages of the Amazon basin. Their language has few words for geometric or spatial concepts, and they lack instruments for spatial measurement or maps. If cultural or linguistic transmission determined the formation of basic geometric concepts, the Munduruku should perform poorly on tasks involving geometric concepts such as parallelism or congruence. By contrast, if the human mind comes equipped with the prerequisites for spatial thought, they should be able to apply such concepts. Dehaene et al. (2006) found that Munduruku children and adults are able to pick out the geometrically odd figure in a picture discrimination task. These results support the idea that humans come equipped with innate, core geometric knowledge. By contrast, the Munduruku performed poorly on tests involving geometric transformations. This suggests that such transformations may not be part of geometric core knowledge. When American children and adults were tested as a comparison group, American adults outperformed Munduruku of any age, as well as American children. This result indicates that culture, language or education builds a more robust structure on the foundation of core geometric knowledge.
The inquiry into what part of geometric knowledge is innate, and what part is determined by culture, can be fruitfully informed by confronting it with research traditions in anthropology and art history. In these disciplines, there is a long tradition of inquiry into the universality of design patterns and the universality of art as a feature of cultures. Franz Boas’ Primitive Art of 1927 is an example of such anthropological research, which after much criticism has recently been taken up again as one of the starting points for new research into the universality of art. Within art history, and particularly architectural history, beginning with the work of Gottfried Semper in the 1850s culminating in Der Stil of 1863, much energy has been devoted to identifying the basic universal patterns used in all human art over the world, which was inspired by recent developments in comparative linguistics. In his wake, art historians such as Alois Riegl (1858-1905) studied cross-cultural patterns in textiles to detect their universal patterns and the formal laws ruling their transformations. Baxandall (1971) and (1972) has pointed to the counting skills of Florentine 15th-century merchants to account for the development of linear perspective, but such connections between mathematical knowledge and artistic developments have not been followed in a systematic way, and certainly not for architecture. In the anthropological tradition, Hardonk (1999) formulates specifically geometric universals for decorative band patterns.
However, such art-historical inquiries never succeeded in formulating a convincing hypothesis on how such universal formal patterns and the laws governing their formation are related to the wider social, religious or practical functions of the objects that they decorated, nor could convincing links be established between these patterns and the psychological make-up of their makers and users (see Hvattum 2003). The results of recent anthropological research such as Hardonk’s have so far not been integrated with art-historical findings.
In the research program proposed here, we wish to combine the study of geometric core knowledge and competences with the various universals in art patterns that have been identified by anthropologists and art historians. In and by itself, the fact that some art patterns are universally attested cross-culturally does not automatically entail that it should be determined by geometric core knowledge, but such universals effectively constrain the search domain by limiting the number of cultural variables involved.
The main research question to be addressed by this subproject can be formulated as follows:
(1) How and to what extent can universals in art patterns be related to the innate
geometrical competences needed to produce and recognize them?
The main hypothesis of this research project is that universals in art patterns will reveal and further refine the complexity of geometric core knowledge. This subproject will be divided into two PhD projects. They will study two basic varieties of artistic patterns that can be found all over the world: two-dimensional geometrical patterns in ceramics, textiles and other flat surfaces; and simple proportional relations such as 1:2 or 3:4 in three-dimensional settings such as buildings. Concentrating on these very basic patterns will enable us to test for the first time a series of hypotheses on the universality of these patterns, the laws that govern their formation and recognition, and their aesthetic appreciation by confronting them with knowledge based on repeatable anthropological and psychological observation and field work.
This subproject will thereby significantly add to the recent new discipline of global art history as developed by John Onians and James Elkins, and in which the Leiden School of Art History plays a significant role. Most present work focuses on the conceptual and methodological issues of world art history (Onians 2004, 2006, Elkins 2007, Zijlmans & Van Damme 2008). By contrast, the project presented here will combine art historical and theoretical discourse with actual fieldwork. It will also be one of the first studies to consider not only ceramics or the visual arts, but also include geometrical patterns in architecture. In this way, the two projects of this sub-program will investigate two basic geometric patterns of increasing complexity which have always been thought to have some universality in light of the hypothesis that they draw on the same geometric core knowledge.
PhD project 1 (Sjoerdieke Feenstra): Basic proportional patterns and geometric core knowledge
For the implementation of this subproject, see Sjoerdieke Feenstra's page.
Proportion is considered in Western architectural history and theory as one of the foremost defining characteristics of human rationality and its expression through artistic design (Wittkower 1949). Simple proportional relations such as 1:2 or 3:4, expressed in geometrical patterns such as a square and its half, have often been considered as one of the main defining characteristics of classical and Renaissance architecture. These proportions have been taken up by Modernists such as Le Corbusier to support their claims for the inherent rationality and universality of their approach to architectural design. Yet, although there has been much research into the psychological aspects of linear perspective, proportional systems in architecture have hardly been the subject of empirical investigation, despite their ideological weight.
Only very recently was a scientific protocol developed for the measurement of buildings that provides an empirical basis for claims about proportion (Cohen 2008). This protocol was tested in measurements of that icon of Renaissance proportion, Brunelleschi’s church of San Lorenzo in Florence, with results that have forced architectural historians to rethink many of their assumptions about the nature and universality of proportional systems.
Cohen’s research provides the basis to start a series of measurements of buildings from a group of carefully selected cultures in order to test whether simple proportional patterns indeed possess the universality that has always been claimed for them, and to study their connection with geometrical core knowledge. One obvious candidate for such fieldwork is the Batammaliba in Benin, who have a well-developed architectural culture characterized by a very rigorous system of anthropomorphic nomenclature of their buildings (Blier 1983 and 2001). Since anthropomorphy has always been associated with proportion (Leonardo’s Vitruvius Man being its iconic image), this community would provide a good candidate for fieldwork that would examine the universality of proportion.
The fieldwork to be carried out in this context will consist of a converse replication of Dehaene’s et al (2006) work with the Munduruku. Where the Munduruku lack precise geometrical terms, the Batammaliba have an abundance of such terms in the architectural lexicon. The specific research question of this project therefore can be formulated as follows:
(2) How does the Batammaliba geometrical lexicon affect their perception of space?
It is to be expected that Batammaliba adults are unusually adept at recognizing proportional shapes that are in line with the anthropomorphic nomenclature of their culture, while Batammaliba children are not yet so equipped. Such work would confirm the thesis that language builds a rich system on top of core geometric knowledge. The geometric idea of proportionality resonates with two other subprojects of the research proposal, more precisely the projects on language and number and particularly poetry, rhythm and meter.
PhD project 2 (Arthur Crucq): Universal decorative patterns and geometric core knowledge
For the implementation of this subproject, see Arthur Crucq's page.
The PhD will first bring together the various universals in artistic patterns that have been formulated in the different traditions of anthropology and art history. One such universal is the following: decorative band patterns that show vertical mirror symmetry (T- or cross-shaped patterns, equilateral triangles) are attested universally, while bands of random figures are never found (Hardonk 1999). Similar notions of symmetry are attested in the world’s writing systems, and can be related to specific neural structures in the brain (Dehaene 2009). The universals will then be compared, analysed, and if necessary reformulated in terms of geometric principles with a cognitive basis, in line with Dehaene’s methodology for letter representation. The main research question can be formulated as follows:
(3) What aspects of geometric core knowledge and of the geometric systems
building on core knowledge can be distilled from universals in decorative patterns?
The universals will be analysed in terms of the principles that recent research has indicated to belong to core geometric knowledge, as well as to the additional cultural geometric knowledge that builds on this core knowledge from an early age. Finally the results will be used to critically assess a number of commonly held views and approaches to the universality of such patterns (Semper 1863/2004, Riegl 1966/2004, Onians 2004 and 2006). This research will therefore be informed by existing research on the acquisition of geometric concepts in young children. (e.g. Casasola 2003 et al.). This project is therefore radically interdisciplinary, bringing together expertise from anthropology, art history, developmental psychology and the acquisition of geometry.