THE MATHEMATICAL APPROACH TO COMPLEXITY
Geometric patterns are inherently amenable to mathematical analysis. As we’ve already noted, 12 of the 17 possible ways to vary a pattern symmetrically on a surface are found in Kasaï textiles, along with examples of 7 of the 7 possible symmetries in borders. However, the main feature of most 20th-century “velvets”, with some exceptions as we’ll see, is represented by a series of asymmetries, variations, improvisations, irregular and broken patterns, transitions, dissonances, creative caos, and entropic degeneration processes. These factors allow the patterns to change and evolve, to tell a story if we want to use a different key, or to represent a unicum in the history of world design if we want to use an even different key. All these factors and the intrinsic dynamism of Shoowa panels pose a challenge to the mathematical analysis and computational approach of shape grammars. Some scholars focus on geometric transformations (e.g., translations, rotations, or scaling) to understand how patterns evolve or change within the panels.
In a 2004 Canadian collaborative study, entitled Textiles, Patterns, and Technology: Digital Tools for the Geometric Analysis of Cloth and Culture, Sushil Bhakar (a computer science researcher), Eric Hortop (a mathematician), Cheryl Kolak Dudek (associate professor of Print Media), Sylvain Muise (Applied Mathematics expert), and Fred E. Szabo (professor of Mathematics) wrote: «Fractal symmetry, including recursion, scaling, self-similarity, infinity, and fractal dimension, is displayed in the Kuba designs through juxtapositions of linear embroidery, velvet forms, and contrasting color that becomes counterpunctual in the composition. According to Ron Eglash, fractals are part of African numerical systems as evidenced in their village planning, decorative motifs and textiles. He specifically describes Kuba designs in the computational terms of a complexity spectrum:”These [Kuba designs] tend to show periodic tiling along one axis, and aperiodic tiling—often moving from order to disorder—along the other. Similar geometric visualizations of the spectrum from order to disorder have been used in computer science». (Textiles, Patterns, and Technology: Digital Tools for the Geometric Analysis of Cloth and Culture, see Bibliography).
Ron Eglash is an American expert in cybernetics and professor at both the School of Information and the School of Design at the University of Michigan (USA). He’s the author of the renowned essay African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press, 1999. In the final BIBLIOGRAPHY, you’ll find all details to get these works as well as the others quoted here.
The most interesting line of research is that of “ethnomathematics”. The leading scholar in this field, the late Paulus Gerdes, a Dutch mathematician and rector of the Pedagogical University of Mozambique, explains what it is: «Ethnomathematics is a discipline that studies mathematics and mathematical education as embedded in their cultural context. (…)». The application of historical and ethnomathematical research methods has contributed to a better knowledge and understanding of mathematics in the history of sub-Saharan Africa, as well as to an awareness of further mathematical elements in African traditions.». (On Mathematics in the History of Sub-Saharan Africa 1, see Bibliography). The development of ethnomathematics is essential to the discovery of “hidden” geometric ideas: « Many mathematical ideas and activities in African cultures are not explicitly mathematical. They are often intertwined with art, craft, riddles, games, graphic systems, and other traditions, so that the mathematics remains ‘hidden’ or ‘implicit’.». (Ibid.). An example of the results of this new approach is shown by Paulus Gerdes himself in relation to the Shoowa textile.
In his volume Ethnomathematics and Education in Africa, Gerdes considers some Kuba patterns and shows how these designs can serve as an interesting starting point for the study of geometry. In particular, he considers a basic pattern called mongo by the Ngongo and mwoong or “elephant defense” by the Bushoong (both are ethnic groups within the Kuba confederation), and shows that by repeating this pattern, one can geometrically discover and easily prove the theorem of Pythagoras.
Below, the mongo pattern according to Georges Meurant and some of the pages dedicated by Gerdes to the demonstration of the theorem mentioned.