Factoring Fractals
Sylvester Okwunodu Ogbechie, University of California Santa Barbara   

Critical Interventions welcomes our readers to a very exciting issue about fractals in African arts and visual cultures, produced by guest editors Ron Eglash and Audrey Bennett. I first met Ron Eglash at the 2007 TEDGlobal Conference in Arusha Tanzania where he presented a talk about fractals and mathematical concepts in African arts and visual culture, drawn from his groundbreaking book African Fractals. I had however heard of him prior to this encounter a couple of years earlier, when I was conducting research for an article on ethno-mathematics and Eglash’s name came up as someone dong interesting work on the subject. I attended his TED talk and it was very well received.

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Fractals in Global Africa
Ron Eglash, Rensselaer Polytechnic Institute, Audrey Bennett, Rensselaer Polytechnic Institute   

Fractals are patterns that repeat themselves at many scales. In the context of African art and design, that simple characterization takes on profound meanings that can move across disciplines and geographic boundaries. Fractal patterns can be found in African architecture, textiles, sculpture, music, and many other places. The means by which computers generate fractal graphics, recursive loops which allow structures to “unfold” or self-generate from an initial state, find parallels in African cultural traditions that might seem distant from math or computing: stories of spiritual rebirth, trickster narratives, the social dynamics of communalism, the “repetition with revision” linking music with oral history, and other ineluctable aspects of lived experience. Rather than imposing an alien analysis from afar, the eclectic mix of contributions in this special issue allow the rich complexity of African culture, in all its global diversity, to enter into dialogue with nonlinear dynamics, complexity theory, and other mathematical and computational frameworks in which fractals occupy a central role.

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Follow the Golden Ratio from Africa to the Bauhaus for a Cross-Cultural Aesthetic for Images
Audrey Bennett, Rensselaer Polytechnic Institute   

The golden ratio, a mathematical relation that often arises in fractals and other scaling geometries, is known for its ability to effect visual beauty.1 As a result, communication designers have used it throughout history to compute a plethora of visual compositions. For instance, many designers use the golden rectangle, a popular compositional grid derived from the golden ratio, to organize verbal and visual information into eye-catching images. However, within the discipline’s literature, our European predecessors, primarily ancient Greece, receive most of the recognition for being the sole contributors to the use of the golden ratio. Very few references in the discipline’s literature acknowledge Africa as the one of the primary contributors to knowledge about the golden ratio. In this paper, I challenge the assumption that the golden rectangle originated solely in ancient Europe and I examine existing evidence within broader interdisciplinary discourse, which suggests that the golden rectangle is more of an outcome of interaction between African and European civilizations.

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Vergange nheitsbewältigung: Recursive Practices and Historical Consciousness among Brass-Casters in Benin City
Joseph Nevadomsky, California State University, Fullerton   

Insights from fractal analysis applied to cultural anthropology are important in reconsidering the contemporary placement of brass-casting in Benin City, Nigeria. The former heart of the Benin Kingdom, Benin City is renowned for producing superb brass-castings and ivory carvings from the 14th to 19th century unlike anything else in sub-Saharan Africa. Benin City’s caster-artists continue to produce artworks in traditional styles, including reproductions, and have introduced innovative new works, often not far removed from earlier styles. Unfortunately, art historians became so focused on the classical corpus after a British Punitive Expedition conquered the kingdom in 1897 that they ignored 20th century productions, relegating them to an historical dustbin of debased artifacts. A reframing of 20th century castings using fractal concepts from geometry allows us to recover some of that past, and to provide a clearer recognition of the importance of contemporary castings as inspired recreations of traditional motifs and reflections of cultural memory through casting patrimony.

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Bricolage : Toward a Scrapture A Proposal of A New Concaept
Shigemi Inaga, International Research Center for Japanese Studies   

For most of the ordinary Japanese public, African contemporary art still remains a remote topic. The dialog between Japan and Africa on cultural issues is far from being intensive, as it is not easy to find common interests that the Japanese can share with African artists. Such is the common understanding of the state of affaires in the Far-Eastern archipelago. And yet, such a perspective loses its relevance once one stays in major metropolises in Europe. From the Western-centered perspective, both Africa and Asia are located at the fringe of the periphery, outside of the Western center. And both Africa and Asia may be classified as the non-Western rest, in opposition to the West. In the so-called world contemporary art scene, both Asians and Africans are searching for their own recognition and fame in the Western market. But, the people from the periphery rarely recognize that they share similar problems in their confrontations with Western criteria. African-Asian cooperation rarely comes to the current agenda.

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Creating Again and Again: Fractal Patterns and Process in Improvisational African-American Quilts
Judy Bales, Artist, Independent Researcher   

An Eye for Fractals

Benoit Mandelbrot, best known as the founder of fractal geometry, described the acclaimed Japanese artist Hokusai Katsushika as having “an extraordinarily refined ‘eye for fractals,’” even though Hokusai could not have been aware of the formal mathematical concept of fractals.1 I do not know if Mandelbrot was familiar with the work of African-American quilters, such as Rosie Lee Tompkins or the quiltmakers of Gee’s Bend, Alabama, but I believe he also would have thought that these artists had a very good eye for fractals. Mandelbrot wrote, “For me, the most important instrument of thought is the eye. It sees similarities before a formula has been created to identify them.”2 As a visual artist, I have been fascinated by Afro-traditional3 quilts for more than 25 years and, in recent years, with fractal geometry. Once I discovered fractals, I began

to see the patterns in improvisational African- American quilts with more refined eyes and I began to understand my strong attraction to this art form.

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Harmonic Fractals in the Music of the Lamellaphone
Martin Scherzinger, New York University   

At stake in this article is a demonstration of the fractal-like logic undergirding harmonic processes found in the archaic lamellaphone music of Zimbabwe, Zambia, and Mozambique. Harmonic practices of African music are a largely neglected aspect of the intellectual discourse on Africa. Most ethnographic writing on the music of this region (no less than the African continent at large) limits its music-analytic findings to general observations, most notably to aspects of rhythm alone, often with a special interest in the music’s kinesthetic aspects. These include analyses of timelines and other asymmetric rhythmic patterns, polymeter and polyrhythm, pulse-based temporal structures, hocketing techniques, shifting metric groupings, etc., and their relation to performance practices (dance steps and the like). Of the technical approaches to African music, very few examine its non-rhythmic dimensions, such as pitch spaces or pitch processes—melody, harmony,

counterpoint, and so on—and, of those that do, most remain within the limits of mere pitch or chord labeling, often hesitantly and provisionally, thereby apparently avoiding the false premises of Eurocentric assignation. There is almost no work on the relation between pitch processes and rhythm.

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Fractal Complexity in Mwalimu Chinua Achebe’s Things Fall Apart: A Mathematical Exploration
Abdul Karim Bangura, Howard University   

While many literary and social-scientific analyses of Mwalimu (Honorable Teacher) Chinua Achebe’s famous book, Things Fall Apart, abound, no work exists that has employed mathematical techniques to examine the text, even though such potential exists. This study is an attempt to fill this gap in the literature. Specifically, I employ the mathematical concepts of fractal dimension and complexity theory to explore the idea of a spectrum progressing from more orderly to less orderly or pure disorder, in the text. The computer-generated results suggest that the combination of negative and positive feedback loops, which form the basis of several African knowledge systems, also form a key mechanism of the general self-organizing systems discussed in Things Fall Apart.

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Accessing the Ancestors: The Re-Mediation of José Redinha’s Paredes Pintadas da Lunda
Delinda Collier, School of the Art Institute of Chicago   

Alguém varreu o fogo a minha infância e na fogueira arderam todos os ancestres.

(Some fire swept through my childhood and the fire burned all of the ancestors.)1

—“Terra Autobiográfica” by Francisco Fernando da Costa Andrade

Júlio Vilhena, scholar and son of the then Delegate Administrator for the Companhia de Diamantes de Angola (Diamang), wrote an article for the Journal of the International Folk Music Council in 1955, in which he presented a folklore project of the Dundo Museum in Lunda North, Angola. He comments on the logistics of recording folk songs and oral culture among the ethnic Chokwe residents in the region, stating, In the future a tape recorder will be used for this work, which will give greater mobility, as it will avoid the transport of the voluminous and fragile stock of virgin discs in the conditions of travel prevailing in the tropics.2

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African Fractals and Culturally Situated Design Tools: Mathematics Education through Self-Empowering Technology
Raymond Lutzky, Rensselaer Polytechnic Institute   

Creating compelling, socially responsible educational technologies for science, technology, engineering, and mathematics (STEM) is one of the greatest teaching challenges for modern American society. Shirley Ann Jackson, noted African-American physicist and university president, has argued that the United States needs to better understand social and cultural experiences in order to abate what she labels America’s “quiet crisis” in mathematics and science.1 The “crisis” is two-fold: a shortage of American students in science and engineering, particularly within under-represented groups, such as women and African-Americans, coupled with the impending retirement of a generation of senior scientists and engineers. In The World Is Flat, journalist Thomas Friedman expands on this concept and notes, “The National Commission on Mathematics and Science Teaching for the Twenty-first Century […] found that two-thirds of the [American] mathematics and science teaching force will retire by 2010.”2 As a result, inspiring America’s youth to be interested in science and engineering in college has become a national priority, particularly among underrepresented groups.

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Fractals of Knowledge: Melaku Center and Central Highlands University
Xavier Vilalta, XV Architects, ETSAB, UPC, Barcelona Tech   

Melaku Center, Ethiopia

Back in the fall of 2009, a Spanish businessman approached me with an idea to help him develop a vocational school in Ethiopia. He proposed a very ambitious project that would involve people with different skills, where each expert would put their experience to use for a main social purpose: the Melaku Center, the first professional education center in Ethiopia. The project name, Melaku, means angel in Amharic (the national language of Ethiopia), and refers to the two people that are the souls of the project, who are both named Angel: the businessman that developed the project, a serial entrepreneur; and another Angel, a missionary from the Basque country in Spain, who lives in Ethiopia and who is the source of inspiration for the former. The project was joined by other professionals, such as a specialist in vocational education and an expert in communication, who made the projection of the project possible: a multidisciplinary team, there each component would create an amazing synergy.

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