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Politics of Africa Reflection Paper 1 The Importance of African Fractals for the Study of African Politics Ron Eglash: “African Fractals” James Addoms

Ron Eglash draws a fascinating parallel between mathematics and African politics in his book,

“African Fractals: Modern Computing and Indigenous Design.” Eglash argues that fractal geometry, as

opposed to linear or Euclidian geometry, is a unique expression of indigenous African social organization,

knowledge systems and an underlying cultural/aesthetic preference. He further argues that these preferences

may manifest in three distinct but overlapping ways: unintentional, intentional but implicit, and intentional

and explicit. Dr. Eglash defines fractal geometry and provides numerous examples in African architecture,

cultural artifacts, body modification and beautification, and social structures which both draw inspiration

from and influence iterations of an inherent application of notions of recursion in African thinking. I will

quarrel with the unique quality of fractals as manifested in the physical world as a peculiarly African

phenomenon and as always indicative of particularly African-style egalitarianism and bottom-up organic

social organization.

Fractal geometry is distinguished from Euclidean geometry by five necessary components: recursion,

scaling, self-similarity, infinity and fractal dimension. Recursion refers to the circular process or feedback

loop by which output from one mathematical or physical “generation” becomes the input for the next

generation. Eglash points out that recursion, while potentially infinite in the abstract, is limited in the

physical world; it will “bottom out,” he says, because, “any physically existing object will only be fractal within

a particular range of scales.” (Eglash 17) This corollary or addendum to recursion will be important as I later

examine non-African fractals in the physical world. Scaling and self-similarity are related concepts referring

to the likeness of a small subset or a particular generation’s growth to the set as a whole. In particular, scaling

is meant to describe the phenomenon of a rough portrait of the whole set corresponding roughly to the

additions of each generation. A craggy rock resembles a craggy shoreline. The vein-like structure of an

individual leaf is evocative of the branching system of the parent plant. Self-similarity refers to a more precise

requirement that, under close examination, at least some of the traits of the parent system may be reproduced

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almost identically in some parts of the generation under examination - for example, a stem of a tree or a

smooth, eroded section of a rock or shoreline. I short, the part looks like the whole. Infinity, the fourth

requirement for the presence of a fractal, is the abstract element by which purely mathematical fractals must,

by definition, continue their recursive and self-similar patterns forever. Given a finite “range of scales”,

perhaps that perceptible to human experience without artificial aid, by definition infinity is not required of

fractals in the physical world. Finally, Eglash concludes that a fifth component, Fractional Dimension, the

curious capacity of fractals to assume, at greater and greater iterations, an infinite dimension in demonstrably

finite space. (Eglash 17-19)

Dr. Eglash argues that one, some or all of these fractal components are observable in the physical

world in general and African indigenous design in particular, and that they may be unintentional, intentional

but implicit, or intentional and explicit. As evidence of unintentional implementation of fractals in African

society, Dr. Eglash includes unconscious activity such as urban sprawl, both in small communities and larger

African cities, and accidental fractals such as a “mirror” portrait of nature, that is a direct reproduction of

natural patterns without the interpretive interventions born of cultural influences one observes in “stylized”

art. (Eglash 49-52)

Eglash also provides ample evidence for an intentional but implicit influence of fractal thinking in

African society. A fractal aesthetic pervades much African art and architecture. It is difficult for Eglash to

establish a direct algorithmic relationship to African art, but he describes a pervasive aesthetic, a sense that

certain patterns were chosen “just for looks” throughout the non-architectural artifacts of African culture.

Eglash cites numerous examples of hairstyles and fabric patterns as indicative of a fractal aesthetic, (and not

specifically African “rules” about pattern construction):

Just as we saw how designs based on nature range from unconscious to intentional, artificial

designs also vary along a range of intention, with some simply the result of an intuitive

inspiration, and others a more self-conscious application of rules or principles. (Eglash 53)

As a mathematician, Eglash set about to uncover these rules or principles his western cultural

understanding primed him to assume were there, but he confesses:

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As far as I could determine from descriptions in the literature and my own fieldwork, there

were no explicit rules about how to construct these designs, and no meaning was attached

to the particular geometric structure of the figures other than looking good. In particular I

spent several weeks in Dakar wandering the streets asking about certain fractal fabric

patterns and jewelry designs, and the insistence that these patterns were “just for looks” was

so adamant that if someone finally had offered an explanation, I would have viewed it with

suspicion. (Eglash 53)

We will see that architecture mirrors social organization, as it must in every part of the world, but

that a particularly African egalitarianism based on a non-western view of family relationships and extended

kin groups leads to the development of particularly African fractals. Architecture and building methods are,

by definition, intentional and explicit, although their reference to fractals in particular may not be. Eglash

makes no claim that indigenous African planners set out to ‘make a fractal.’ That is not the point of his writing.

However, Eglash cautions that “the greatest danger of this book is that readers may misinterpret its meaning

in terms of primitivism… The geometric thinking that goes into these examples may be simple. But it is quite

intentional.” (Eglash 53) This relationship between intentional and explicit design based in African political

and social structure and the emergence of fractals, with their clearly defined 5 characteristics within this

intentional and explicit construct is the essence of Eglash’s concept.

African civil society demonstrates recursion, scaling, self-similarity, infinity and fractal dimension in

surprising ways. Recursion, or a hearkening back to what has come before and reproducing new generations

with that knowledge is seen explicitly in the privileged place ancestors hold in the spiritual life of African

communities. Most importantly, however, is the concept of scaling in reproductive patterns and ancestral

computation. Ancestral mounds in some African communities represent the (growing) presence of the souls

of kinsfolk. A concept of the tiny human passed on in reproduction is also a continuation of this process. The

whole continuum of life stretches in both directions in a uniquely African fashion. The deceased lend more

weight to the current spiritual zeitgeist as they add physical space to the ancestral mounds. Non-African

societies have largely abandoned this sense of history, of institutional memory, just as they have abandoned

the idea of the homunculus, the tiny human transferred at the moment of conception. (Eglash 127)

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A sense of the infinite, or a move along this ancestral and descendent continuum, is also present in

the recursion in each generation. While non-African religions still stress the timelessness and the durability

of creation, this view is very often at odds with non-African concepts of the arrow of time, which flows in only

one direction. Non-African ancestral history is more about positioning in the here-and-now, rather than

honoring what has come before. Preoccupation with the problem of entropy in non-African societies persists

with the realization that the universe’s capacity to aggregate itself into higher states of order is at constant

odds with the tendency of the miniscule to disaggregate itself into higher states of disarray. Indeed, all

modern concepts of energy and mechanical and electronic motion are dependent on the knowledge that

thermodynamic systems always lose energy, that is, the act of performing any action produces entropy, or

disarray. Thus, non-African knowledge systems seem preoccupied with “ultimate fates” or ultimate entropy,

rather than current existence as a point within a circular, infinite existence.

Most fascinating and most critical to an analysis of non-African and African political structures may

be the concept of the family unit or kin group itself, and the way in which descent and belonging are

calculated and reproduced. The Cantor Set is a fascinating abstraction, in many ways, of African and non-

African descent calculations, as a linear fractal may be considered for diagramming descent. 1 generation

makes two, two makes four, four makes eight until an infinity of generations may be imagined within a finite

diagramming framework. While containing many similar concepts, the non-linearity of kin groups in African

society supplements the Cantor Set by changing the rules of the fractal in each generation. The active line is

changed slightly, so that each successive “generation” is not merely a reflection of the previous, but may be

enhanced by additions of active lines not directly produced by the previous “generation.”

Non African societies may also exhibit fractal characteristics, as Eglash admits in the case of certain

artistic movements which stylize nature-based interpretation born of distinct cultural traits rather than

mimic it in its entirety. The Art Nouveau movement at the turn of the twentieth century and the arts and

crafts movement slightly later both reproduce natural forms based on particular “rules” stemming from

cultural preference. In the unintentional category, non-African settlements exhibit urban sprawl in much the

same way that African societies’ equilibrium may be punctuated by spurts of growth beyond centrally,

traditionally and explicitly meaningful forms. It is in the intentional and explicit category, however, that non-

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African architecture and social organization may soundly lay claim to fractal generation, albeit of a decidedly

non-African variety; intentional culture, across human communities, demonstrates surprising similarity in

its demonstration of fractal characteristics.

The Cantor set may describe family units organizing, merging and regrouping in predictable, but

non-linear ways, but is that family structure particularly indigenous to African society, or is it merely pre-

industrial? Family planning is entirely the province of modern society. Indeed, modern societies may

implicitly demand family planning by creating an urban, industrial economic model in which supporting a

large brood of children until adulthood is economically unviable, creating an implicit freezing effect on

childbearing, or may impose explicit restrictions on the number of children born to each successive

generation, like the current Chinese one-child policy, in which a state has deemed it impossible to sustain a

positive birthrate given the demands of existing members of society. One would expect to see small family

units centered around others - likely kin, for protection - in many pre-industrial societies.

Indeed, the early Neolithic settlement known as Skara Brae, located on the Bay of Skaill on the west

coast of Scotland demonstrates this non-linear family grouping while also demonstrating recursion and self-

similarity (figure 1: A,B,C). Additionally, the overall schema of the site is decentralized indicating an

egalitarian government structure and an accretion style of construction centered on family units. 1

Eglash also discusses African fractal architecture which is explicitly designed to highlight

political/religious patterns. Much architecture is designed with the experience of the visitor in mind. This

explicit choice in architecture is mentioned by Eglash in several of his descriptions of fractal design in

administrative and royal dwellings. (Eglash 24) Similarly, the 12th Century royal temple known as Angkor Wat,

in present day Cambodia, a textbook example of Khmer architecture and a lasting Cambodian national

symbol, demonstrates multiple recursions across “generations” meant to mirror a spiritual ascendancy,

(figure 2), and are evocative of the “spiral path taken by visitors to the throne” of the Miarre in the Kotoko

culture, shown in Eglash’s figure 2.1 (Eglash 23).

1 While it is also true that non-African political architecture is unlikely to resemble a perfect fractal – that is, it is unlikely to display all five characteristics of fractals or to resemble a conch shell or a fern frond – important architectural patterns often reflect one or more “essential components of fractal geometry.”

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FIGURE 1 FIGURE 2

Fractal geometry reflecting social harmony and imposing a social order has existed throughout non-

African architectural history. The Parthenon of Athens, a lasting symbol in western society and nearly

universally regarded as the height of Classical artistic and architectural achievement is designed explicitly

with the knowledge of and intent to display the golden ratio, (figure 3: A), an aesthetically pleasing fractal

and natural measurement based on human anatomy, “rediscovered” during the European Renaissance. Like

Angkor Wat, the temple represents recursion between its inner and outer colonnades, clearly demarcating

an outer area for offerings and pilgrimages, a middle area for worship by clerics, and an inner area in which

the spirit of the goddess was given shelter, (figure 3: B).

In early modern temples of state - the palaces and government halls of Europe – clearly demarcated

living and socializing areas structured a top down social order, but provided the real estate on which bottom

up, (or at least middle-up until the revolutions of the 18th and 19th century), social positioning could take

place. The term “inner temple” remains a common term in British legal regulation for one of the four “Inns

of Court” to which a British barrister must nominally belong to practice law. The palace of Versailles, a much

later addition to the architecture of France than the Knights Templar’s contribution to the architectural and

lexicographical landscape of English society, is an obvious example of recursion through literally several

“generations” of construction, often reproducing ‘active lines’ on a grander but more internally restricted

scale. This architecture is also replete with at least three levels of scaling, from a grid of square gardens and

broad boulevards, (reproduced in the crosshatching of paths through smaller courtyards and within the

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gardens themselves), through to the exterior of the palace demonstrating reflections of the symmetry and

outline of the bordering boulevards and through to the staterooms, private rooms and anterooms distributed

among grand interior passages and public and private halls, stairs and graduated receiving spaces (figure 3:

D).

Of the scaling fractal component evident in the Ba-ila settlement, Eglash’s figure 2.3, he writes:

The settlement as a whole has the same shape: it is a ring of rings. The settlement, like the

livestock pen, has a front/back social distinction: the entrance is low status, and the back

end is high status. At the settlement entrance there are no family enclosures at all for the

first twenty yards or so, but the farther back we go, the larger the family enclosures become.

(Eglash 26)

Eglash argues that this architecture is mostly egalitarian between families. The individual subset of

the fractal is the result only of practical physical arrangements between family quarters (in the rear), and

animals and the means of production and sustenance (in the front). If this were true, however, one would

imagine a different arrangement at Ba-ila, in which all buildings and agglomerations were identical in size

and position, with only the individual subset showing notions of hierarchy based on animal-human divisions.

Instead Eglash admits:

At the back end of the interior of the settlement, we see a smaller detached ring of houses,2

like a settlement within the settlement. This is the chief’s extended family ring, the chief has

his own house. And if we were to view a single house from above, we would see that it is a

ring with a special place at the back of the interior: the household altar.” (Eglash 26)

2 “At the back end of the interior of the settlement, we see a smaller detached ring of houses, like a settlement within settlement.” These are not individual buildings smaller than other individual buildings, but a smaller circle, compared to the community circular structure as a whole. This does not, in my view, necessarily represent egalitarian architecture, no matter the actual political arrangement. As I will demonstrate, many non-African examples of political architecture specifically place the “chief’s” residence or the central political establishment in a privileged place, but necessarily smaller than the whole of the community. No one argues that to establish dominance, as seen in an aerial view, Buckingham Palace, the White House, the Capitol Building, Vatican City, the Kremlin, or the Forbidden City be equivalent to the size of the city or population over which the government rules, reigns, or exercises authority.

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Once again, we see the Palace of Versailles offers a similar social organization sketched out in its

floor plan. The concentric rooms spread out around an external courtyard focus the intrepid visitor - delving

deeper and deeper into the palace and thus higher and higher into the echelons of government authority –

into ever greater spaces of more intimacy and more attenuated appreciation of power. The inner rooms of

the King’s chamber, (figure 3: C), in this way reproduce the graduated experience of the visitor to the Miarre

in the Kotoko culture and their presence in the center of the “community” – the court – resemble the chief’s

dwelling in the Ba-ila settlement:

The reason for choosing scaling rectangles as a symbol of royalty becomes clear when we

look at the passage that one must take to visit the Miarre. The passage as a whole is a

rectangular spiral. Each time you enter a smaller scale, you are required to behave more

politely. By the time you arrive at the throne you are shoeless and speak with a very cultured

formality. Thus the fractal scaling of the architecture is not simply the result of unconscious

social dynamics; it is a subject of abstract representation, and even a practical technique

applied to social ranking. (Eglash 24)

A telling passage recounts a similar experience in the external and internal spaces of

Versailles – an experience in the creation of which Louis XIV was intimately involved:

The sun King’s experience as garden-tour planner in 1689 evidently rekindled his interest in

the problem of how visitors should best be conducted through his vast estates… The tour

began by moving the visitor from the Château [after a similarly rigid conveyance from outer

to inner courtyard and back again] directly westward between the long, rectangular basins

of the Parterre d’Eau, halting at the steps overlooking the Latona Fountain, with the great

view west down the Allée Royale, past the Apollo Fountain to the Grand Canal and beyond.

This seemingly inevitable curtain raiser, which was guaranteed to impress every visitor,

looped back within itself, and from the colonnade, the visitor returned to the Allée Royale

from which their return to the Château would commence.3

3 Berger, Robert W and Thomas F Hedin. Diplomatic Tours in the Gardens of Versailles. Philadelphia, Pennsylvania: University of Pennsylvania Press, 2008 (58-59).

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FIGURE 3

Eglash makes much use of the quincunx, (figure 4), as representative of African art in a Euclidean, if

“fractalized” vein. (Eglash 55) Non-African cultural artifacts similar to the quincunx can be seen in puzzles,

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(figure 6), as well as architecture. The building shown below, (figure 5), represents an intentional

architectural rendering of a decentralized entrance plan to a centralized center. Unlike the rigid power-

attenuation schema of the Versailles fractal, this building welcomes visitors into a shared space as holders of

equal status. This building, the Schine Sudent Center on the Syracuse University campus, affords no special

entry to professors or administrators. All entrants must ascend or descend passageways to access central

programming, events and common spaces.

FIGURE 4 FIGURE 5 FIGURE 6

My counterargument to the existence of fractal geometry in non-African societies is meant only as a

mild critique of the only slightly sub-conscious argument of the book that it is a particularly African concept

to represent notions of civil society with fractal geometry. Eglash’s argument about the particularly African

starting points, the seed shapes and their underlying methodology, forming an end result evocative of fractal

geometry is sound, and fascinating.

Let is examine two seed shapes in the abstract. The first is a family home modeled after sustainable,

indigenous architecture and similar to other architecture within the set. We will define the set as all buildings

within a community. I imagine the Ba-ila settlement but perhaps also the ancient Scottish Skara Brae

settlement as representative of this indigenous community. The second seed shape is quite different. Let us

imagine two family units, from which the Versailles “fractal” might emerge. One family unit is the

peasant’s shack, the other is the chateau of the noble family. The Ba-ila family home, (figure 7: A), an abstract

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model of generations of recursion in the architecture of the Bamileke settlement, (figure 7: B), and the family

unit in Skara Brae, (figure 2: A, C), are compared easily between these three cultures, but they are poorly

compared to the unequal architecture within the society which produced the Versailles fractal, (figure 3, C,

D), and the agglomeration of ghetto architecture - a ghetto “fractal” seen in numerous early drawings of

medieval and early modern era peasant villages and poor sections of larger towns and state capitals, (figure

7: C).4

FIGURE 7

The next “generation” of all of these seed shapes yields predictable results. An agglomeration of

peasant huts produces a ghetto, an agglomeration of noble houses, again, in the abstract, yields court

4 This illustration is taken from a website comparing a reproduction of medieval Paris – displaying what I could not help but recognize as a pre-Haussmann fractal dimension - with intended modern construction in centralized and geometrically similar block housing. The housing itself hints of at least three “generations” of recursion: the intersections between blocks, the cross shaped structures themselves, and the cross hatching on each of the four wings of the main structure.

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architecture, itself requiring self-sorting and hierarchical statements reinforced through the architecture and

the power attenuation demonstrated in the Versailles fractal. The inequality between the seed shapes is

critical, because it is the underlying community equality or stratification, expansive or limited concepts of

family and kin group relationships, and bottom-up or top-down political structures which inform

architecture and art – in short, culture. Yet it is the recursive process of culture, seed shapes undergoing

generation change in non-physical space, in notions of culture, that demonstrate that recursion and self-

similarity exist across cultures and may reproduce themselves across fractal “generations”.

Eglash’s most powerful statement is not that fractal architecture – that is, the presence of the five

necessary components: recursion, scaling, self-similarity, infinity and fractal dimension – is a uniquely African

development, but that fractal motion appears consistent across a largely egalitarian society, rather than in a

punctuated fashion as seen in the top-down hierarchical structures of non-African societies. Political and

cultural notions shape the next generation of political and cultural notions across generations. This recursion

is not unique to Africa, but is indicative of a stable sense of a cultural self. It is evident that many non-African

cultures have undergone great transitions, from fractal thinking to linear and, occasionally, back again. A

normative judgment is beyond the scope of the paper, but it is clear that the underlying cultural assumptions

in African and non-African cultures inform the type of fractal components and fractal dimensions created

and expressed in class-based and classless, hierarchical and bottom-up, and extended or limited kin group

societies.