Sacred geometry, cosmological theories, and spiritual beliefs rooted in ancestry and the afterlife have profoundly influenced African architectural design since the beginning of civilizations. There is a planned organic as well as a perfectly geometrical lean toward the architectural inventions of this land. The article intends to shed light on the following topics:

  1. The Ancient Pyramids of Egypt: A study focusing on The Great Pyramid of Giza in the Giza Necropolis.
  2. A study on fractal geometry displayed by indigenous settlements in Africa. 
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Giza Necropolis_©Dario Morandotti on Unsplash

1. The Ancient Pyramids of Egypt

The Egyptians held close to the concept of an afterlife, considering death as the start of another life. Pyramids were massive tombs built for the Pharaohs (the ruling Kings and Queens) by the masons of that time. Today, a total of 118 pyramids exist in Egypt. The materials used in pyramid construction were limestone, granite, basalt, gypsum (mortar), and baked mud bricks. The limestone blocks were excavated from Giza, along with possibly other sites. Granite presumably came from upriver at Aswan. Alabaster was obtained from Luxor, and basalt came from the Fayoum depression. Since iron tools were not available back then, it is presumed that the workers made use of copper and stone-cutting tools to carve out the blocks in the quarries. This process was followed by using levers to move the stone blocks from the quarry site to the construction sites. Out of these African architectural marvels, the Giza Necropolis is the birthplace of various theories and explorations that unravel new secrets to date.

Mathematical significance of The Great Pyramid of Giza

  • The sacred geometry involved in the concept of The Pyramid of Giza is a scaled representation of the earth, the moon, and the human body.
  • One of the mathematical properties of The Pyramid is π (pi) (the value 3.142). It is a topic of debate and argument because the Greeks are credited with discovering π (pi) years later. Graham Hancock quotes, “We find π (pi) incorporated in The Great Pyramid of Giza; it is found in precisely the ratio needed to calculate the circumference of a sphere from its radius. If you take the height of the Pyramid and multiply it by 2π (pi), you get an exact value of the perimeter of its base.”
  • In his book ‘The Great Pyramid of Giza: Decoding the Measure of a Monument’ (2012), Eckhart R. Schmitz mentions that The Great Pyramid is a 43200:1 scaled model of the earth.
  • Eckhart R. Schmitz credits the authors Robert Bauval and Adrian Gilbert, who studied the relationship between the Great Pyramid, Orion, and Leo in their book ‘The Orion Mystery’ (1993). He concludes that the builders of the Great Pyramid were mindful of the effect of the Precession of the Earth’s Axis over time. The Cosmology of the Ancient Egyptians holds fast to the idea that Orion is associated with their god, Osiris, thus aligning the three pyramids of the Giza Necropolis namely the Pyramid of Khufu, Pyramid of Khafre, and Pyramid of Menkaure along the Orion belt at the time they were built.
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Cross section of the Great Pyramid of Giza_©The Great Pyramid of Giza- Decoding the Measure of a Monument by Eckhart R. Schmitz
  • The Fibonacci numbers are a numbering system found everywhere in nature: in the leaf arrangement in plants, the nautilus shells, the seeds in a sunflower head, or the human body. Dividing any number in the Fibonacci sequence by the one before it, for example, 89/55 or 21/13, the solution obtained is always a constant, close to 1.61803, known as the Golden Ratio.
  • The Golden ratio (Phi) is employed in The Great Pyramid of Giza. The Pyramid’s height divided by one of its base’s sides yields a ratio that roughly equals Phi.
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The nature of geometrical shapes formed by Giza Necropolis are in a Golden Ratio and are thus mystically unique among geometric forms_©The Arch Insider

2. A Comprehensive Overview of Fractal Geometry in African Settlements

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Formation of fractal geometry from polyline_©African Fractals- Modern Computing and Indigenous Design by Ron Eglash

Fractals are distinguished by the repetition of similar patterns at infinitely diminishing scales. Fractal geometry is mainly found in the massing of houses in certain rural settlements in the vast expanse of traditional African architecture. It exhibits ‘self-similar’ characteristics: circles of circular dwellings, rectangular walls enclosing ever-smaller rectangles, and roads in which broad avenues branch down to small pathways with striking geometric repetition. According to Ron Eglash, author of the book ‘African Fractals: Modern Computing and Indigenous Design,’ the presence of mathematics in cultures can be considered a spectrum from unintentional to self-conscious. The physical construction of these designs makes use of scaling and iteration.

The fractal palace of Logone-Birni

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Breakdown of the structure of the city of Logone-Birni in Cameroon_©Ron Eglash- Fractals in African settlement architecture. Complexity, Volume 4 (2) via ResearchGate

Image 05 displays Figure 1(a-e), providing an aerial view of the city of Logone-Birni in Cameroon, built by the Kotoko people. The chief’s palace is the most prominent building complex (Figure 1b). Since this architecture can be described in terms of self-similar scaling, it uses the same pattern at several scales. Figures 1(c–e) portray a simulation using a computer-generated fractal. The seed shape (original/initial shape) of the model is a rectangle, but each side is made up of active lines (gray) and passive lines (black). After the first iteration, we see how each active line reproduces a miniature version of the original rectangle. One more iteration gives a range of scales about the same as that of the palace, shown enlarged in Figure 1e.

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Fractal geometry in Ba-ila settlement_©Ron Eglash- Fractals in African settlement architecture. Complexity, Volume 4 (2) via ResearchGate

Image 06 portrays Figure 2(a,b), providing an aerial view of the Ba-ila settlements of Southern Zambia, which look like a gigantic ring. The livestock occupied the front (smaller circles), the humans nested at the back, in the larger circles. The village chief took charge of the largest unit, located at the end, and looked after the settlements with the authority of a father.

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Fractal geometry in Mokoulek vilage, Cameroon_©Ron Eglash- Fractals in African settlement architecture. Complexity, Volume 4 (2) via ResearchGate

Image 07 unveils the fractal geometry of the Mokoulek village of Cameroon. It is an example of fractals growing inward. There exist small and large granaries spiraling towards the center in a precise way. The square at the nucleus of this arrangement is the building of the village altar, which possessed religious and political authority; it was also the location for generating cycles of agricultural fertility and ancestor successions.

African architectural design covering settlements, villages, and cities distinctly uses fractal geometry. While pure mathematical fractal figures can be infinite in their iterations, examples of fractal shapes with limited scales are found in their architecture, as discussed in the above examples. Understanding fractals through computational iterations and nature to decipher complex adaptive systems is essential. The cultural code in precolonial African societies shows the absence of political hierarchy and the presence of decentralized social groups. These societies presented a pyramidal approach to communal organization, which did not impose a fixed urban plan but instead societal values for architectural interventions. Individual families from settlements explored designs with different iterations. It is crucial to remember that culture can play a pivotal role as a self-organizing phenomenon in planning architectural designs. The fractal structures of traditional African settlements reveal indigenous knowledge systems with valuable insights into complexity, which polarize them from the linear ways of planning of the Western world.

References

  1. Books

Citations for books with one author:

Schmitz, E. (2012). The Great Pyramid of Giza- Decoding the Measure of a Monument. Canada: Roland Publications.

Eglash, R. (1999). African fractals: modern computing and indigenous design. New Brunswick, N.J.: Rutgers University Press.

Rossi, Corinna. (2003). Architecture and Mathematics in Ancient Egypt. Cambridge: Cambridge University Press.

Citations for books with two or three authors:

Shujaa, M. and Shujaa, K. (2015). The SAGE Encyclopedia of African Cultural Heritage in North America. Thousand Oaks: SAGE Publications, Inc.

Bauval, R. and Gilbert, A. (1995). The Orion Mystery- Unlocking the Secrets of the Pyramids. New York: Three Rivers Press.

  1. Articles

Citations for Journal Articles accessed on a website or database:

Eglash, Ron. (1998). Fractals in African settlement architecture. Complexity, Volume 4 (2), pp. 21-29. Available at: https://www.researchgate.net/publication/242911162_Fractals_in_African_settlement_architecture [Accessed: 22 August 2023].

Citations for Magazine Articles – Print or Online:

Yakubu, P. (2023, June 02). The Fractals at the Heart of Indigenous African Architecture. ArchDaily. Retrieved from: https://www.archdaily.com/1001808/the-fractals-at-the-heart-of-indigenous-african-architecture

  1. Online sources

Citations for websites:

Hoskins, Andrew (2015). Pyramids and Sacred Geometry. Elemental Orgone. [online]. (Last updated 5 May 2015). Available at: elementalorgone.co.uk/pyramids-sacred-geometry/. [Accessed 24 August 2023].

Freudenrich, C. (1970). How Pyramids Work. HowStuffWorks Science. [online]. (Last updated 08 March 2023). Available at: https://science.howstuffworks.com/engineering/structural/pyramid4.htm. [Accessed 25 August 2023].

Fidanci, E.A. (2022). Golden Ratio in Architecture. Illustrarch. [online]. (Last updated 04 December 2022) Available at: https://illustrarch.com/articles/13472-golden-ratio-in-architecture.html. [Accessed 25 August 2023].

The Stemettes Zine. (2020). How Many Times Have You Spotted Fibonacci In Nature? Here Are 7 Examples For You…- The Stemettes Zine. [online]. (Last updated 01 June 2020) Available at: https://stemettes.org/zine/articles/fibonacci-in-nature/#:~:text=In%20trees%2C%20the%20Fibonacci%20begins [Accessed 26 August 2023].

csdt.org. (n.d.). African Architecture | CSDT. [online] Available at: https://csdt.org/culture/africanfractals/architecture.html [Accessed 26 August 2023].

Citations for Social Media:

Judah-El, F. (2017). Excerpt of: The Science of Sacred Geometry in Ancient Kemet the Golden Ratio Student’s Edition Faheem Judah-El D.D. [LinkedIn]. Written 18 November 2017. Available from: https://www.linkedin.com/pulse/excerpt-science-sacred-geometry-ancient-kemet-golden-ratio-d-d-/ [Accessed 24 August 2023].

Jueco, S. (2021). Sacred Geometry, Symbolism and Architecture. [PeakD]. Written 11 July 2021. Available from: https://peakd.com/hive-178708/@juecoree/sacred-geometry-symbolism-and-architecture. [Accessed 26 August 2023].

  1. Images/visual mediums

Citations for YouTube videos:

TED (2007). Ron Eglash: The fractals at the heart of African designs. [YouTube video]. Available at: https://www.youtube.com/watch?v=7n36qV4Lk94&t=187s [Accessed 24 August 2023].

Abstract Away. (2020). Geometry of The Great Pyramid || The Mathematics Of Pyramids Of Egypt. [YouTube video]. Available at: https://www.youtube.com/watch?v=bTv4cEfA5Ls [Accessed 24 August 2023].

Citations for images/photographs – Print or Online:

mohamed (2021). The pyramids of Giza Egyptian edifice giant and immortal over the years Stock Photo. [Photograph]. (Document number 262353872, Adobe Stock).

Schmitz, E. (2012). Cross section of Great Pyramid. [Photograph]. (The Great Pyramid of Giza- Decoding the Measure of a Monument- Figure 1, Canada: Roland Publications).

Eglash, R. (1999). The Koch curve. [Photograph]. (African fractals: modern computing and indigenous design- Figure 1.2. New Brunswick, N.J.: Rutgers University Press).

Eglash, R. (1998). Inside Logone Birni. [Photograph]. [Complexity, Volume 4 (2), pp. 22- Figure 1(a-e), University of Michigan: ResearchGate]

Eglash, R. (1998). Ba-ila. [Photograph]. [Complexity, Volume 4 (2), pp. 24- Figure 2 (a,b), University of Michigan: ResearchGate]

Eglash, R. (1998). Mokoulek. [Photograph]. [Complexity, Volume 4 (2), pp. 25- Figure 3 (a-d), University of Michigan: ResearchGate]

Author

Rutuja is an inquisitive architect and a sustainability enthusiast; she loves to decipher the languages of built spaces and takes an interest in storytelling through photographs and poetries.