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Inca Insistence, the Relevance of Symmetries

What could possibly be meant by the term The Inca Insistence? I will get to this! First, I have to explain how patterns are categorized and how this is culturally relevant. Previously in this blog, we have discussed basic pattern repeats. An ogee, such as this image below (and which we defined here), is an example of a half-drop pattern repeat. The motif moves a half step over in each subsequent horizontal line.

But if you look more closely at this ogee, you can see that there is a lot more going on than the simple half-step. There are also both horizontal and vertical symmetries. This actually puts this ogee into a different repeat category group than the simple half-drop.

Pattern Categories

What are the motions that govern motif repetition? Amazingly, there are only four basic ones: translation (sliding across, up, or down), reflection (mirror axis), glide reflection (a combination of translation and reflection), and rotation. Furthermore, the way these motions can be combined to create a repeating plane is mathematically finite too. There are only 17 groups of motions in which a motif can be repeated across a 2-dimensional plane.

Every single repeating pattern that you have ever looked at – whether it is an intricate damask or an Escher etching – can be classified within one of these 17 categories.  And if one is considering a 1-dimensional plane (think a frieze, or the border of a towel), there are only 7 different categories of repetition. How many different motions can you see in this category of a monkey frieze from Peru?

Credit: Stevens page 137

How the Experts do it

Now, and this is where it gets interesting, every ancient culture only practices a subset of these categories. Furthermore, the ones that they use in their textiles are the same ones that they use in their pottery and other artifacts. Indeed, anthropologists train themselves in the different symmetrical categories in order to identify tribes and timelines from archeological digs. It only takes a few examples, or even shards of pottery with repeated motifs, to come up with the tribal marker.

Credit: Californian, Arizonian, and Peruvian Baskets; Washburn/Crowe pages 186 – 188

The Inca Insistence

As for the Incas, their empire only lasted about a hundred years around the 15th century. They were proficient weavers; most of the world’s weaving techniques were known to them and practiced in some form or another. Yet, the Incas also only relied on a limited subset of possible patterns. These patterns were repeated in their textiles, their pottery, their housing, and even the layout of their streets. Here is an example of symmetrical stones (highlighted in green) that can be seen in the masonry of a building in Machu Picchu.

Credit: Kubicka page 112

This cultural coherence is what is referred to as the Inca Insistence in the book Mathematics of the Incas: Code of the Quipu. These repeated elements underlined the very basis of their empire as they imposed their organization on each of the peoples that they conquered.  

As a textile designer, my imagination was captured by the alliteration of the term “The Inca Insistence” and the concept that symmetries are culturally relevant. I hope to have inspired your reverie as well.

Sources and Further Study

  • Asher, Marsha & Asher, Robert. Mathematics of the Incas: Code of the Quipu. Mineola, New York. Dover Publications Inc. 1981.  The Inca Insistence is the name of the third chapter within this book. The authors cite Gertrude Stein as the inspiration of the term insistence as per this quote:  “’ I am inclined to think that there is no thing as repetition’ she said ‘The inevitable seeming repetition in human expression is not repetition, but insistence’” 
  • Christie, Archibald H. Pattern Design; An Introduction to the Study of Formal Ornament. New York. Dover Publications Inc., 1969. For an analysis of ornament that predates (originally published in 1910) the mathematical exploration of pattern, but which is organized along the same principles.
  • Eglash, Ron. African Fractals: Modern Computing and Indigenous Design Rutgers University Press, 1999. For further study of the cultural significance of symmetry within an African context.
  • Stevens, Peter S. Handbook of Regular Patterns: An Introduction to Symmetry in Two dimensions. Cambridge. The MIT Press. 1981. For an introduction to pattern symmetries using the comma (think the paisley) as the base element and illustrated with many examples redrawn from artifacts.
  • Washburn, Dorothy K. & Crowe, Donald W. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle, University of Washington Press, 1988. For a lavishly illustrated primer written as a training manual for anthropologists and archeologists.
  • Header photo of Machu Picchu by David Hansen

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Daniela Venezia

Originally from New York City, Daniela spent eight years in Paris, France as a freelance textile designer after receiving a Fulbright grant to study at l’École nationale supérieure de création industrielle. Upon returning to New York City, she worked in home furnishings for several years before relocating to Montréal and working with the Quebec-based contract mill Duvaltex. She is currently launching a consultation business as a textile professional offering her services in pattern design, jacquard and dobby construction, color forecasting, business intelligence, and technical writing. She loves to do research and enjoys sharing her finds writing for this blog.


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