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Keys to Mathematical Treasure Chests: Andean Khipus – Quantitative Research and Digital Analysis

Manuel Medrano (Harvard University)


Beyond documentation and conservation, an active vein of the last 50 years has been the subjection of khipus to various mathematical, observational, and—most recently—digital analyses. Such efforts, often motivated by the goal of khipu decipherment, are widely available online. They are surveyed chronologically here.

Between the 1960s and 1980s, the identification of elaborate patterns and mathematical relationships within individual samples was a hallmark of khipu research. Carlos Radicati di Primeglio, an Italian-born scholar, dedicated decades of his life to the endeavor, identifying color schemas and numerical phenomena in his collection of specimens that revealed their purportedly “extranumerical” properties [Radicati di Primeglio 1964]. Most of Radicati’s published work was compiled for a 2006 volume, arranged by the Universidad Nacional Mayor de San Marcos, that is available online. Radicati’s successors would be an American duo, Marcia Ascher and Robert Ascher, who from the early 1970s cataloged hundreds of khipus in the Americas and Europe [Ascher and Ascher 1981]. A mathematician (Marcia) and an anthropologist (Robert), the Aschers produced two “Databooks,” today preserved on a Cornell University webpage, that list cord-by-cord observations and internal arithmetic relationships for over 200 samples. Another webpage preserves some of the Aschers’ own photographs of the specimens. An overview of the Aschers’ mathematical investigations appears in a 2021 lecture of mine to Gresham College, London.

Image of a Treasure Chest and Key. The Ascher Databooks provided most of the initial entries for the first digital khipu database, which was designed and implemented by Carrie Brezine at Harvard University beginning in 2002. Now managed at the University of Chicago, the Open Khipu Repository (OKR) is open source and stores detailed, downloadable information on 630 khipus from around the world. A newly-formed independent advisory board for the OKR, presently consisting of Carrie Brezine, Jon Clindaniel, Ivan Ghezzi, Sabine Hyland, and me, is working to ensure that khipu research is transparent, inclusive, and accountable going forward; we have recently published a number of concrete steps toward this end [OKR Team 2022].

Combining these records with an additional database stored locally at the MNAAHP (the second-largest khipu collection, see above), today some 866 khipus are stored in full digital format, about 60% of the known total [Medrano 2021b, p. 117]. The history of their cataloging and digitization since 1920 is reproduced in Figure 5. Note the significant spike in the bolded line in the late 1970s, coinciding with the Aschers’ research. More fine-grained analyses are possible by subdividing geographically; the history of European khipu cataloging reproduced in Figure 6 is one such example. The OKR’s relational database uses a standardized khipu recording schema described in a recent video from the British Museum’s preparation of OKR data in 2021. Peruvian archaeologist Alejo Rojas Leiva [2009] and others have contributed to ongoing discussions surrounding best practices in the digital encoding of khipus.

Graph of worldwide khipu cataloging and digitization efforts.
Figure 5. Worldwide khipu cataloging and digitization efforts, 1920–2020 (graph by Manuel Medrano).

Figure 6. Dynamic map of khipu cataloging in European museums, 1920–2020. Note the world’s largest
khipu collection at the Ethnologisches Museum, Berlin (animation by Manuel Medrano).

The widespread availability of khipu data has both enabled and encouraged a wide variety of new digital initiatives in the last five years. An exciting entry point for the uninitiated is The Khipu Keepers [2021], an online compilation of research overviews produced by Google Arts & Culture and the Museo de Arte de Lima (MALI), in consultation with multiple specialists. High-quality images of Wari, Inka, colonial, and modern khipus appear throughout. Additionally, Cecilia Pardo of the British Museum has presented a thoughtful khipu overview in connection with the Museum’s first major Peruvian exhibition [2021], as has Kylie Quave in a thoroughly illustrated essay for the art history platform Smarthistory [2022].

Image of a Treasure Chest and Key. The Khipu Keepers project emerged in part from a major khipu exhibition at the Museo de Arte de Lima in 2020–2021, which has been digitally preserved in 3D. See the annotated exhibition online on the Google Arts & Culture platform, or the original scan maintained on Matterport.

Readers interested in visualizations and computational analyses of OKR data are encouraged to explore the Khipu Field Guide (KFG), a new project by independent researcher Ashok Khosla. In addition to highly-detailed symbolic renderings of over 600 OKR khipus (e.g., Figure 7), users can explore interactive graphics accompanying Khosla’s exploratory data analysis. The KFG investigations are a welcome addition to previous computational work on the OKR carried out by Carrie Brezine (published in Urton [2006]) and Jon Clindaniel [2019].

Symbolic diagram of khipu VA 42554, held by the Ethnologisches Museum in Berlin.
Figure 7. Symbolic diagram of khipu VA 42554, Ethnologisches Museum, Berlin, rendered from OKR data (diagram courtesy of Ashok Khosla). The khipu itself can be found online in the Museum’s collections database.

Finally, those seeking more exploratory visualizations can peruse a series of 3D models of Wari and Inka khipus in the Dumbarton Oaks collection created by archaeologist Alexandre Tokovinine (see,,, and Though it remains in development, the 3D imaging of khipus holds significant promise for future analyses, capturing the multidimensionality of their strings and knots in visually striking ways.

Manuel Medrano (Harvard University), "Keys to Mathematical Treasure Chests: Andean Khipus – Quantitative Research and Digital Analysis," Convergence (August 2022)