An Ethnomathematics Adventure in Rapa Nui
CSHPM Notes brings scholarly work on the history and philosophy of mathematics to the broader mathematics community. Authors are members of the Canadian Society for History and Philosophy of Mathematics (CSHPM). Comments and suggestions are welcome; they may be directed to the column’s editors:
Amy Ackerberg-Hastings, independent scholar (aackerbe@verizon.net)
Nicolas Fillion, Simon Fraser University (nfillion@sfu.ca)
In 2019, on a very cold July winter morning, we boarded a plane from Santiago, Chile, to Rapa Nui, the indigenous name of Easter Island, a Chilean dependency in the eastern Pacific Ocean. It is one of the most remote inhabited locations in the world, famous for its giant stone statues called moai.
Our group, led by renowned archaeologist Dr. Ed Barnhart from Ancient Explorations, was composed of archaeologists, anthropologists, mathematicians, and historians. The group included three mapping experts from Tukuh Technologies (now known as Tepa Companies), a tribally-owned business located in Kansas City, MO. The mapping of an assigned area, with permission from the Chilean government, was conducted with two fixed-wing unmanned aircraft systems (UAS), commonly known as drones (Figure 1). The drones captured high-resolution orthoimagery of the archaeological sites. The laser beams of LiDAR technology (Light Detection and Ranging) were used to provide 3-D point cloud data of some of the caves on the island. We have traveled with Dr. Barnhart’s archaeological team for over 17 years to remote places to do archaeological studies associated with mathematics. What we have been doing all these years falls under a broad area of study called Ethnomathematics, the intersection of culture, history, and mathematics [5].
Figure 1. One of the drones used in the study is on the left. The area mapped by our group is enclosed in red in the map on the right. Images supplied by the authors.
During our trip we explored a mathematical mystery revealed by the foundations of some of the ancient Rapa Nui houses (Figure 2). The stone-base houses, called hare paenga and used until the mid-19th century by the elite, have an elliptical shape. After the trip we obtained a temporary license from Tukuh Technologies that allowed us to analyze some of the blocks (groups of imagery) and locate archaeological artifacts. By using GeoGebra and linear algebra on the drone images of hare paenga foundations, we were able to show that they are indeed elliptical.
Figure 2. Remnants of hare paenga foundations. Photographs by Sebastián Melin, son of Ximena Catepillán.
Once we had convinced ourselves that the hare paenga foundations were ellipses, new questions arose. “Why?” seemed to be answered by the name hare paenga itself, since it translates to boat house, and the elliptical shape resembles a canoe. But how could the early Rapa Nui have laid out an ellipse? Were sophisticated tools needed? To answer this question, all one needs to do is to consider the definition of an ellipse—the set of points in a plane such that the sum of the distances from the point to two given points, called foci, is a constant. By analyzing data from twelve images, we discovered the distance from a focus to the nearest end of a major axis was three to four inches, roughly the width of a hand or three or four fingers, even though the length of the hare paenga ruins varied from 30 to 46 feet. (Having the foci this close to the ends of the major axis is what makes the ellipse so narrow and pointed looking.) Therefore, a method the Rapa Nui could have used to lay out a hare paenga foundation, based on the definition of an ellipse, was to start with a rope tied to two poles so that, when pulled tight, one had the desired length of the foundation. Then, move the poles in towards the center one hands-width. Keeping the rope taut and the poles immobile, use a stick to trace out the ellipse.
Even though we don’t know exactly how the Rapa Nui people laid out the foundations, there is evidence that they used rope or string for measurements. The following quote is from an expedition to Easter Island in 1786: “The care they took to measure our vessel convinced me, that they had not contemplated on arts with stupidity. They examined our cables, our anchors, our compass, and our steering wheel; and in the evening they returned with a string to take their measure over again.” [6, p. 328, emphasis added]
The lead archaeologist, Ed Barnhart, wrote in his trip report that we found fewer hare paenga foundations than expected. Sadly, many of the foundation stone bases were reused later in the construction of walls, stone houses, and other dwellings. In Figure 3, which depicts the interior of an underground shelter called ana kionga or hare kionga, we can see how hare paenga bases were used to build an interior wall, providing us with a good reason to return to the island soon.
Figure 3. The exterior of a hare paenga replica is shown on the left. On the right, we see the view from the interior of a hare paenga. Photographs by Sebastián Melin, son of Ximena Catepillán.
For additional information regarding our work in Rapa Nui, we have published two book chapters and two articles [1; 2; 3; 4].
Figure 4. The authors at Ahu Tongariki with some of the participants on the trip. Photograph supplied by the authors.
Figure 5. Structure made from reused hare paenga stones as depicted in [7].
References
[1] Catepillán, Ximena, and Cynthia Huffman. (2024) Investigating foundations of ancient Rapa Nui houses. In Teaching Mathematics Through Cross-Curricular Projects, edited by Elizabeth A. Donovan, Lucas A. Hoots, and Lesley W. Wiglesworth, 1–10. Providence, RI: MAA Press, an imprint of the American Mathematical Society.
[2] Catepillán, Ximena, and Cynthia Huffman. (2024, February). Mathematics in a Faraway and Forgotten Place. Math Horizons 31(3), 8–11.
[3] Catepillán, Ximena, and Cynthia Huffman. (2024) Two Examples of Ethnomathematics: The intersection of culture, history, and mathematics. In Research in History and Philosophy of Mathematics: The CSHPM 2023 Volume, edited by Maria Zack and David Waszek, 109–121. Annals of the Canadian Society for History and Philosophy of Mathematics / Société canadienne d’histoire et de philosophie des mathématiques. Cham, Switzerland: Birkhäuser.
[4] Catepillán, Ximena, Cynthia Huffman, and Scott Thuong. (2021, March) Mathematical Mysteries of Rapa Nui with Classroom Activities. MAA Convergence, DOI:10.4169/convergence20210405. In September 2021 the article was translated into Spanish for MAA Convergence by Ximena Catepillán with the help of Samuel Navarro from Universidad de Santiago de Chile: Misterios Matemáticos de Rapa Nui con Actividades para el Aula de Clases. In 2023 the Spanish version of the article was reprinted in the magazine Morfismo of the Department of Mathematics and Computer Sciences of Universidad de Santiago de Chile.
[5] D’Ambrosio, Ubiratan. (2019, February) The Program Ethnomathematics: Basic Ideas. CMS Notes 51(1), 10–11.
[6] La Pérouse, Jean-François, comte de Galaup, L. A. Milét-Mureau, and L. Destouff. (1798) The voyage of La Pérouse round the world, in the years 1785, 1786, 1787, and 1788, with the nautical tables. 3 vol. London: Printed for John Stockdale.
[7] McCoy, Patrick Carlton. (1976) Easter Island Settlement Patterns in the Late Prehistoric and Protohistoric Periods. Bulletin 5, Easter Island Committee. New York: International Fund for Monuments, Inc.
Ximena Catepillán is a professor emerita at Millersville University of Pennsylvania, and Cynthia Huffman is a university professor at Pittsburg State University in Kansas. They have been traveling in the summers for several years—to do research in ethnomathematics with a group of archaeologists, historians, and mathematicians—to remote places such as Tikal and the Highlands of Guatemala, Rapa Nui in Chile, Native American sites along the Mississippi River, the temples of Angkor Wat in Cambodia, and Greece.
