#quipu

Astronomers have discovered what may be the largest-scale structure in the known universe — a group of galaxy clusters and clusters of galaxy clusters that spans roughly 1.3 billion light-years across and contains a mind-boggling 200 quadrillion solar masses. The newfound structure is dubbed Quipu after an Incan system of counting and storing numbers using knots on cords. Like a Quipu cord, the structure is complex, made up of one long filament and multiple side filaments. It spans roughly 1.3 billion light-years (more than 13,000 times the length of the Milky Way), potentially making it the largest object in the universe in terms of length, beating out previous record-holders such as the Laniākea supercluster.

National Poetry Month: Cecilia Vicuña

Today we highlight the work of Chilean poet and artist Cecilia Vicuña (b. 1948). Exiled in the early 1970s, after the military coup against Salvador Allende, Vicuña fled to London and co-founded Artists for Democracy in protest against the brutality of the Pinochet regime. Her writing, visual art, and performance work center ecology, human rights, precarity, and the re-interpretation of ancient culture – including the pre-Columbian quipu system of knotted communication – as a guard against “cultural homogenization.”

The images here are drawn from three of her collections: quipoem (1997: Wesleyan University Press, Hanover); Word Weapons (2023: The Wattis Institute; RITE Editions, San Francisco), and Deer Book (2024: Radius Books, Santa Fe). Vicuña examines the power of language – word weapons – laments the imagination lost under authoritarian rule, and considers those disappeared. In a recent interview with Brooklyn Rail, she describes her imagination as “Amerindian”:

The Amerindian scale is a scale that you find throughout the Americas and you also find throughout the entire world, until the takeover of what is now called Western culture. It is this concept of the body in relation to the cosmos…I was not in the scale of a person. I was in the scale of the sun, of the ocean, you see?

See more posts about poetry!

--Amanda, Special Collections Graduate Intern

Cecilia Vicuña (b. 1948) is a poet, artist, activist and filmmaker. She was born and raised in Santiago de Chile, but she has been in exile since the early 1970s after the military coup against then president Salvador Allende. While exiled in London, Vicuña focused on political activism, demonstrating in peaceful protests for human rights in Chile and other countries. She was a co-founder of Artists for Democracy in 1974.

Vicuña also lived in Bogotá, Colombia, and traveled extensively including throughout Venezuela and Brazil. In 1980, Vicuña moved to New York City.

Throughout her long career, Vicuña has been creating work that addresses ecological destruction, human rights, and cultural homogenization. Quipu, her ongoing project since the 1960s, is inspired by the Pre-Columbian record-keeping system that uses knots on threads. After making various types of quipus with found materials and threads, Vicuña also began creating quipus that were composed of people. She claimed that “weaving people” is “a political act of joining people together.” These acts eventually became something deeper than political act when she realized that it was about “connecting our collective body to the movement of water.”

Apologies for such a delayed response to your question!

By Professor Chris Staecker, PhD

Department of Mathematics

The quipu is a beautiful thing. It was a mystery, but we figured it out. It’s a system of knots tied in cords that the Inca and their predecessors used to keep track of numbers. The Inca had no written language, so the only records we have from the old days were written with fingers in quipus. Over 1,000 pre-columbian quipus still exist today, mostly in museums.

One old document, Felipe Guaman Poma de Ayala’s El primer nueva corónica y buen gobierno (1615), includes a famous drawing of an Inca recordkeeper—a quipucamayoc—holding a quipu. For historians, the quipus became a source of frustration: the old sources describe it, but none of them explain exactly how it worked. That’s where the mathematicians came in.

Left: Drawing of the Quipucamayoc, from El primer nueva corónica y buen gobierno by Felipe Guaman Poma de Ayala, 1615. He holds a quipu, with a diagram of the yupana at lower-left. 

Right: A replica quipu on the wall of the author’s office in Bannow Science Center. (Come see it!)

In 1912, American mathematician and historian Leslie Leland Locke was the first to propose a specific interpretation. He believed, without much evidence, that the Inca were using a base-10 positional number system exactly like our own. So the number 74 would be represented by a 7-turn knot followed by a 4-turn knot. It seemed reasonable, if a bit familiar, but really there was no way to test Locke’s theory.

Above: The number 74, as it would be represented on a quipu. (Photo and knotting by the author.)

Later work by Marcia and Robert Ascher in the 1970s discovered that many quipu bundles include a special cord which shows the sum of all the others. So the quipu functions like a spreadsheet! Several numbers grouped together, with the sum written at the bottom. This work finally proved that Locke’s interpretation of the numbers was right: the correctness of the sums only makes sense if the Inca really were using Locke’s proposed system.

Anyone with an appreciation for mathematics will recognize a particular thrill here: we encounter mysterious symbols, see strange patterns, and apply mathematical analysis to figure it all out! This “decoding” of the quipu represented one of the first great successes in what we now call “ethnomathematics:” the study of the mathematical practices of traditional cultures.

But the quipu was not the only mathematical instrument used by the Inca. There was also the yupana, shown in the lower left of Guaman Poma’s quipucamayoc. It seems to have been some sort of counting board, where counters would be moved around to represent calculations—like an abacus without bars. However, details in the old sources are very scarce concerning the yupana, even more so than for the quipu. No one described exactly how it worked, and Guaman Poma’s simple illustration is the only drawing of a yupana that exists in any primary source.

But in the late 1800s, archaeologists began to find rectangular boards divided into compartments, often with different shapes and heights (see photos). These artifacts were quickly identified as the mysterious yupana, and again the mathematicians took an interest.

Above: Author’s photo of a “yupana” artifact in the Museum of Natural History, New York City. The museum describes this item as: “carved stone block of unknown purpose.”

Below: Author’s 3D-printed replica of the “yupana” artifact of Chordeleg, excavated 1869. (Come see it in my office!)

The yupana artifacts are beautiful objects, but they didn’t come with instructions. How exactly were they used as calculators? Over the decades, many people have announced their own “decoding” of these things. But serious scholarship has reached other conclusions.

The problem is, these artifacts don’t actually match the historical descriptions of the yupana. The striking feature of the artifacts is their three-dimensionality, but Guaman Poma’s drawing is a flat grid of squares. Additionally, the old texts describe the specific geographical region where the yupana was used, which doesn’t correspond to the locations where the artifacts have been found.

We also know that the Inca loved to use regular geometric shapes in their art, and they played board games. So these artifacts could very well be artistic carvings or gameboards. The scholars generally agree: the Inca probably did use a counting board, but they are lost today. And there’s no good evidence that these “yupana” artifacts have anything to do with calculation at all. 

Unfortunately, the mystique of the Inca is too strong for this kind of dry academic conclusion. Especially in our time, a lifetime of serious study is no match for the idle notions of a confident hobbyist who “does their own research.”

A European engineer made headlines in 2004 when he announced his own “decoding” of the yupana artifacts. His explanation assumes a base-40 number system (the Inca used base-10). He boasted that it took him less than an hour to “solve the riddle,” and that he did so with no knowledge of the Inca culture (no surprise there). Media reports did not point out how deeply unserious it all was. This guy probably felt a certain cosmic kinship with the Inca, a respectful connection across the centuries when he finally cracked their code.

But it is not respectful to trample so casually on another culture’s ground. Respect begins with seeing the Inca not as a puzzle, but as a people. They were a complex people, and yes, a mathematical people, but they did not exist in order to be deciphered. They created their own mathematics for their own reasons, and their knowledge was not lost by accident: it was purposefully destroyed. And who are we, after all this time, to demand answers from the Inca? Their yupana, at least, will probably remain forever unconquered.