Go Geometry Machu Picchu, Inca City

The Quipu, the Pre-Inca Data Structure

Caral, the Oldest Quipu

Above center: An illustration from 1615 showing the Fibonacci series 1, 2, 3, 5.

The Quipu is a system of knotted cords used by the Incas and its predecessor societies in the Andean region to store massive amounts of information important to their culture and civilization.

The colors of the cords, the way the cords are connected together, the relative placement of the cords, the spaces between the cords, the types of knots on the individual cords, and the relative placement of the knots are all part of the logical-numerical recording. For example, a yellow strand might represent gold or maize; or on a population quipu the first set of strands represented men, the second set women, and the third set children. Weapons such as spears, arrows, or bows were similarly designated.

The combination of fiber types, dye colors, and intricate knotting could be a novel form of written language, according to Harvard anthropologist Gary Urton. He claims that the quipus contain a seven-bit binary code capable of conveying more than 1,500 separate units of information.

Quipus were knotted ropes using a positional decimal system. A knot in a row farthest from the main strand represented one, next farthest ten, etc. The absence of knots on a cord implied zero.

Quipucamayocs, the accountants of the Inca Empire (called Tahuantinsuyu in old spelling Quechua) created and deciphered the quipu knots. Quipucamayocs were capable of performing simple mathematical calculations such as adding, subtracting, multiplying, and dividing information for the indigenous people.

In the absence of written records the quipus served as a means of recording history and passed on to the next generation, which used them as reminders of stories. An thus these primitive computers - quipus - had knotted in their memory banks the information which tied together the Inca empire.

Hiram Bingham, the American explorer who found the ruins of Machu Picchu in 1911, wrote:

"The Incas had never acquired the art of writing, but they had developed an elaborate system of knotted cords called quipus. These were made of the wool of the alpaca or the llama, dyed in various colors, the significance of which was known to the magistrates. The cords were knotted in such a way to represent the decimal system and were fastened at close intervals along the principal strand of the quipus. Thus an important message relating to the progress of crops, the amount of taxes collected, or the advance of an enemy could be speedily sent by the trained runners along the post roads." ‘Lost City of the Incas, The Story of Machu Picchu and its Builders’ by Hiram Bingham.

Hiram Bingham at Machu Picchu, 1911
The inspiration for Indiana Jones?



The Quipus and The Royal Commentaries of the Inca

In 1609, the Inca Garcilaso de la Vega published the first volume of his Royal Commentaries of the Incas in Lisbon. He wrote:

The word quipu means both knot or to knot; it was also used for accounts, because they were kept by means of the knots tied in a number of cords of different thicknesses and colors, each one of which had a special significance. Thus, gold was represented by a gold cord, silver by a white one, and fighting men by a red cord.

When their accounts had to do with things that have no color - such as grain and vegetables - they were classified by categories, and, in each category, by order of diminishing size. Thus, to furnish an example, if they had had to count the various types of agricultural production in Spain, they  would have started with wheat, then rye, then peas, then beans, and so forth. In the same way, in order to make an inventory of the arms of the imperial army, they first counted the arms that were considered to belong in a superior category, such as lances, then javelins, bows and arrows, hatchets and maces, and lastly, slings, an any other arms that were used. In order to ascertain the number of vassals in the Empire, they started with each village, then with each province: the firs cord showed a census of men over sixty, the second, those between fifty and sixty, the third, those from forty to fifty, and so on, by decades, down to the babes at the breast.

Occasionally other, thinner, cords of the same color, could be seen among one of these series, as though they represented an exception to the rule; thus, for instance, among the figures that concerned the men of such and such an age, all of whom were considered to be married, the thinner cords indicated the number of widowers of the same age, for the year in question: because, as I explained before, population figures, together with those of all the other resources of the Empire, were brought up to date every year.

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they were all as well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger. Indeed, those men, called quipucamayus, who were in charge of the quipus, were exactly that, imperial accountants.

The number of quipucamayus scattered throughout the Empire, was proportional to the size of each place. Thus the smallest villages numbered four, and others twenty, or even thirty. The Incas preferred this arrangement. even in places where one accountant would have sufficed, the idea being that, if several of them kept the same accounts, there was less risk that they would make mistakes.

Every year, an inventory of all the Inca's possessions was made. Nor was there a single birth or death, a single departure or return of a soldier, in all the Empire, that was not noted on the quipus. And indeed, it may be said that everything that could be counted, was counted in this way, even to battles, diplomatic missions, and royal speeches. But since it was only possible to record numbers in this manner, and not words, the quipucamayus assigned to record ambassadorial missions and speeches, learned them by heart, at the same time that they noted down the numbers, places and dates on their quipus; and thus, from father to son, they transmitted this information to their successors. The speeches exchanged between the Incas and their vassals on important occasions, such as the surrender of a new province, were also transmitted to posterity by the amautas, or philosophers, who summarized them in simple, clear fables, in order that they might be implanted by word of mouth in the memories of all the people from those at court to the inhabitants of the most remote hamlets. The harauicus, or poets, also composed poems based on diplomatic records and royal speeches. These poems were recited for a great victory or festival, and every time a new Inca was knighted.

When the curacas and dignitaries of a province want to know some historical detail concerning their predecessors, they asked these quipucamayus, who were, in other words, not only the accountants, but also the historians of each nation. The result was that the quipucamayus never let their quipus out of their hands, and they kept passing their cords and knots through their fingers so as not to forget the tradition behind all these accounts. In fact, their responsibility was so great and so absorbing, that they were exempted from all tribute as well as from all other kinds of service.

All laws, ordinances, rites, and ceremonies throughout the Empire were recorded by these same means.

When my father's Indians came to town on Midsummer's Day to pay their tribute, they brought me the quipus; and the curacas asked my mother to take note of their stories, for they mistrusted the Spaniards, and feared that they would not understand them. I was able to reassure them by re-reading what I had noted down under their dictation, and they used to follow my reading, holding on to their quipus, to be certain of my exactness; this was how I succeeded in learning many things quite as perfectly as did the Indians.



Illustrations from 1615 by The "Indian Chronicler" Felipe Guaman Poma de Ayala about the quipu.
Finding his most persuasive medium to be the visual image, he organizes his 1200-page Nueva corónica y buen gobierno (New Chronicle and Good Government) around his 398  pen-and-ink drawings, all skillfully executed by his own hand. For the archaeologist, Guaman Poma's drawings of native life under the Incas are like photographs of the past.


An encounter at a "Collca" or "Warehouse of the Inca": Tupac Inca Yupanqui (left) interviews his accountant or warehousekeeper (right). The warehousekeeper is extending a cord record or quipu, which contains records of goods in the storage chambers.


Chief accountant and treasurer,  authority in charge of the quipu of the kingdom.
In the lower left corner, there is an abacus counting device used with maize kernels on which computations were performed and later transferred to the quipu.

The maize kernels are the first numbers of the Fibonacci series, in which each number is a sum of two previous: 1, 2, 3, 5. 


The native administrator of resources, with the book and quipu he uses for accounting.


The Inka’s secretary and accountant who records the dispositions of the royal lords.


Reference: Guaman Poma - 'El primer Nueva corónica y buen gobierno'.


Caral: Ancient Peru city reveals 5,000-year-old 'writing'
July 19, 2005, 22:45, SABC News

Archeologists in Peru have found a "quipu" on the site of the oldest city in the Americas, indicating the device, a sophisticated arrangement of knots and strings used to convey detailed information, was in use thousands of years earlier than previously believed. Previously the oldest known quipus, often associated with the Incas whose vast South American empire was conquered by the Spanish in the 16th century, dated from about 650 AD.

But Ruth Shady, an archeologist leading investigations into the Peruvian coastal city of Caral, said quipus were among a treasure trove of articles discovered at the site, which are about 5,000 years old. "This is the oldest quipu and it shows us that this society ... also had a system of "writing" (which) would continue down the ages until the Inca empire and would last some 4,500 years," Shady said. She was speaking before the opening in Lima today of an exhibition of the artifacts which shed light on Caral, which she called one of the world's oldest civilizations.

The quipu with its well-preserved, brown cotton strings wound around thin sticks, was found with a series of offerings including mysterious fiber balls of different sizes wrapped in "nets" and pristine reed baskets. "We are sure it corresponds to the period of Caral because it was found in a public building," Shady said. "It was an offering placed on a stairway when they decided to bury this and put down a floor to build another structure on top."

Pyramid-shaped public buildings were being built at Caral, a planned coastal city 180km north of Lima, at the same time that the Saqqara pyramid, the oldest in Egypt, was going up. They were already being revamped when Egypt's Great Pyramid of Keops (or Khufu) was under construction, Shady said. "Man only began living in an organized way 5,000 years ago in five points of the globe - Mesopotamia (roughly comprising modern Iraq and part of Syria), Egypt, India, China and Peru," Shady said, adding Caral was 3,200 years older than cities of another ancient American civilization, the Maya.



Knotty Incan Accounting Untangled

Source: Science News, August 12, 2005

  A ball of string tied into countless knots could very well be seen as a source of frustration. But for the ancient Inca civilization, carefully tied knots formed the basis of a method of record-keeping known as khipus. Now researchers report that the ledger system is more complex than previously believed and includes a way of communicating information to higher-ups in the well-categorized Incan chain of command between workers and administrators with higher rank.

Hundreds of khipus, each consisting of a single strand of wool from which hundreds to thousands of other knotted strings hang, have been discovered to date. Gary Upton and Carrie J. Brezine of Harvard University designed a computer program to analyze the patterns in 21 khipus recovered from a site in Puruchuco, an Inca administrative center on the coast of Peru near modern day Lima. They discovered that certain patterns within the strings of varying colors and lengths appear to contain numerical data that represent summations. What is more, the information is arranged among the khipus in a ranked pattern with three levels of authority. Information is passed between them by including the sum from a khipu in one level on a khipu representing a higher level.



Quipus of Rapaz

Left, A collection of Quipus in San Cristobal de Rapaz, Oyon, Lima-Peru.

A project of research and conservation began in January 2004 with an agreement between the village and the anthropologist Frank Salomon, of the University of Wisconsin in the USA. The village agreed to give access for scientific study in exchange for conservation help to make the quipus and their environment safer against deterioration.

Source: The Khipu Patrimony of Rapaz, Peru


Quipu as a central metaphor
In Quipu, Arthur Sze’s eighth book of poetry, he writes with imaginative rigor and urgency poems that move across cultures and time, from elegy to ode, to find a precarious splendor.

Source: Quipu by Arthur Sze

Quipu was a tactile recording device for the pre-literate Inca, an assemblage of colored knots on cords. Sze utilizes quipu as a central metaphor, knotting and stringing luminous poems that each have an essential place and integrity, each contribute to the recurrent “knotting” in the book. Sze’s language is taut and startling, and what appears stable may actually be volatile. In Quipu Sze harnesses the particulars of our lives and spins them into something enduring. He makes us envision the terrors and marvels of our contemporary world in this collection of crucial poems of our time.

What is “Quipu?”

Quipu means knot in Quechua, the native language of the Andes. The Incas had a system of accounting and data recording that relied on the quipu, a devise in which cords of various colors were attached to a main cord with knots. The number and position of knots as well as the color of each cord represented information about commercial goods and resources. Quipu-makers were responsible for encoding and decoding the information. The messages included information about resources in storehouses, taxes, census information, the output of mines, or the composition of work forces. Archeologists have recently suggested that authors used the quipu to compose and preserve poems and legends. Because there were relatively few words in Quechua, the cords of a quipu could be used as pronunciation keys.

In "The Angle of Reflection Equals the Angle of Incidence," he writes:

Quinoa simmers in a pot; the aroma of cilantro
on swordfish; the cusp of spring when you

lean your head on my shoulder. Orange crocuses
in the backyard form a line. Once is a scorched site;

we stoop in the grass, finger twelve keys with

interconnected rings on a swiveling yin-yang coin,
dangle them from the gate, but no one claims them. 


Conversations: String Theorist
Unraveling a knotty Inca puzzle

Source: Archaeology, A publication of the Archaeological Institute of America, Volume 58 Number 6, November/December 2005 .

Khipu, the enigmatic and still undeciphered record-keeping system of elaborately knotted strings used by the Inca Empire, has long intrigued anthropologist Gary Urton. Since 2002, he and his colleague Carrie Brezine, a mathematician, have maintained the Khipu Database Project at Harvard University, which corrals all existing khipu scholarship in one online repository. Most recently, they announced they may have found the meaning of a particular sequence of knots. ARCHAEOLOGY spoke with Gary Urton about mystified Spanish colonials, teaching Harvard students how to make khipu, and bringing tax records to the afterlife. See more.

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Last updated: November 13, 2007