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Yupana the Fibonacci Number Grid Based Calculator of Inka Empire

by Liliana Usvat

The Incan culture is probably one of the most mysterious of South American indigenous civilizations. Maybe the reason for that is the Incans’ strange writing system known as Quipu. This system is extraordinary for several reasons. It can’t be compared with any other writing system in the world and is yet to be entirely deciphered.

Math is regarded as a universal language. No matter where you are in the world, math will always be the same. Mathematics is crucial to the sustainability of a great society, the ability to conduct commerce, track time, and record data all play a part in the growth of a nation.

The Incas utilized a device known as the yupana, a Fibonacci number, grid based, calculator capable of computing complex mathematical equations. Such devices were used by the Inca to conduct tribute, taxes, accounting, and trade.

It's known that yupana were used for mathematics from accounts such as Pomo de Ayala stating in 1615 "The counters count with tablets. They number 100,000 and 10,000 and 100 and 10 until arriving at 1" and father Jesuit Jose de Acosta stating "...they take the corn and put one here, three there, eight from another part; they move from a finally get the result without error".


José de Acosta

The father José de Acosta wrote:

... they take the corn and put one here, three there, eight from another part; they move from a box and exchanged three other grains from one to another to finally get the result without error

Father Juan de Velasco wrote:

... these teachers were using something like a series of tables, made of wood, stone, or clay, with different separations, in which they put stones of different shapes, colors and angular shapes

The term yupana refers to two distinct classes of objects:

  • table-yupana (or archaeological yupana): a system of trays of different sizes and materials, carved at the top of the device into geometric boxes, into which seeds or pebbles were placed, presumably for performing complex arithmetic calculations. The first of these tables was found in 1869 in the province of Azuay (Ecuador) and marked the beginning of systematic studies on these objects. All archaeological finds are very different from each other.
  • yupana of Poma de Ayala: a picture on page 360 of El primer nueva corónica y buen gobierno written by the chronicler of the Indies Felipe Guaman Poma de Ayala, representing a 5x4 chessboard.The picture, although having some similarities with the majority of table-yupana, presents several differences from these, first of all, the shape of the boxes (rectangles), when those of table-yupanas are polygons of varying shape.

Although very different from each other, most of the scholars who have dealt with table-yupana, have then extended its reasoning and theories to the yupana of Poma de Ayala and vice versa, perhaps in an attempt to find a unifying thread or a common method. The Nueva Coronica was discovered only in 1916 in the library of Copenhagen and that part of the studies on it were based on previous studies and theories regarding table-yupanas.

Several chroniclers of the Indies described, unfortunately approximately, the Incan abacus and its operation.

Emilio Mendizabal

Emilio Mendizabal was the first to propose in 1976 that the Inca were using, as well as the decimal representation, also a representation based on the progression 1,2,3,5. Mendizabal in the same publication pointed out that the series of numbers 1,2,3 and 5, in the drawing of Poma de Ayala, are part of the Fibonacci sequence, and stressed the importance of "magic" that had the number 5 for civilization the north of Peru, and the number 8 for the civilizations of the south of Peru.

I found one video with explanation Yupana. I will present screenshots ( In case youtube is not visible or will be removed) and youtube video that explain it





The Inca did not have any alphabetic writing to fulfill the purpose of communication and store knowledge. What they did make use of was the Quipu system, a simple and very mobile system that has striking capacities to store various data. The device would normally be composed of different colored threads that were knotted in many combinations. The most complicated quipu would integrate several hundred knots.

Primarily, the Incas used the quipu to keep a record of significant information of a statistical character. However, more research has shown that some of the devices were employed to memorize some of the most compelling stories and songs of the Inca folklore.

A normal quipu would consist of a string going horizontally, to which other strings in different colors were knotted so that they hang down. Both cotton and wool ones were used. The biggest quipu would have up to 1,500 such strings, all of which were woven in any way possible, having a distinct reference in reality.

Everything added meaning on the device: the color of the string, the specific shade of a color, the kind of knot tied, where it was positioned on the composition, how many knots in total were used, and so on. When different quipus were put together, they potentially revealed more meanings too.

Mathematics had a crucial role in the entire method. The quipu took advantage of a decimal positioning, and the most substantial decimal of all to be used was 10,000.

What helped to read the device was a portion of strings positioned the farthest from the principal string. That was the key to the entire encryption. A knot was capable of telling which number in between one and nine were represented. Depending on the type of knot, the meaning further changed and varied, for example a granny knot stood for number ten.

The Inca communities also had experts capable of thoroughly memorizing what a certain quipu meant. Reading the quipu, these people could tell which was the most significant information it contained, and what was of secondary importance. The person to undertake this task would usually pass their knowledge within the family, and so it went generation after generation. It was a well-respected job.

In the capital of Cuzco, experts who read quipu used it to store information of outstanding character for the empire, whether these were a conquest of the emperor or the bloodlines within the ruling family. Across the provinces, the quipu found more pragmatic usage by storing data about the population: how many people lived in the area and how many of them were male or female, how many children or how many served the army–all of it was inscribed in the strings and knots.

Today, there are only a few hundred surviving quipus, and we are limited in the extent to which we can understand the knots. Surprisingly enough, some shepherds who thrive on the Andes still make use of quipu to keep count of their cattle.



Inka Counting Borards


"Chance favors the prepared mind." - Louis Pasteur