[Guaman Poma de Ayala]
"Primer Nueva Coronica y Buen Gobierno", 1615
/ref> The chessboard, though resembling a table ''yupana'', differs from this style in most notably in each of its rectangular trays have the same dimensions, while table ''yupanas'' have trays of other polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
al shapes of differing sizes.
Although very different from each other, most scholars who have dealt with table ''yupanas'' have extended reasoning and theories to the Poma de Ayala ''yupana'' and vice versa, perhaps in an attempt to find a unifying thread or a common method of creation. For example, the ''Nueva coronica'' (New Chronicle) discovered in 1916 in the library
A library is a collection of materials, books or media that are accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vir ...
of Copenhagen
Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan ar ...
contained evidence that a portion of the studies on the Poma de Ayala yupana were based on previous studies and theories regarding table yupanas.
History
Several chroniclers of the Indies described, in brief, this Incan abacus and its operation.
Felipe Guaman Poma de Ayala
The first was Guaman Poma de Ayala around the year 1615 who wrote:
In addition to providing this brief description, Poma de Ayala drew a picture of the yupana: a board of five rows and four columns with each cell holding a series of black and white circles.
José de Acosta
Predating Pomo de Ayala's writings, in 1596 The Jesuit father José de Acosta
José de Acosta (1539 or 1540 in Medina del Campo, Spain – February 15, 1600 in Salamanca, Spain) was a sixteenth-century Spanish Jesuit missionary and naturalist in Latin America. His deductions regarding the ill effects of crossing over the ...
wrote:
Juan de Velasco
In 1841, Father Juan de Velasco wrote:
Table yupana
Various table yupana have been found across Ecuador and Peru.
The Chordeleg Yupana
The earliest known example of a table yupana was found in 1869 in Chordeleg
Chordeleg () is a town and parish and seat of Chordeleg Canton, Azuay Province, Ecuador. The parish covers an area of and according to the 2001 Ecuadorian census it had a population total of 5,478.
Chordeleg is well known for its jewelry industr ...
, Azuay Province
Azuay (), Province of Azuay is a province of Ecuador, created on 25 June 1824. It encompasses an area of . Its capital is Cuenca. It is located in the south center of Ecuador in the highlands. Its mountains reach above sea level in the national ...
, Ecuador
Ecuador ( ; ; Quechua: ''Ikwayur''; Shuar: ''Ecuador'' or ''Ekuatur''), officially the Republic of Ecuador ( es, República del Ecuador, which literally translates as "Republic of the Equator"; Quechua: ''Ikwadur Ripuwlika''; Shuar: ' ...
. A rectangular
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
table (33x27 cm) of wood
Wood is a porous and fibrous structural tissue found in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin ...
consisting of 17 compartments, 14 of which are square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, 2 are rectangular
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...
, and one of which is octagonal
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, w ...
. Two edges of the table contain other square compartments (12x12 cm) raised and arranged side by side, upon which two square platforms (7x7 cm), are superimposed. These structures are called "towers". The table's compartments are symmetrical with respect to the diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
of the rectangular compartments. The four sides of the board are also engraved with images of human heads and a crocodile. As a result of this discovery, Charles Wiener
Charles Wiener (1851–1913) was an Austrian-French scientist-explorer. Born in Vienna, he is perhaps best known as the explorer who traveled extensively in Peru, climbed the Illimani and came close to re-discovering Machu Picchu.
Biography
Hi ...
a systematic study of these objects in 1877. Wiener concluded that the table yupanas served to calculate the taxes
A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer (an individual or legal entity) by a governmental organization in order to fund government spending and various public expenditures (regional, local, o ...
that farmers paid to the Incan empire.
The Caraz Yupana
Found at Caraz
Caraz is a town in the Caraz District in the southeastern part of Huaylas Province of the Ancash Region in Peru.
Political Creation
Recent investigations suggest that its political creation happened on 12 February 1821 when General San Martín, ...
between 1878 and 1879, this table yupana differs from that of Chordeleg as the material of construction is the stone
In geology, rock (or stone) is any naturally occurring solid mass or aggregate of minerals or mineraloid matter. It is categorized by the minerals included, its Chemical compound, chemical composition, and the way in which it is formed. Rocks ...
and the central octagonal compartment is replaced with a rectangular one; towers also have three shelves instead of two.
The Callejón de Huaylas Yupana
A series of table yupanas much different from the first, was described by Erland Nordenskiöld
Baron Nils Erland Herbert Nordenskiöld (19 July 1877 – 5 July 1932) was a Swedish archeologist and anthropologist. Nordenskiöld's research focused on the ethnography and prehistory of South America.
Biography
He was born in Stockholm, t ...
in 1931. These yupana, made of stone, boast a series of rectangular and square compartments. The tower has two rectangular compartments. The compartments are arranged symmetrically with respect to the axis of the smaller side of the table.
The Triangular Yupana
These yupana, made of stone, have 18 triangular compartments. On one side there is a rectangular tower with one level and three triangular compartments. In the central part there are four square compartments.
The Chan Chan Yupana
Identical to the yupana of Chordeleg, both for the material and the arrangement of the compartments, this table yupana was found in the Chan Chan
Chan Chan was the largest city of the pre-Columbian era in South America. It is now an archaeological site in La Libertad Region west of Trujillo, Peru.
Chan Chan is located in the mouth of the Moche Valley and was the capital of the historic ...
archaeological complex in Peru
, image_flag = Flag of Peru.svg
, image_coat = Escudo nacional del Perú.svg
, other_symbol = Great Seal of the State
, other_symbol_type = National seal
, national_motto = "Firm and Happy f ...
in 1967.
The Cárhua de la Bahía Yupana
Discovered in the Peruvian province of Pisco
Pisco is a colorless or yellowish-to-amber colored brandy produced in winemaking regions of Peru and Chile. Made by distilling fermented grape juice into a high-proof spirit, it was developed by 16th-century Spanish settlers as an alternative ...
, these are two table yupana in clay
Clay is a type of fine-grained natural soil material containing clay minerals (hydrous aluminium phyllosilicates, e.g. kaolin, Al2 Si2 O5( OH)4).
Clays develop plasticity when wet, due to a molecular film of water surrounding the clay par ...
and bone
A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, ...
. The first is rectangular (47x32 cm), has 22 square (5x5 cm) and three rectangular (16x18 cm) compartments, and has no towers. The second yupana is rectangular (32x23 cm) and has 22 square compartments, two L-shaped compartments and three rectangular compartments in the center. The compartments are arranged symmetrically with respect to the axis of the longer side.
The Huancarcuchu Yupana
Discovered in Northern Ecuador
Ecuador ( ; ; Quechua: ''Ikwayur''; Shuar: ''Ecuador'' or ''Ekuatur''), officially the Republic of Ecuador ( es, República del Ecuador, which literally translates as "Republic of the Equator"; Quechua: ''Ikwadur Ripuwlika''; Shuar: ' ...
by Max Uhle
Friedrich Max Uhle (25 March 1856 – 11 May 1944) was a German archaeologist, whose work in Peru, Chile, Ecuador and Bolivia at the turn of the Twentieth Century had a significant impact on the practice of archaeology of South America.
Biogra ...
in 1922, this yupana is made of stone and its compartments are drawn onto the surface of the tablet. It has the shape of a pyramid consisting of 10 overlapping rectangles: four on the first level, three on the second, two in the third and one in the fourth. This yupana is the one that is closest to the picture by Poma de Ayala in Nueva Coronica, while having a line fewer and being partially drawn.
The Florio Yupana
C. Florio presents a study [C. Florio, "Recovering memory - The Inca Key as Yanantin"](_blank)
/ref> which does not identify a yupana in these archaeological findings, but an object whose name has been forgotten and remains unknown. Instead, this object is used to connect to the tocapu
Tocapu (Tocapo or Tokapu) was a decorative artwork with discrete geometrical motifs. It was associated with Andean textiles, especially for the use of the Royals' clothing. Tocapu was also painted on wooden boards.
Motifs
Tocapu was an integra ...
(an ideogram already used by pre-Incas civilizations) called “llave inca” (i.e. Inca key) to the yanantin-masintin philosophy. The scholar justifies this based on from the lack of objective evidence that recognizes this object as a yupana, a belief that consolidated over years without repetition or demonstration of this hypothesis, and with the crossing of data from the Miccinelli Documents and the tocapu catalogued by Victoria de la Jara.
Image:Teoria Florio su yupana a casetta - B.png, Fig. A - Structure of a “Chordeleg” table-yupana. Colouring to differentiate the compartments.
Image:Teoria Florio su yupana a casetta - C.png, Fig. B - Identication of a stereotyped color
Image:Teoria Florio su yupana a casetta - D.png, Fig. C - Really existing tocapu catalogued by Victoria de la Jara
Image:Teoria Florio su yupana a casetta - E.png, Fig. D - Other tocapu pattern, possible stylization of the previous one
Image:Teoria Florio su yupana a casetta - F.png, Fig. E - Tocapu called “llave inca”, Inca key
Supposing to color the different compartments of the table yupana (fig. A), C. Florio identifies a drawing (fig. B) very similar to an existing tocapu (fig. C) catalogued by Victoria de la Jara. In addition, in the tocapu reported in figure D, also catalogued by V. de la Jara, Florio identifies a stylization of tocapu C and the departure point for creating the tocapu “llave Inca” (Inca key). She finds the relation between the table yupana and the Inca key also similar in their connection with the concept of duality: the table yupana structure is clearly dual and Blas Valera in “Exsul Immeritus Blas Valera populo suo” (one of the two Miccinelli Documents) describes the "Inca key" tocapu as representing the concept of the “opposite forces” and the “number 2”, both strictly linked to the concept of duality.
According to C. Florio, the real yupana used by the Incas is that of Guáman Poma, but with more columns and rows. The Poma de Ayala yupana would have represented just the part of the yupana useful for carrying out a specific calculation, which Florio identifies to be multiplication (see below).
Theories Based On the Poma de Ayala Yupana
Henry Wassen
In 1931, Henry Wassén studied the Poma de Ayala yupana, proposing for the first time a possible representation of the numbers on the board and the operations of addition and multiplication. He interpreted the white circles as gaps carved into yupana into which the seeds described by chroniclers would be inserted: so the white circles correspond to empty gaps, while the blacks circles correspond to the same gaps filled with a black seed.
The numbering system at the base of the yupana was positional notation in base 10 (in line with the writings of the chroniclers of the Indies).
The representation of the numbers then followed a vertical progression such that the numbers 1-9 were positioned in the first row from the bottom, the second row contained the tens, the third contained the hundreds, and so on.
Wassen proposed a progression of values of the seeds that depends on their position in the table: 1, 5, 15, 30, respectively, depending on which seeds occupy a gap in the first, second, third and fourth columns (see the table below). Only a maximum of five seeds could be included in a box belonging to the first column, so that the maximum value of that box was 5, multiplied by the power of the corresponding row. These seeds could be replaced with one seed of the next column, useful during arithmetic operations. According to the theory of Wassen, therefore, the operations of sum and product were carried out horizontally.
This theory received a lot of criticism due to the high complexity of the calculations and was therefore considered inadequate and soon abandoned.
The following table shows the number 13457 as it would appear on Wassen's yupana:
This first interpretation of the Poma de Ayala yupana was the starting point for the theories developed by subsequent authors, into the modern writing. No researcher moved away from the positional numbering system until 2008.
Emilio Mendizabal
Emilio Mendizabal was the first to propose in 1976 that the Inca
The Inca Empire (also known as the Incan Empire and the Inka Empire), called ''Tawantinsuyu'' by its subjects, (Quechua for the "Realm of the Four Parts", "four parts together" ) was the largest empire in pre-Columbian America. The admin ...
a representation based on the progression 1,2,3,5 in addition to the decimal representation. In the same publication, Mendizabal pointed out that the series of numbers 1,2,3 and 5, appear in Poma de Ayala's drawing, and are part of the Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
, and stressed the importance of the "magic" that the number 5 contained for civilizations of Northern Peru
, image_flag = Flag of Peru.svg
, image_coat = Escudo nacional del Perú.svg
, other_symbol = Great Seal of the State
, other_symbol_type = National seal
, national_motto = "Firm and Happy f ...
, similar in significance to the number 8 for the civilizations of Southern Peru
, image_flag = Flag of Peru.svg
, image_coat = Escudo nacional del Perú.svg
, other_symbol = Great Seal of the State
, other_symbol_type = National seal
, national_motto = "Firm and Happy f ...
.
Radicati di Primeglio
In 1979, Carlos Radicati di Primeglio emphasized the difference of table yupana from that of Poma de Ayala, describing the state-of-the-art research and advanced theories so far. He also proposed the algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for calculating the four basic arithmetic operations
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
for the Poma de Ayala yupana, according to a new interpretation for which it was possible to have up to nine seeds in each box with a vertical progression of powers of ten. Radicati associated each gap with a value of 1.
The following table shows the number 13457 as it would appear on Radicati's yupana:
William Burns Glynn
In 1981, the English textile engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...
William Burns Glynn proposed a positional base 10 solution for the yupana of Poma de Ayala.[William Burns Glynn]
"Calculation table of the Incas"
''Bol. Lima No. 11'', 1981, 1-15.
Glynn, as Radicati, adopted Wassen's idea of full and empty gaps, as well as a vertical progression of the powers of ten, but proposed an architecture that allowed yupana users to greatly simplify the arithmetic operations themselves.
The horizontal progression of the values of the seeds in its representation is 1, 1, 1 for the first three columns, such that in each row is possible to deposit a maximum of ten seeds (5 + 3 + 2 seeds). Ten seeds in any row corresponds to a single seed in the line above it.
The last column in Glynn's yupana is dedicated to the "memory''"'', a place that can hold up to ten seeds before they are moved to the upper line. According to the author, this is very useful during arithmetic operations in order to reduce the possibility of error.
Glynn's solution has been adopted in various teaching projects all over the world, and even today some of its variants are used in some schools
A school is an educational institution designed to provide learning spaces and learning environments for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is sometimes compulsor ...
of South America
South America is a continent entirely in the Western Hemisphere and mostly in the Southern Hemisphere, with a relatively small portion in the Northern Hemisphere at the northern tip of the continent. It can also be described as the sout ...
.[Mora & Valero "La Yupana come strumento pedagogico alle elementari"](_blank)
/ref>[Fiorentino, "La yupana elettronica: uno strumento per la didattica interculturale della matematica"](_blank)
/ref>
The following table shows the number 13457 as it would appear on Glynn's yupana:
Nicolino de Pasquale
In 2001, the Italian
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional Ita ...
engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...
Nicolino de Pasquale proposed a positional solution in base 40 of the Poma de Ayala yupana, taking the representation theory of Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
already proposed by Emilio Mendizabal and developing it for the four operations.
De Pasquale's yupana also adopts a vertical progression to represent numbers by powers of 40. The representation of the numbers is based on the fact that the sum of the values of the circles in each row is 39, if each circle takes the value 5 in the first column, 3 in the second column, 2 in the third and 1 in the fourth one; it is thus possible to represent 39 numbers, united to neutral element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
( zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
or "no seeds" in the table); this forms the basis of 40 symbols necessary for the numbering system.[N. De Pasqual]
"Il volo del condor", ''Pescara Informa'', 2001
/ref>
The following table shows one of the possible representations of the number 13457 in De Pasquale's yupana:
After its publication, De Pasquale's theory sparked great controversy among researchers who fell into two primary groups: a group supporting the base 10 theory and another supporting the base 40 theory. The Spanish chronicles written of the conquest of the Americas
During the Age of Discovery, a large scale European colonization of the Americas took place between about 1492 and 1800. Although the Norse had explored and colonized areas of the North Atlantic, colonizing Greenland and creating a short ter ...
indicated that the Incas used a decimal system and since 2003 the base 10 theory has been proposed as the basis for calculating both with the abacus and the quipu
''Quipu'' (also spelled ''khipu'') are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.
A ''quipu'' usually consisted of cotton or camelid fiber strings. The Inca people ...
[Lorenzi, Incan counting system as easy as 1,2,3,5 (2004)](_blank)
/ref>
De Pasquale has recently proposed the use of yupana as astronomical calendar running in mixed base 36/40[N. De Pasquale, "The Saved Kingdom"](_blank)
/ref> and provided his own interpretation of the Quechua
Quechua may refer to:
*Quechua people, several indigenous ethnic groups in South America, especially in Peru
*Quechuan languages, a Native South American language family spoken primarily in the Andes, derived from a common ancestral language
**So ...
word ''huno'', translating it as "0.1".[N. De Pasquale, "Decimal Guaman Poma"](_blank)
/ref> This interpretation diverges from all chroniclers of the Indies, especially Domingo de Santo Tomas who in 1560 translated ''huno'' into ''chunga guaranga'' (ten thousand).
Cinzia Florio
In 2008 Cinzia Florio proposed an alternative and revolutionary approach compared to all the theories proposed so far. Florio's newer theory deviated from the positional numbering system and adopted additive, or sign-value notation
A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty (50 + 10 + 10 ...
.[C. Florio, "Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala", Salerno, 14-15 maggio e 10-12 Dicembre 2008 - Oédipus Editore, 2009](_blank)
/ref>
Relying exclusively on Poma de Ayala's design, Florio explained the arrangement of white and black circles and interpreted the use of the yupana as a board for computing multiplications, in which the multiplicand
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
is represented in the right column, the multiplier in the two central columns, and the product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
in the left column, illustrated in the following table:
The theory differs from all the previous in several aspects: first, the white and black circles would not be gaps that could be filled with a seed, but rather different colors of seeds, representing respectively tens and ones (this according to the chronicler Juan de Velasco).
Secondly, the multiplicand is entered in the first column respecting the sign-value notation: so, the seeds can be entered in any order and the number is given by the sum of the values of these seeds.
The multiplier is represented as the sum of two factors, since the procedure for obtaining the product is based on the distributive property of multiplication over addition.
According to Florio, the multiplication table drawn by Poma de Ayala with provision of the seeds represented the calculation: 32 x 5, where the multiplier 5 is decomposed into 3 + 2. The sequence of numbers 1,2,3,5 would be causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
, contingent to the calculation done and unrelated to the Fibonacci series.
Key: ◦ = 10; • = 1; The operation represented is: 32 x 5 = 32 x (2 + 3) = (32 x 2) + (32 x 3) = 64 + 96 = 160
The numbers represented in the columns are, from left to right:
* 32 (the multiplicand),
* 64 = 32 x 2 and 32 x 3 = 96 (which together constitute the multiplicand, multiplied by the two factors in which the multiplier has been broken down)
* 151 (the product)
The final number in this computation (which is incorrect) is the basis for all possible criticisms of this interpretation, since 160, not 151, is the sum of 96 and 64. Florio notes, however, that the mistake could have been on the part of Poma de Ayala in the original drawing, in designing a space as being occupied by a black circle instead of a white one. In this case, changing just one black circle into a white one in the final column gives us the number 160, the correct product.
Poma de Ayala's yupana cannot represent every multiplicand either, it is necessary to extend the yupana vertically (adding rows) to represent numbers whose sum of digits exceeds 5. The case is the same for the multipliers: to represent all the numbers is necessary to extend the number of columns. Apart from the supposed erroneous calculation (or erroneous representation by the designer), this is the only theory that identifies in Poma de Ayala's yupana a mathematical and consistent message (multiplication) and not a series of random numbers as in other interpretations.
Andrés Chirinos (2010)
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (ACEID), revised older drawings and descriptions chronicled by Poma de Ayala, and finally deciphered the use of the yupana: a table with eleven holes which Chirinos calls a "Pre-Colombian Calculator", capable of adding, subtracting, dividing, and multiplying, making him hopeful of applying this information to the investigation of how quipus were used and functioned.
See also
* Quipu
''Quipu'' (also spelled ''khipu'') are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.
A ''quipu'' usually consisted of cotton or camelid fiber strings. The Inca people ...
* Inca Empire
The Inca Empire (also known as the Incan Empire and the Inka Empire), called ''Tawantinsuyu'' by its subjects, ( Quechua for the "Realm of the Four Parts", "four parts together" ) was the largest empire in pre-Columbian America. The adm ...
* Numbering System
References
External links
Gilsdorf - Ethnomathematics of the Inkas
Heliane Seline - Mathematics through cultures
Chroniclers of the Indies
*
*
José De Acosta - Historia Natural y Moral de las Indias
*
Velasco - Historia del reyno de Quito del America del Sur
Theory by Wassen and table-Yupana
*
Radicati di Primeglio - El sistema contable de los Incas: Yupana y Quipu
Theory by Glynn Burns and school projects
*
Mora & Valero - La Yupana come strumento pedagogico alle elementari
Leonard & Shakiban - The Incan Abacus
*
Fiorentino - La yupana elettronica: uno strumento per la didattica interculturale della matematica
Theory by De Pasquale
Università Bocconi di Milano - La Matematica nelle civiltà pre-colombiane
* ttp://it.geocities.com/newsarcheo/precolombiane4.htm Notizie sulla numerazione Inca e sulla yupana
Un italiano scopre l'enigma della matematica inca
* ttp://4.bp.blogspot.com/_K8aa26UngiU/SEXI6P-odEI/AAAAAAAABQI/OqPjrzw84rY/s1600-h/yupane.jpg L'unione Sarda - I numeri della natura nella scacchiera degli Inca - di Andrea Mameli
"Guaman Poma Game, by N. De Pasquale, D. D'Ottavio
Theory by C. Florio
*
Florio - Incontri e disincontri nella individuazione di una relazione matematica nella yupana in Guaman Poma de Ayala
*
Florio - Encuentros y Desencuentros en la identificación de unarelación matemática en la yupana de Guaman Poma de Ayala
{{Authority control
Inca mathematics
Mathematical tools
Quechua words and phrases