THE INKA QUIPU ENIGMA

OLIVER KNILL

1. Introduction
1.1. The history, mathematics and database technology of the quipu 1, the “talking
knots” of the Inka empire is a fascinating subject. Quipu are an original approach
to number systems, database structures. Unlike marks on bones, tally sticks or clay
tablets, ink on wood, papyrus or paper, it is a topological encoding, similarly in
nature than genetic code is woven from protein knots. The first scientific study of
quipus began by L. Leland Locke. His important article [11] documented in a very
clear way how knots were used for recording numbers. In his introduction, Locke
also pointed out that also in other parts of the world, like China, knot records have
preceded the knowledge of writing.
74.104.146.187 on Sat, 29 Jan 2022 10:15:54 UTC
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Figure 1. A page from [11].

Date: 10/22/2018, updated 1/31/2022 for Math E 320.

1
Quipu and Khipu are equivalent spelling variations
1
 Inka quipu

1.2. Recent research pointed to Rosetta stone break-through discoveries lead-

ing to publications for a general audience like [6, 14, 5, 20]. In 2018, when this
document started, there there was also quipu exhibit at the Boston museum of fine
arts. Naturally, these popularizations or reports hide the work which is needed to
investigate the topic. There is the field work of digging out, cleaning, reading and
then cataloging the information, then to place the data into the context of the his-
tory, linguistic, and culture of the time and finally to translate interpret and cross
referencing the data. In the quest to decode the quipu cypher, there has been spec-
tacular progress for post-colonial quipus [7, 13] and progress in better understanding
non-numerical pre-conquest quipus [4].
2. Knots, Links and Graphs
2.1. Strictly speaking, for a mathematician, a quipu is neither a knot (a closed loop
in space) nor a link, a collection of non-intersecting knots in space. But they are
links in a generalized sense in that they would be links if the ends of the individual
ropes were connected. It is not so much the topology of quipu which is of interest
for researchers but the information content which is encoded topologically. Because
only three different type of knots appear in Inka style data (simple knots, figure eight
knots and long knots) (whose topology is well understood), these entities could be
replaced by symbols like L4, S, E standing for a long knot with 4 turns, a single
knot or a figure eight knot. A quipu can be described as a graph on which scalar
and vector data are attached. The scalar data assign to a node the knot type or the
attachment type, if the node is branching off there. The vector data which describe
the connecting strings are determined by ply and spin direction, attachment type,
color and the material of the knot. For a computer scientist a quipu is an example
of a graph database.

Figure 2. The trefoil knot and a figure eight knot.

 3. Crypto riddles
3.1. The Inca code is a cypher which has not yet cracked. While the numerical
data encoding are quite well understood, the problem lies in understanding non-
numerical signs. Crypto riddles have always attracted the interest of the general
public. Examples are the Maya code, the Egyptian hieroglyphs, the German
Enigma during the second world war or the fictional alien pictorial language which
is at the center of the movie “Arrival”. An other example of an outstanding riddle is
the “Antikythera” instrument which is believed to be an early analogue com-
puter used for astronomical computations. More riddles have presented themselves
when trying to decode texts from Palimpsests, texts which are hidden beneath
other texts. An example is the Archimedes Palimpsest. In those cases, reading the
text requires first to reveal the structure as the text had been erased and written
over.

Figure 3. The Anticitera, the Maya and the Rosetta stone.

4. Rosetta stone moments

4.1. Also in the case of the German enigma code which was cracked at Bletchley
Park, the problem was not entirely mathematical. One had to wait for Rosetta
stone moments, clues like knowledge of the weather code, or rely on planted
information which allowed the cryptographers to attack the code using crib-based
decryption techniques. In the case of the quipu, the task is harder because there are
no known cribs (at least from the pre-colonial time) and many of these documents
were destroyed in the wake of the colonial conquest and because quipu from regions
with high precipitation deteriorated rapidly if not preserved as they are made of
organic material like wool.
4.2. There are less than 1000 quipu known today. The Berlin collection contains
about a third. The decoding problem has linguistic, historical and anthropological
context. Understanding the content of a coded text or new language needs “Rosetta
stone moments” like in the case of the hieroglyphs, where Champoleon and Young
have been able to crack the code of the hieroglyphs. The quipu form a cryptological
riddle in which plain text information is missing. Since the information is believed
 Inka quipu

to be non-phonetic, the problem is harder than in the case of hieroglyphs, the

cuneiforms or the Maya code.

Figure 4. Some quipu researchers: Leland Locke, Marcia and Robert

Ascher, Sabine Hyland.

5. Algebra with strings attached

5.1. For a mathematician, quipu can be fascinating in various ways. One knows
already quite a bit about the numerical aspects [2]. But mathematics can be under-
stood as a more general concept, not only as the science of numbers, or the quest
to understand algebraic or geometric objects but more generally as a science of
structure. For a mathematician, a language is a mathematical structure, usually
a subset of a monoid of words, in which a grammar and axiom systems define what
is meaningful in this language. [10].

5.2. In formal language theory, a language is a set of strings over some finite alpha-
bet A. There is an operation on the set of string, which is concatenation. This is
associative. Together with the zero element, the empty string, one has a monoid.
A linear order on A defines a lexicographical ordering on the language. If we look
at language encoded on a quipu however, then the monoid structure is gone. There
are algebraic operations on graphs, like disjoint union or joins (which both can serve
as additions) or product operations which complement them rendering the category
of finite simple graphs into rings, but these structures have no meaning in language.
The addition of strings to a quipu needs more information as strings can be attached
in different ways.

5.3. Communicating with knots is a completely different approach to writing. The

sentences are not elements in a monoid because there is a spacial nonlinear ap-
proach. One can encode a quipu as a weighted graph, where the nodes are the
knots, which are labeled by the value of the knot, the spin or attachment direction.
The edges can be equipped with color, ply direction and hierarchy data too. Num-
bers are encoded using three different type of knots, but they can also be arranged
in different ways leading to more information content than anticipated.

6. Nonlinear languages
6.1. But having content written down in a linear narrative way is not unique. This
has also be developed by other cultures. We use pictures for example to represent
mathematical statements, we use tables represent data, we use graphs to represent
relations, mind maps are examples of graph information containers which are non-
linear. In our time of electronic documents, we can add a parameter “detail” to a
mathematical text. Varying the detail level allows then to zoom in and out in the
knowledge landscape, similarly as we do when we look at a map of the earth. A
map is a highly non-linear representation of data.

Mathematical notation
Axioms of Mathematics

Linguistics Grammar
Quechua
Data structure
Quechua

Mathematics Knots
Peirce

Symbiotics Graphs
Saussure

Khipu
Inkas
Graph Database
History

Relational Database Database

Statistics
Americas
Quipu Database
Inka Code
Management Antropology
Cryptology

Language Crib based attacks

Rosetta moment

Figure 5. A mind map as an example of a non-linear language.

7. Popular culture
7.1. Intersections of linguistic and mathematics appear frequently in pop culture.
The reason is that mathematics related to linguistics is more approachable than
mathematics related to algebraic structures. We can mention the novels of Dan
Brown, in which a Harvard symbiologist Robert Langdon is the hero. The field of
 Inka quipu

Symbology does not exist. We should also mention the movie “Contact”, in which
an alien language is broadcast from an other planetary system to us. The decoding
of the language needed spacial insight as it was a three-dimensional document. Also
remarkable is the movie “Arrival” in which a linguist and physicist work together
to get access to a strange smoke ring based language spoken by two aliens “Abbot
and Castello”.

Figure 6. Linguistic in pop culture: Arrival and the Dan Brown

story, Inferno and Contact.

8. Topological writing
8.1. First of all, the mathematical approach of the quipu is unique. It is a three
dimensional writing, dealing with topological objects known as knots and links which
are of interest to mathematicians, and physicists. For a computer scientist it is a
graph database. Using spacial, material and color information, the Inkas have
placed information onto the strings. An introduction about this fascinating topic is
[20].
9. Seeing real samples
9.1. The museum of fine arts in Boston currently has an exhibit showing off some
of the quipus from the Peabody museum at Harvard. While quipu are always men-
tioned in the context of the origins of number systems, there had been much progress
recently in understanding more about these Inka code. This and some art installa-
tion must have prompted the exhibit at the museum.
 Figure 7. Photos of quipus from the exhibit at the Boston museum
of fine arts.

10. Mathematical notation

10.1. The quipu research sheds light onto the origins of mathematical notation, the
origins of number systems and even the philosophy of mathematics [17]. Talking
knots are a highly original approach to language and naturally are of extreme interest
in linguistics, semiotics, sociology and anthropology. There are also relations to
education because the way human cultures develop mathematics is similar to how
students acquire it. Recent battles about notation syntax (the PEMDAS wars)
illustrate how “antropological” mathematical notation is: computers and humans
empirically disagree with reading mathematical content like 6/2(1+2). Most humans
get 1 while computers get 9. The syntax laws are ambiguous [3]. The PEMDAS
wars are silly because it is a battle in a realm where no consensus has been built by
authority. It is a heated battle because there are some, who religiously defend their
own interpretation of mathematical syntax.

11. Management
11.1. An efficient record keeping system was necessary for building the Inca em-
pire. Engineering projects involved calculations, recording of data, calculating ratios
and proportions. As the largest empire in the pre-Columbian new world between
Ecuador and Columbia, it stretched 5000 km along the Andes. The Tawantin-
suyu, “the four parts divided together” or “land of the four quarters” as it was
called covered a complex environment reaching deserts, the Andes or the Amazon.
 Inka quipu

11.2. The quipu story leads to insight in management and organization theory [20].
The Inka empire which lasted only for a short time (1400-1532) was able to develop
and run effectively because of technology. (Of course also using military force but it
appears that the power of organization can complement military conquests. This has
been proven to be true even up to very recent times. Failed military adventures failed
at providing management.) The quipu technology was essential for management
and administration of such a complex structure. In some sense it must have enabled
progress similarly as the modern internet now does: the Inka road network [19, 1]
compares to the internet backbone and the quipu are the files.
11.3. To conclude, the story of the Inkas is an allegory for our time: investement
in infrastructure, in language, in organization can be as powerful as military power.
It is modern because in our time, power is also more and more established by entities
which know how to gather process and understand information.
12. Genetic code
12.1. Interestingly, the Inkas stored information similarly than our genetic code
is stored, on knots: our DNA consists of strands of twisted DNA molecules while
quipus use twisted rods. It appears that modern bioinformatics is getting inspired
by quipus [16]. Color encodings of electronic parts like resistors encode numbers in
colors.
12.2. Mind-boggling is also that the Inkas used binary encoding [18] using spinning,
plying and knot directionality and the markednes theory in linguistic. Binary steps
were also done in Chinese hexagrams where the binary encoding had six bits (26 =
64).
12.3. Binary symmetries appear also in biology and modern physics, where chi-
rality and parity are important. In physicist it is the weak force which shows an
asymmetry, in biology it is the orientation of the DNA, which is dominant. There
is also a left handed Z-DNA. In the Urton terminology of markedness, this would
be the marked version.
 Figure 8. DNA and Quipu both have orientations.

13. Graph Database

13.1. Organization through inka decimal administration, required time account-
ing, census data to be organized on quipu by quipukamayuq (professional quipu
writers), who had an information technology (IT) structure with a consolidated
database in Cuzco, a prerunner of the “cloud”. The cloud is just a modern word for
a decentralized “main frame”. If one would compare the quipu with files of modern
computing, the analogue of the internet was realized by Chaskis, the quipu runners.

13.2. In our time of information technology, we deal a lot with three dimensional
physical space: we have augmented reality, computer vision, 3D scanning or
3D printing technologies. Usual writing is two dimensional. The quipu system
therefore appears very modern. Our own genetic code is encoded on knotted de-
vices, the DNA, we use graph based databases, like Neo4J. Graph databases are
an alternative to relational databases. They appear to be superior if the data
structures are complicated.
 Inka quipu

14. The Unix philosophy

14.1. But one does not have to go far. One of the most popular databases used is the
Unix file system, which organizes information in a tree. This technology allows
comfortably to work with a half a dozen tera bytes of data at the finger tips and
split different things into different tree branches (directories). This smaller quipu
project (which occupied me over a few weeks) is an independent tree in my Unix
database. Like every course, every website I maintain, my library with thousands
of electronic documents, programming parts etc, they are all comfortably separated
and organized like on a quipu.
14.2. One of the most important insights could be that Like the Unix file system,
the quipu database system is a paradigm. It is a gem in simplicity and
efficiency and very close to the UNIX idea. This principle was adopted also for
our AI experiment of 2003 [9], where the AI bot was just a Unix file system and
the intelligent agent just parses a sentence the travels the file system to do things
as the nodes of the file system can be programs or little scripts which look things
up. One advantage of this quipu way is that it is highly scalable. The industry
uses it even with peta bytes of data while any conventional data base would get
challenged. Extending a quipu data base is very easy, just add an other strang of
nodes or produce more subsidary nodes. Similarly, a Unix file system can virtually
have unbounded capacity.

15. The problem of backup

15.1. The only limitation in scale is the size of the harddrives. I personally currently
have my files on 5TBytes external drives which are then stored in a frozen and of
course encrypted form also in different locations (as the Inkas did). There are
currently about 8million files there. They can be tiny text or program fragments, or
larger documents like books, pictures or movies. But I would not store this in the
cloud as one can also learn from history. The most obvious one is that services and
companies die or change their focus, cutting off things which are no more profitable.
Companies are no charity. The Inkas saved things in the “cloud” which was then
their main capital “Cusco”. And we all know what happened when the Spaniards
invaded the place. Many major databases were destroyed and less than 1000 quipus
survived.
15.2. My own data would even survive if Boston would be annihilated by a nuclear
catastrophe (the analogue of a colonialization disaster) or all cloud services would
have bit the dust. [One can easily imagine scenarios in which they could disappear
in the near future. Examples are CPU leaking concerns, lawsuits due to copy rights
or then that companies running the business will simply die or forced by some rogue
government to make things accessible.] In the past, the surviving quipu were stored
and backed up in hidden decentralized places. Unfortunately for us, we can not read
most of the non-numerical data. The Inkas somehow used to encrypt things (even
so this had not been the main intention it had the advantage of some privacy as the
quipus contained what we would call today bank information or services owned).
Also this is a lesson: never store information in a form which is not accessible by
simple tools for which public domain or at least open source tools exist to read it.
15.3. My own small quipu project is a small branch in a bigger Unix tree of my
work stations (which are synced regularly). My “quipu pendant string” contains
only 500 MBytes of data currently but it includes scanned books, documents, the
Harvard khiup database, articles and pictures as well as texts about quipu. If in
future, more things should appear, I would add it as “subsidiaries” as the Inkas did
when adding more information to a primary cord. I have absolutely no problem to
find things like that as it is in of of the 3 major project branches in my Unix file
hierarchy. It is nice to see that this simple but efficient storage paradigm is actually
Inka technology.
16. Reversed Polish Notation
16.1. An important feature of the Quechua language is agglutination, which al-
lows that operators often can be found at the end. Like ”ni=I” appears at the
end of Runasimi-ta yacha-ku sa-ni. (People language, learn, now, I) or Oliver,
Wasi-Ta ruwan which translates as Oliver house builds. [8]. Despite that linguists
call Quechua a SOV language (Subject, Object, Verb), the aggultinative part makes
it possible to put a subject suffix and have the subject at the end.
16.2. This reminds of the reverse polish notation RPN (still used in stack oriented
programming languages like Poscript or Bibtex). One sees also reverse order in
numbers like Quepchua: 13, “ten, possessor of three”, while we say “thirteen”. In
a stack oriented language, you say 23x rather than 2x3 =. We don’t need
the equal sign. Operators come to the end, which is more efficient and does not
need equal signs. So, it appears that at least for addition and multiplication, no
computing device is needed. And unlike for pebbles (bad for transportation) and
tally sticks (we can not subtract), the computation with knots can do that.
16.3. The advantage of RPN is also that no brackets are needed. We use the RPN
often when doing quick computations. For example, to compute the sum of the
squares of the square roots of the first 100 primes, one can use the RPN notation
in Mathematica: Range[100] // Prime //Sqrt //N // Total which has the
advantage that I see in each step what has been computed. The traditional (written
way) is to write: Total[N[Sqrt[Prime[Range[100]]]]] which gives just the end
result but requires to write a nested sequence of brackets.
 Inka quipu

Figure 9. Non-RPN and RPN calculators.

17. Semiotics
17.1. The Swiss Ferdinand de Saussure (1857-1913) was a pioneer in linguistic
and semiotics. Saussure was eclipsed vastly both in scope and originality by his
contemporary Charles Sanders Peirce (1939-1914) who only later would be rec-
ognized as one of the greatest thinkers and philosophers of his time. Frank Salomon
suggests the quipus reference system to be a general purpose semasiography [15].
Semasiographic signs were present in multiple Andean systems.
17.2. Highly successful and persistent non-phonetic scripts are not only used in
math notation (figures or combinatorial diagrams like Dynkin or Ferrers diagrams
or commutative diagrams) but also in physics (Feynman diagrams for example), they
also are common in music notation, programming flow charts, chemical formulas,
choreographic notation, and knitting and weaving codes.
17.3. Notation is important in mathematics and it is linked to mathematics itself:
Barry Mazur was cited in ”Enlightening Symbols” [12] that A seemingly modest
change of notation may suggest a radical shift in viewpoint. Any new notation may
ask new questions. This also applies to the quipu language. It is a completely
new angle to the origin of mathematical language and illustrates the richness and
diversity with which the art of expressing mathematical thought has begun.
 Figure 10. Two pioneers in linguistics: Ferdinand de Saussure
(1857-1913) and Charles Sanders Peirce (1839-1914).

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 Inka quipu

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