The Mayan Mat: Mathematical Modeling of an Ancient Number Pattern
Daniel Clark Orey, Ph.D.
Professor of Mathematics and Multicultural Education
Office: (916) 278 5531 FAX: (916) 278 6643
http://www.csus.edu/indiv/o/oreyd/
Key
words / palavras-chaves: ethnomathematics / etnomátematica, mathematical
modeling / modelagem matemática, Mayan mathematics / matemática dos maias
Os conhecimentos sobre a astronomia e os calendários,
a surpreendente arquitetura e as descobertas matemás, realizadas pelos
descendentes dos povos maias que vivem na América Central, estão sendo
redescobertos e catalogados diariamente. A maioria dos estudos da matemática
maia focaliza um sistema único de numeração, utiliza o valor-posicional e o
zero e, possui um sofisticado sistema de calendário. Neste estudo, o autor
descreve a aplicação de uma metodóloga alternativa que pode ser facilmente
transferida para o sistema decimal. Em 1998, o autor esteve como professor
visitante com bolsa pela Fulbright na Pontificia Universidade Catolica de
Campinas, onde aprendeu os fundamentos da etnomatemática e da modelagem num
programa coordenado por Geraldo Pompeo Júnior.
Introduction
The
detailed discussion related to the architectural, astronomical, calendrical and
mathematical discoveries made by Mayan peoples of Meson-America is prolific as
it is well documented. Much of the work related to their mathematics has
focused on their unique numbering system and calendar. In the early 80’s as a
mathematics and English teacher in
I
was fortunate enough to come across these ideas in a book that I found in
I have since that time, had the opportunity to
take these activities and work with students and teachers in
Ethnomathematics
Ideas
for the use of ethnomathematics have been developed in many countries over many
years. However, the actual theory of ethnomathematics was first introduced by
Professor Ubiratan D'Ambrosio of
Ethnomathematics is the art or technique (techne), that is,
to explain, to understand, of playing in reality (mathema), inside of a proper
cultural context (ethno).
Ethnomathematicans readily recognize
that all cultures and all people have developed unique and sophisticated ways
to explain, to know and to modify their own reality. They recognize that these
ideas, as are the cultures that they are embedded in, are part of a natural,
constant, and dynamic process of evolution and growth. It is not the premise of
ethnomathematics to disdain non-western or non-European models or traditions,
but to consider the validity of all explanations of reality as constructed by
all peoples, cultures, and historical backgrounds. These forms of knowledge are
not considered static or dead; they are part of a process of constant mutation,
evolution, and growth, as part of their own unique cultural dynamism, and so
therefore should be studied and catalogued.
Using
an idea borrowed from cultural anthropology, no single form of problem solving
is any better or worse than any other form (Hall, 1976). Each has evolved
unique problems to resolve. Ethnomathematics is not solely mathematical; it is
broader and inclusive of other disciplines. By making use of the diverse
methods that cultures and peoples use to find explanations to increase their
own understanding of their world and time, ethnomathematics seeks to understand
these unique realities.
Alternative
forms of mathematics often come about as people’s work to explain and resolve
practical problems in their daily lives. What is universal is that all cultures
have found ways to search for knowledge. All cultures have the necessity, in
fact have found, unique ways to quantify, compare, classify, measure and
explain day to day phenomena (Borba, 1990). This is no less true about the
Mayans and their use of numbers. What is difficult for us to understand is that
the paradigm that they lived in created an entirely different mind set, which
allowed at least the elite, to use mathematics as a method for defining their
universe.
The
Mayans (Nichols, 1975, Grattan-Guiness, 1997) made use of a series of number
patterns that held a certain sacredness to them. The rattlesnake, Crotalus
durissus, is found throughout the region, and may have inspired early American
mathematics to an extent not fully appreciated or fully understood. The pattern
that is found on rattlesnake skin (forming a diamond in some species) may have
inspired much of the Mayan arts, geometry and architecture. Though
Grattan-Guiness (1997), does not appreciate the connection between the sacred
and scientific found outside of the Euro-western scientific paradigm, he says,
contemplation of this pattern may even have helped Mayan geometry in the first
place and perhaps also sacred arithmetic (p.112).
One
example that is found at
In
Mayan contexts, each number from one to nine had a sacred value, and since the
Mayan number system was based on the number 20, there is given meanings to
numbers, and the figures and patterns that they were inscribed upon. It seems
that it was may have been like poetry written in Arabic, in that symbols, words
and numbers held a meaning and significance towards other symbols, words and
numbers of similar value, adding a multidimensional aspect to their
mathematics, art and literature. In
other words, it was a form of numerology. Though Nichols is unclear as to his
sources, he left no bibliography, and being a respected North American artist
was interested primarily in the aesthetic and metaphysical value of the number
pattern he observed. He wrote that each number, from one to nine had a specific
value and meaning:
- God, Goddess
- The Maker,
Parents
- The Created,
Life
- Venus, called
Kulkulcan
- The Priest,
The Hand of God
- Life and
Death
- God and
Divine Power
- Body and Soul
- The Nine
Drinks
What
we do know is that these patterns were carved into stone, worn as jewelry, and
woven into cloth, and are fairly universal in much of Mayan art and culture
found up to this day. Many of these patterns can still be seen in the weaving
still done by the Mayan people of
Mayans
considered certain patterns, similar to magic squares, called mats, as sacred.
The Mayans designed their mats in many patterns; their craftsmen wove and
carved these patterns into stone and cloth. The patterns became known for their
specific numbers, power and significance. For example, Nichols thought that
whatever a person saw an X or a XX pattern the person was able to decode a
certain message. Numbers are used consecutively; staring at the top left
corner. For example:
4
= 5
X
3
= 5
10
= 1+0 = 1 (The Goddess)
Further
study of the mats led me to experiment; seeking the solution for numbers one to
nine for several mat patterns (Appendix I & II), each one X wide. The
ensuing patterns have been found quickly enough, there is more than one
possibility for some mats and none at all for others. Most importantly, many
learners who have been unmotivated with basic drill have been interested and
successful at these activities, eagerly finding all the possible combinations
for each number through trial and error and practice. The time spent discussing
and filling in the charts and activity sheets has led to a number of further
discoveries with numbers not used by Mayans?
For example: What happens when negative numbers (something unknown of by
the ancient Mayans) are used? A sequence of -5, -4, -3, -2, -1, 0, 1, 2, 3, 4,
5 led to a great deal of interest for a group of fifth and sixth graders.
Sculptures
found in
Motifs
of many kinds, birds, animals, flowers, and significant abstract designs conform
to a grid pattern, another type of mat, which, undoubtedly, also was
sacred. Whether or not we believe that
any form of counting results in a subliminal sense of pleasure because,
theoretically such act is a ritual of universal attunement, we have only to
observe the serene look in the face of a Mayan weaver to realize that the act
of weaving is, indeed, pleasurable. Why do we tend to tap our foot to a lively
tune? It is it because it is rhythmical? Transposed mathematics? Cant’ we
associate weaving with music? Whatever we may answer, the fact is that Mayan
weavers count, and various numbers create their designs (Nichols, 1975, p.5).
One
legend from
By
weaving, people are still able to integrate the magic into their daily and
ceremonial dress. Mats not only have numerical values but so do certain animal
patterns. Here is an example of a bird that has a numerical equivalent, and is
a very popular design.
<
Insert Mayan bird picture here >
Aztec
and Mayan pyramid steps are both step and shallow at the same time. The easiest
way in which to ascend or descend many of these pyramids is to climb or descend
in a diagonal or zigzag fashion that was in the same form as the mats, and
found on the skin of a rattlesnake. The criss-cross pattern on its
(rattlesnake) skin may have inspired the Mayan geometry and their architecture
(Grattan-Guiness, 1997). Nichols felt that this was evidence that the priests
ascended or descended in patterns that looked like various designs that we
explored on the on X sheets. If we look closely at the photo of the
(Insert
picture of
Suggestions
for Classroom Use
Students
work cooperatively in small groups or pairs at this activity. I began the
activity by showing learners a map of
Examples
are given to students to find. A large (but empty) example of the sheet in
Appendix II, either on the board or overhead. As a class we could fill in at
least a couple of cells. Then the students are encouraged to go ahead and work
as a group to complete the activity. After the matrix is filed in the class as
whole compares and contrasts findings. There will be a number of cells that
have more than one answer, a few cells where there are only one answer, and a
few where as far my students have not been able to find answers. I hope that
you will enjoy
Conclusion
/ Summary
Where
is the math here? This activity has the following properties, related to
algebra found in the one by two patterns which p missing solutions in a
pattern.
In
order to get a cross section of 1 0r 2, we have these choices 1, 2, 10, 11, 20.
Setting algebraic sum equal to each
possibility, we have:
3(2B+5A)
= 1, becoming 2B+5A = 1/3;
3(2B+5A)
= 2, becoming 2B+5A = 2/3;
3(2B+5A)
= 10, becoming 2B+5A = 10/3;
3(2B+5A)
= 11, becoming 2B+5A = 11/3; and finally
3(2B+5A)
= 20, becoming 2B+5A = 20/3.
Each
equation therefore has no solution of A, B, N.
SAMPLE ACTIVITY – MAYAN MAT PATTERNS
Framework Strands: Number, Geometry, Patterns/Functions, Algebra
Objective: Each student pair will successfully complete ultimate values matrix
The Mayan people use a series of sacred number patterns. Priests were responsible for keeping all the knowledge, spiritual, or scientific. Indeed the delineation was far less discrete as it is in our time. They considered certain patterns, similar to magic squares in many ways, called mats, as sacred. Each number, from one to nine had a specific value and meaning:
1. God, Goddess
2. The Maker, Parents
3. The Created, Life
4. Venus, called Kulkulcan
5. The Priest, The Hand of God
6. Life and Death
7. God and Divine Power
8. Body and Soul
9. The Nine Drinks
Patterns were carved into stone, worn as jewelry, and woven
into cloth. Many of these patterns can still be seen today in the colorful
weaving still done by the people of
X X X X
X X X
X X
X
Activity Sheets: Students may choose consecutive number patterns (i.e. 1, 2, and 3). But other number patterns work just as well, for example counting by even numbers, or odd, or by fives.
MAT
PATTERNS – ONE X WIDE
PATTERN
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ULTIMATE VALUES |
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2 |
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3 |
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4 |
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5 |
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6 |
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7 |
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8 |
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9 |
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Bibliographical References
Borba, M. (1990). Ethnomathematics and education. For The
Learning of Mathematics 10(1).
D'Ambrosio, D. (1998). Ethnomathematics:
the art or technique of explaining and knowing.
_____________. Etnomatemática: Um Programa. A educação matemática: Revista da
Sociedade Brasileira de Educação Matemática (SBEM). Blumenau, Brazil. Ano 1 (PP
5- 11), 1993.
_____________. (1996). Educação
matemática: da teoria à prática.
Diaz-Bolio, J. The geometry of the
Maya and their rattlesnake art. Mérida, México (In Grattan-Guiness, 1997),
1987.
Grattan-Guiness, I. The rainbow of
mathematics: A history of mathematical sciences.
Hall, E. T. Beyond Culture.
Orey, D. "Mayan Math." The
______. Chapter. "Geometry of the Tipi
and Cone: Using Mathematical Modeling as Applied Ethnomathematics. In
Mathematics Across Culture: the History of Non-Western Mathematics. (Selin, H.
Ed.).
Nichols, D. The Lords of the Mat of