Mayan mathematics
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Hernán Cortés, excited by stories of the lands which Columbus had
recently discovered, sailed from Spain in 1505 landing in Hispaniola
which is now Santo Domingo. After farming there for some years he sailed
with Velázquez to conquer Cuba in 1511. He was twice elected major of
Santiago then, on 18 February 1519, he sailed for the coast of Yucatán
with a force of 11 ships, 508 soldiers, 100 sailors, and 16 horses. He
landed at Tabasco on the northern coast of the Yucatán peninsular. He
met with little resistance from the local population and they presented
him with presents including twenty girls. He married Malinche, one of
these girls.
The people of the Yucatán peninsular were descendants of the ancient
Mayan civilisation which had been in decline from about 900 AD. It is
the mathematical achievements of this civilisation which we are
concerned with in this article. However, before describing these, we
should note that Cortés went on to conquer the Aztec peoples of Mexico.
He captured Tenochtitlán before the end of 1519 (the city was rebuilt as
Mexico City in 1521) and the Aztec empire fell to Cortés before the end
of 1521. Malinche, who acted as interpreter for Cortés, played an
important role in his ventures.
In order to understand how knowledge of the Mayan people has reached us
we must consider another Spanish character in this story, namely Diego
de Landa. He joined the Franciscan Order in 1541 when about 17 years old
and requested that he be sent to the New World as a missionary. Landa
helped the Mayan peoples in the Yucatán peninsular and generally tried
his best to protect them from their new Spanish masters. He visited the
ruins of the great cities of the Mayan civilisation and learnt from the
people about their customs and history.
However, despite being sympathetic to the Mayan people, Landa abhorred
their religious practices. To the devote Christian that Landa was, the
Mayan religion with its icons and the Mayan texts written in
hieroglyphics appeared like the work of the devil. He ordered all Mayan
idols be destroyed and all Mayan books be burned. Landa seems to have
been surprised at the distress this caused the Mayans.
Nobody can quite understand Landa's feelings but perhaps he regretted
his actions or perhaps he tried to justify them. Certainly what he then
did was to write a book Relación de las cosas de Yucatán (1566)
which describes the hieroglyphics, customs, temples, religious practices
and history of the Mayans which his own actions had done so much to
eradicate. The book was lost for many years but rediscovered in Madrid
three hundred years later in 1869.
A small number of Mayan documents survived destruction by Landa. The
most important are: the Dresden Codex now kept in the Sächsische
Landesbibliothek Dresden; the Madrid Codex now kept in the American
Museum in Madrid; and the Paris Codex now in the Bibliothèque nationale
in Paris. The Dresden Codex is a treatise on astronomy, thought to have
been copied in the eleventh century AD from an original document dating
from the seventh or eighth centuries AD.
The Dresden codex:
Knowledge of the Mayan civilisation has been greatly increased in the last thirty years (see for example [3] and [8]).
Modern techniques such as high resolution radar images, aerial
photography and satellite images have changed conceptions of the Maya
civilisation. We are interested in the Classic Period of the Maya which
spans the period 250 AD to 900 AD, but this classic period was built on
top of a civilisation which had lived in the region from about 2000 BC.
The Maya of the Classic Period built large cities, around fifteen have
been identified in the Yucatán peninsular, with recent estimates of the
population of the city of Tikal in the Southern Lowlands being around
50000 at its peak. Tikal is probably the largest of the cities and
recent studies have identified about 3000 separate constructions
including temples, palaces, shrines, wood and thatch houses, terraces,
causeways, plazas and huge reservoirs for storing rainwater. The rulers
were astronomer priests who lived in the cities who controlled the
people with their religious instructions. Farming with sophisticated
raised fields and irrigation systems provided the food to support the
population.
A common culture, calendar, and mythology held the civilisation together
and astronomy played an important part in the religion which underlay
the whole life of the people. Of course astronomy and calendar
calculations require mathematics and indeed the Maya constructed a very
sophisticated number system. We do not know the date of these
mathematical achievements but it seems certain that when the system was
devised it contained features which were more advanced than any other in
the world at the time.
The Maya number system was a base twenty system.
Here are the Mayan numerals.
Almost certainly the reason for base 20 arose from ancient people who
counted on both their fingers and their toes. Although it was a base 20
system, called a vigesimal system, one can see how five plays a major
role, again clearly relating to five fingers and toes. In fact it is
worth noting that although the system is base 20 it only has three
number symbols (perhaps the unit symbol arising from a pebble and the
line symbol from a stick used in counting). Often people say how
impossible it would be to have a number system to a large base since it
would involve remembering so many special symbols. This shows how people
are conditioned by the system they use and can only see variants of the
number system in close analogy with the one with which they are
familiar. Surprising and advanced features of the Mayan number system
are the zero, denoted by a shell for reasons we cannot explain, and the
positional nature of the system. However, the system was not a truly
positional system as we shall now explain.
In a true base twenty system the first number would denote the number of
units up to 19, the next would denote the number of 20's up to 19, the
next the number of 400's up to 19, etc. However although the Maya number
system starts this way with the units up to 19 and the 20's up to 19,
it changes in the third place and this denotes the number of 360's up to
19 instead of the number of 400's. After this the system reverts to
multiples of 20 so the fourth place is the number of 18 × 202, the next the number of 18 × 203 and so on. For example [ 8;14;3;1;12 ] represents
12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203 = 1253912.
As a second example [ 9;8;9;13;0 ] represents
0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 =1357100.
Both these examples are found in the ruins of Mayan towns and we shall explain their significance below.
Now the system we have just described is used in the Dresden Codex and
it is the only system for which we have any written evidence. In [4]
Ifrah argues that the number system we have just introduced was the
system of the Mayan priests and astronomers which they used for
astronomical and calendar calculations. This is undoubtedly the case and
that it was used in this way explains some of the irregularities in the
system as we shall see below. It was the system used for calendars.
However Ifrah also argues for a second truly base 20 system which would
have been used by the merchants and was the number system which would
also have been used in speech. This, he claims had a circle or dot
(coming from a cocoa bean currency according to some, or a pebble used
for counting according to others) as its unity, a horizontal bar for 5
and special symbols for 20, 400, 8000 etc. Ifrah writes [4]:-
Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.
Let us say a little about the Maya calendar before returning to their
number systems, for the calendar was behind the structure of the number
system. Of course, there was also an influence in the other direction,
and the base of the number system 20 played a major role in the
structure of the calendar.
The Maya had two calendars. One of these was a ritual calendar, known as
the Tzolkin, composed of 260 days. It contained 13 "months" of 20 days
each, the months being named after 13 gods while the twenty days were
numbered from 0 to 19. The second calendar was a 365-day civil calendar
called the Haab. This calendar consisted of 18 months, named after
agricultural or religious events, each with 20 days (again numbered 0 to
19) and a short "month" of only 5 days that was called the Wayeb. The
Wayeb was considered an unlucky period and Landa wrote in his classic
text that the Maya did not wash, comb their hair or do any hard work
during these five days. Anyone born during these days would have bad
luck and remain poor and unhappy all their lives.
Why then was the ritual calendar based on 260 days? This is a question
to which we have no satisfactory answer. One suggestion is that since
the Maya lived in the tropics the sun was directly overhead twice every
year. Perhaps they measured 260 days and 105 days as the successive
periods between the sun being directly overhead (the fact that this is
true for the Yucatán peninsular cannot be taken to prove this theory). A
second theory is that the Maya had 13 gods of the "upper world", and 20
was the number of a man, so giving each god a 20 day month gave a
ritual calendar of 260 days.
At any rate having two calendars, one with 260 days and the other with
365 days, meant that the two would calendars would return to the same
cycle after lcm(260, 365) = 18980 days. Now this is after 52 civil years
(or 73 ritual years) and indeed the Maya had a sacred cycle consisting
of 52 years. Another major player in the calendar was the planet Venus.
The Mayan astronomers calculated its synodic period (after which it has
returned to the same position) as 584 days. Now after only two of the 52
years cycles Venus will have made 65 revolutions and also be back to
the same position. This remarkable coincidence would have meant great
celebrations by the Maya every 104 years.
Now there was a third way that the Mayan people had of measuring time
which was not strictly a calendar. It was an absolute timescale which
was based on a creation date and time was measured forward from this.
What date was the Mayan creation date? The date most often taken is 12
August 3113 BC but we should say straightaway that not all historians
agree that this was the zero of this so-called "Long Count". Now one
might expect that this measurement of time would either give the number
of ritual calendar years since creation or the number of civil calendar
years since creation. However it does neither.
The Long Count is based on a year of 360 days, or perhaps it is more
accurate to say that it is just a count of days with then numbers
represented in the Mayan number system. Now we see the probable reason
for the departure of the number system from a true base 20 system. It
was so that the system approximately represented years. Many
inscriptions are found in the Mayan towns which give the date of
erection in terms of this long count. Consider the two examples of Mayan
numbers given above. The first
[ 8;14;3;1;12 ]
is the date given on a plate which came from the town of Tikal. It translates to
12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203
which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.
The second example
[ 9;8;9;13;0 ]
is the completion date on a building in Palenque in Tabasco, near the landing site of Cortés. It translates to
0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203
which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.
We should note some properties (or more strictly non-properties) of the
Mayan number system. The Mayans appear to have had no concept of a
fraction but, as we shall see below, they were still able to make
remarkably accurate astronomical measurements. Also since the Mayan
numbers were not a true positional base 20 system, it fails to have the
nice mathematical properties that we expect of a positional system. For
example
[ 9;8;9;13;0 ] = 0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 = 1357100
yet
[ 9;8;9;13 ] = 13 + 9 × 20 + 8 × 18 × 20 + 9 × 18 × 202 = 67873.
Moving all the numbers one place left would multiply the number by 20 in
a true base 20 positional system yet 20 × 67873 = 1357460 which is not
equal to 1357100. For when we multiple [ 9;8;9;13 ] by 20 we get 9 × 400
where in [ 9;8;9;13;0 ] we have 9 × 360.
We should also note that the Mayans almost certainly did not have
methods of multiplication for their numbers and definitely did not use
division of numbers. Yet the Mayan number system is certainly capable of
being used for the operations of multiplication and division as the
authors of [15] demonstrate.
Finally we should say a little about the Mayan advances in astronomy. Rodriguez writes in [19]:-
The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.
The Mayans carried out astronomical measurements with remarkable
accuracy yet they had no instruments other than sticks. They used two
sticks in the form of a cross, viewing astronomical objects through the
right angle formed by the sticks. The Caracol building in Chichén Itza
is thought by many to be a Mayan observatory. Many of the windows of the
building are positioned to line up with significant lines of sight such
as that of the setting sun on the spring equinox of 21 March and also
certain lines of sight relating to the moon.
The Caracol building in Chichén Itza:
With such crude instruments the Maya were able to calculate the length
of the year to be 365.242 days (the modern value is 365.242198 days).
Two further remarkable calculations are of the length of the lunar
month. At Copán (now on the border between Honduras and Guatemala) the
Mayan astronomers found that 149 lunar months lasted 4400 days. This
gives 29.5302 days as the length of the lunar month. At Palenque in
Tabasco they calculated that 81 lunar months lasted 2392 days. This
gives 29.5308 days as the length of the lunar month. The modern value is
29.53059 days. Was this not a remarkable achievement?
There are, however, very few other mathematical achievements of the Maya. Groemer [14]
describes seven types of frieze ornaments occurring on Mayan buildings
from the period 600 AD to 900 AD in the Puuc region of the Yucatán. This
area includes the ruins at Kabah and Labna. Groemer gives twenty-five
illustrations of friezes which show Mayan inventiveness and geometric
intuition in such architectural decorations.
References (26 books/articles)
Other Web sites:- Kevin Brown (Mayan numeration)
- Rhonda Robinson (Mayan numbers)
- Mayan World Study Center (Mayan Mathematics)
- Mayan World Study Center (The Mayan calendar)
- Michiel Berger (Mayan Astronomy)
- B and V Böhm(The Dresden codex)
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Mayan_mathematics.html
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