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\centerline{MONT 109N -- Mathematics Across Cultures}
\centerline{Discussion/Problem Set 1 -- Ancient Mayan mathematics from the {\it Dresden Codex}}
\centerline{February 4, 2011}
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\noindent
{\it Background}
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The ancient Mayans used a fully
positional {\it vigesimal}, or base-20, number system.  They wrote numbers
in an extremely economical way using just three different
symbols -- a round dot for 1, a longer bar for 5, and
a ``shell-shaped'' symbol for 0.  

Today, we want to look at pages from one of the four surviving 
ancient Mayan {\it codices} -- folding books made from fig tree bark,
with writing created by brush and ink on a surface prepared with a kind of plaster.
The one we will look at is called the {\it Dresden Codex}.  See the photocopy
of three of the pages attached to this sheet.   Part of what we
want to do in this assignment is to decipher and try to interpret
a portion of the contents.  

This precious record of the ancient Mayan past was created some time after
the end of the Classic period, likely copying an earlier source.  
It was one of the few Mayan books to escape being 
burned by the Spanish during the period they ruled this part of Central
America.  A major goal of Spanish colonial rule was to stamp out the cultures 
of the indigenous peoples in their territories while converting them to 
Christianity.  

The codex is kept in a museum in the German city of Dresden, hence the name.  
(By the way, the damage across the top of the pages is not ancient -- it was 
caused by water following the fire-bombing of Dresden by the Allies in WWII in 
February 1945.  Amazingly enough, the Germans cared enough about preserving this 
artifact to save it even though most of the city of Dresden was destroyed
and perhaps 25,000 civilians lost their lives in these attacks.)  

Take a look at the contents of these pages to start, and note that
a great proportion of the contents consists of numbers(!)  How
should we read them and what do they mean, though?

We are going to concentrate on the material on the {\it lower half}
of the first ({\it left-hand}) page.
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\noindent
{\it Questions -- Note:  You will start work on the following questions in
(new) discussion groups in class on February 4.  The assignment will consist,
though, of individual writeups, due in class on Friday, February 11.}
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\item{A)}  Let's start with the numbers below the two rows of
other glyphs.  Let's make the assumption that each is a three-digit
base-20 number, with the digits stacked {\it vertically}, knowing
that's the way the Mayans generally did things.  (The other possibility,
  that the digits are from base-20 numbers listed horizontally
  leads to improbably huge numbers, so we can reject that possibility,
even though other known Mayan texts did deal with truly huge numbers.)
But of course, there is
a remaining question -- {\it which is the ones digit and which is
the $20^2 = 400$'s digit?}  Do the numbers read ``up'' or ``down?''
Of course, the number in the first column is not much help here -- 
it reads {\it the same} ``up'' or ''down(!)''
But, for instance, is the second from the left to be read as
$$15 + 7 \times 20 + 11 \times 20^2 = 4555,$$ 
or
$$ 11 + 7 \times 20 + 15\times 20^2 = 6151?$$
Try reading the numbers {\it both} ways and see whether one of them looks 
more likely to you, based on some order or rudimentary pattern you might observe.
Explain how you are deciding which alternative makes more sense.
\item{B)}  One way to explore a series of numbers is to combine
or work with them arithmetically in various ways.  A very useful thing to do
 with a list of numbers in a lot of situations is to find 
{\it the differences} between successive pairs in the list.  What do you 
get if you do that here?   What could these mean?  Well, the Mayanists
who deciphered the Dresden Codex noticed something interesting.  
As we know, the {\it lunar month} 
(the time from say one new moon to the next new moon is approximately 
29.5 days).  How are the number 177 that appears among the differences 
and 29.5 related?  
\item{C)}  The successive differences from part B do not all work
out to be whole number multiples of 29.5, though.  What's going on?
To get a glimpse of something new, look at the numbers in the {\it last
two rows at the bottom of the page}.  Decipher them.  There should be 
a tantalizing but incomplete overlap between these numbers and
the successive differences from part B.
\item{D)}  To explain the discrepancies, 
we need to make one further ``correction.''  It is known that the
ancient Mayans used {\it a slightly odd system} for recording
{\it numbers of days in calendar calculations}(!)
In the notation for numbers of days, instead of a pure base 20 system, 
a modified or {\it mixed} system was used in which the digits were numbers of 
$1$'s, $20$'s, $18 \times 20 = 360$'s,  $18\times 20^2$'s, etc.  
This system is probably best explained by the fact that $18\times 20 = 360$ 
is closer to the number of days in a solar year than $20^2 = 400$ is(!)  
Since we think that the numbers here might have something to do with numbers 
of days, let's go back and recalculate the numbers using this ``modified'' base-20 
system.  For instance, the number in the second column might ``really'' be
$$11 + 7 \times 20 + 15\times 18 \times 20 = 5551.$$
Recompute all the numbers this way and find the new successive differences.
You should see a nice agreement now with the last rows of this page
of the codex.
\item{E)} The glyphs on the rows {\it in between} the rows of numbers
from the previous parts are names of particular days in
the Sacred Round or ``tzol'kin'' portion of the Mayan calendrical 
system we will look at in detail next week.  Here's how they translate top to bottom
within each column:
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\item{} Column 1: 4 Ik,  5 Akbal, 6 Kan.
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\item{} Column 2: 12 Cauac, 13 Ahau, 1 Imix.
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\item{} Column 3: 7 Cib, 8 Caban, 9 Eznab.
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\item{} Column 4: 2 Ben,  3 Ix, 4 Men.
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\item{} Column 5: 10 Oc, 11 Chuen, 12 Eb. 
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\item{} Column 6:  2 Eznab, 3 Cauac, 4 Ahau.
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What is the pattern in each column?  How are the two dates on the ``ends''
related to the one in the middle?  
\item{F)}  {\it What does the table mean?}   The page we looked at
is part of a section of the table that is thought to deal with a particular
astronomical phenomenon that has interested people of many 
different cultures.  Each of the ``middle'' dates here is correlated
with a Long Count date on which it is known that a solar or lunar {\it eclipse} (in most cases
partial, not total) occurred in the years 770 - 772 C.E. that would
have been visible in the Mayan lands.
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\item{} Solar eclipses occur at a new moon, when the moon is between the earth 
and sun and blocks out the sun's light for a certain period.  Lunar eclipses occur
similarly at a full moon when the earth is between the sun and the moon and 
appears to block out the moon's light.  If the moon's orbit was not inclined slightly to 
the plane of the earth's orbit about the sun (called the {\it ecliptic} in astronomy!)
then eclipses would occur every month (and would probably have been much 
less alarming to humans historically!).  But because of the inclination of the moon's orbit,
eclipses can only happen when the moon is crossing the ecliptic, and 
this happens twice in each lunation, at the ``ascending node'' and ``descending node.''
Now, if the moon is not ``lined up'' with the sun and the earth at that time, there
is no eclipse.  However, if the moon is near the ascending node at the time of an 
eclipse, it will be near the decending node about half a year later.  There is 
{\it a good chance that an eclipse will occur again} at the nearest new or full moon.  
This will usually be six lunar months, or approximately 177 days, after the first 
eclipse (this can be closer to 178 days in some cases).  
Several eclipses may follow at 177- (or 178-) day intervals.   The lunar
month is not exactly 29.5 days long, though, so eventually the differences build
up enough and another eclipse in the series cannot occur.  At the time of the last 
eclipse in a series, though, the closest new moon to the node may arrive after five 
lunar months, approximately 148 days.  (It is important to realize that the Earth 
is rotating while all this is going on, so the eclipses in a series like this might 
not all be visible from the same location on the Earth.) 
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\item{}  Based on this astronomical information, an
astronomer named {\it Martin Meinshausen} came up with an interpretation of what
this table in the Dresden codex probably was for the Mayans in 1913.  
By thinking about this information and your transcriptions of the 
numbers from the codex page carefully, you should be able to see what
his interpretation was.   Write a paragraph describing  
{\it Meinhausen's interpretation of what this part of the Dresden 
Codex is about.}  Start
with a clearly stated topic sentence, then explain how the calculations
you did in the previous parts and the given information about 
eclipses support your claim.  
\bye
