Archive for May 5th, 2009

The Actual Dresden Codex

May 5, 2009

dresden_codex_p09Happy Cinco de Mayo!    A year ago we celebrated the day here by talking about some Cinco de May math, branching out into Spanish Colonial Mathematics [including a really cool multiplication technique called The Method of the Cup].

But that was a pretty broad geographical region, so today it’s back to Mexico, with Guatemala, Belize, and maybe a little bit of Honduras and El Salvador thrown in.  (Aside:  I often make students learn where countries are when I teach The History of Mathematics, and a few times I’ve had them memorize all the countries of Central America for the first test.   One year I had the great idea that I’d test them on a map with no political boundaries, reasoning that there is some value in being able to identify countries by other landmarks.   Although I’d warned them about this, it turned out to be a truly awful test to grade — some people started off wrong, so while the countries were right in relation to each other they were all either too far north or too far south, and figuring out partial credit was a bear. I’ve never given a test like that again.)

So anyway, here’s the region we’re talking about:

Yep, it’s the Mayas!  I’ve seen a lot of math books that talk about Mayan mathematics, because it’s pretty straightforward (until it isn’t).

A dot means 1:  •
A line means 5:  ______
And a shell means zero.

The Mayans used a modified Base 20 system, so that the group of numbers at the bottom counted the 1s, the next group up counted the 20s, the next group up counted the 360s [and yes, it should really be 400 because twenty 20s is 400, but for some reason they used 360 here.  I’ve seen speculation as to why — it might have something to do with the fact that there are just over 360 days in a year — but no one knows for sure], the next group up counted the 7200 [because that is twenty 360s], the next group up counted the 144000 [because that is twenty 7200s], etc.

But the point of this post is that because the Internet is such an amazing place, you can see images of the actual original sources that use this!  Sadly, there aren’t that many:  most of the writings were destroyed, and only about four codices remain.  Codices are folded books, like this one (from the St. Andrew’s web site).

This is the Dresden Codex, and it’s about a thousand years old.  It was made of Amatl paper [fig bark covered with a lime paste], and although the picture above is in black and white it was actually done in color.  Indeed, when it came to numbers the groups (1, 20s, 360s, etc) were written in alternating black and red so that it was easy to tell where one group ended and other began.

dresden_codex_annotated1Look at that number in the lower left-hand corner.   There are 7 (1s), then 1 (20), 4 (360s), 6 (7,200s), 3 (144,000s), 11 (2,880,000), 3 (57,600,000s), and 6 (1,152,000,000s), giving a sum of over 7 billion.  It’s a pretty big number, and only uses 2 symbols [since the zero wasn’t  necessary in this case].

When I started teaching about Mayan numbers, there weren’t many pictures of the Dresden Codex available so I made do with a photocopy of the one of the pages.  But the Internet is an amazing place, and now you can actually see the entire thing in color in not one but two places!

The first place I found it is this site (archaeoastronomie).  It has the best pictures; my favorite for looking at numbers is page 51 and the pages around it [the zooming is really impressive, although it doesn’t necessarily zoom to the page you want].

And more recently there’s the Foundation for Advancemet of Mesoamerican Studies, Inc. They also provide downloadable .pdf files here [on the order of 100 Mb, but the download page offers the option of downloading in sections].  Hooray for the Internet!