buried mirror: latest reflections

mesoamerica and the maya world

Category: mathematics

Maya multiplication

I don’t know anything about this. Is it really Maya?


Adding up the bones

and arms, and hearts, and hands, and arrows . . .

Geographer Barbara Williams and mathematician Maria del Carmen Jorge y Jorge have, after three decades of labor, deciphered an Aztec code used to calculate the areas of land plots.

The Aztecs needed to calculate the area of irregular shaped parcels of land for tax purposes. Their calculations were recorded in two books, the Codex Vergara and the Codice de Santa Maria Asuncion, in which older documents written on tree bark or cotton cloth were presumably transcribed onto paper brought by Spanish conquistadors. According to an article by Alan Zarembo in the L. A. Times,

The pages of the books are filled with tiny property maps. For each plot, there are two drawings — one showing the lengths of the sides and another showing the area. The measurements are represented by seven symbols: lines, dots, arrows, hearts, hands, arms and bones. Each map also includes the name of the property owner and the soil type.

Researchers already knew what each map represented and the value of some of the measurements. A line, for example, was the standard unit of length, which was known as a tlalquahuitl, or rod, and in modern units would measure a little more than 8 feet.

When the researchers knew the values of the units in roughly rectangular plots, they could easily follow the logic of the Aztecs and reproduce their calculations by multiplying lengths and widths.

But they were stymied in calculating many plots because they didn’t know the value of the units. The breakthrough came when Jorge y Jorge, a professor at the National Autonomous University of Mexico, found that the values of some areas were prime numbers.

So now we know, a hand equaled 3/5 of a rod, an arrow was 1/2 , a heart was 2/5 , an arm was 1/3 , and a bone was 1/5.

But I would like to know more about how the Aztecs classified soil types.


Maya Architecture and the Golden Mean

the golden mean at copan honduras

Mathematicians define the golden section as a relation in which the smaller unit is to te larger unit as the larger is to the sum. In other words, a:b = b:(a+b). The name for this relation is phi. Its numeric value is 1.618034. Phi is an interesting number. If you add 1 to it you get its square. If you subtract 1 from it, you get its reciprocal (1/phi). If you keep multiplying it by itself you get an infinite series that retains the phi proportion.

The thirteen-century Italian mathematician Leonardo Fibonacci discovered that the phi proportion often manifests itself in nature as a spiral of increase found in snails, seashells, pinecones, and so forth. And so, it is said, did Maya mathematicians.

One element contributing to the beauty of Maya architecture may be its use of the golden section. (In the image above, I have drawn an approximate golden section over the opening atop the east court at the Maya ruins of Copan in Honduras.) Of course, we must beware of what a professor of mine called “the blueberry principle” — if you are out gathering blueberries you tend not to notice anything else, and you tend to see blueberries wherever you look. If we go looking for the golden section, we are likely to find it. But does this mean the Maya consciously employed it?

A researcher named Christopher Powell concluded that the answer to this question is “yes.” He says that the fundamental shape of Maya geometry is the golden section, and that the Maya composed such sections using a procedure that is brilliant in its simplicity. Using a cord, it is easy to construct a square. If the cord is doubled back on itself it obviously becomes half the length, and that halved cord can be used to find the midpoint of one of the sides of the square. Next, if the cord is placed on the midpoint and extended to one of the opposite corners, it can be swung like a compass in an arc that will define the length of a golden section, from which the final rectangle can be constructed.

Powell observed modern Yucatec Maya using this very technique. He was told that the use of the cord makes houses that are like flowers because of the relations of their proportions. His theory appears to have been confirmed by red marks that remain on some structures at Copan and Tikal and suggest sizing via this cord method.

In the Popul Vuh it is written that gods used the following method to lay out the cosmos:

Its four sides
Its four cornerings
Its measurings
Its four stakings

Its doubling-over cord measurement
Its stretching cord measurement
Its womb sky
Its womb earth
Four sides
Four corners as it is said


  • Linda Schele and Peter Mathews, The Code of Kings (Scribner)
  • Alan Christenson, tr., The Popul Vuh, translation adapted by Schele and Mathews
  • See also the Dennis Tedlock version of the Popul Vuh with his comment on this section in which he says that it is based on the cord approach to layout, and he reports that a source informed him that the passage “describes the measuring out of the sky and earth as if a cornfield were being laid out for cultivation.”
  • Christopher Powell

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