History of Mathematics Project | Aztec Dates from Codex Mendoza

History of
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1541

Aztec Dates from Codex Mendoza

Aztec calendrical dates using pictograms and dots

Folio 5v from the Codex Mendoza (a history of Aztec rulers, conquests and daily life compiled around 1541) depicts conquests by Itzcōhuātl, fourth king of the Aztec capital Tenochtitlan. The vertical column on the left of this page shows years in the Aztec sacred almanac calendrical system, which consisted of pictograms representing day signs plus number coefficients that cycled in a fixed order depicted using an arrangements of dots.

The Aztec sacred calendar was constructed using a sequence of 20 day names paired with year numbers running sequentially from 1 to 13. In folio 5v from the Codex Mendoza, a vertical column of years is shown with the dates running from top to bottom as 1 Flint, 2 House, 3 Rabbit, 4 Reed, down through 13 Flint. The years express the reign of Itzcōhuātl, fourth ruler of Tenochtitlan, who is depicted as the seated figure and ruled from 1 Flint (1428 CE) to 13 Flint (1440 CE). As can be seen, the year-number dots are arranged in groups of five with the exception of 11 Rabbit.

Artifact dimensions

22.7 cm × 31.0 cm (71 pages; European paper)

Artifact origin

Mexico

Catalog number

fol. 005 v, Bodleian Library MS. Arch. Selden. A. 1

Timeline

Interactive Content

Computational Explanation

The Aztec calendar contained both an annual calendar that approximated the tropical year and a sacred almanac called the tonalpohualli. The tonalpohualli was constructed from a sequence of 20 day names represented by glyphs (Crocodile, Wind, House, etc., reading sequentially downward then across to the right):

Out[]=

Crocodile	Death	Monkey	Vulture
Wind	Deer	Grass	Movement
House	Rabbit	Reed	Flint
Lizard	Water	Ocelot	Rain
Serpent	Dog	Eagle	Flower

… paired with a number from 1 to 13, where the latter was represented using a series of dots drawn around the central day-name glyph.

The tonalpohualli began with a certain glyph and number of dots and incremented the dots until 13 were reached. At that point, the calendar proceeded to the next glyph in sequence and reset the dot count to 1. When the final glyph in the list was reached with 13 dots, the calendar began again with the first glyph and a single dot. Since 13 and 20 are relatively prime (i.e. their greatest common divisor is 10):

In[]:=

GCD[13,20]

Out[]=

… they have a least common multiple of 260:

In[]:=

LCM[13,20]

Out[]=

260

As a result, the sacred calendar year consisted of 260 days.

When the annual 365-day calendar and sacred calendar were used together, the combination of annual and sacred dates repeated only after 52 years of 365 days each:

In[]:=

LCM[365,260]/365

Out[]=

The 52-year calendrical cycle was known as the xiuhmolpilli (sacred bundle) and was important in Aztec religious practice.

Rather than using the first day of a full 365-day year to designate the year, the Aztecs instead used the sacred almanac name for its 360th day.

Interestingly, as a result of the fact that the greatest common divisor of 365 (corresponding to the number of days in a year) and 20 (corresponding to the number of day-name glyphs) is 5:

In[]:=

GCD[365,20]

Out[]=

… only five possible glyphs can appear in a 365-year name. Furthermore, these must be equally spaced by five glyphs in the sequence, thus corresponding to one of the rows in the preceding table. As it turns out, the particular calendrical starting point used by the Aztecs gives the only possible year glyphs as House, Rabbit, Reed and Flint.

To construct the dot arrangements representing year numbers in the sacred almanac of the Aztecs, begin by encoding the positions of dots for each possible year number 1 to 13. While the dot positions were not necessarily completely standardized, they commonly followed a pattern close to the following. In particular, construct a list of 13 triples

(r,t,l)

in which r, t and l specify the number of dots appearing on the right, top and left, respectively, of each calendar for a given number:

In[]:=

DotData={{1,0,0},{0,2,0},{0,3,0},{4,0,0},{5,0,0},{1,5,0},{2,5,0},{3,0,5},{4,0,5},{5,5,0},{4,6,1},{5,2,5},{5,3,5}};

As an example, the last element (5, 3, 5), corresponding to year number 13, encodes five dots on the right, three on the top and five on the left.

The number of dots on all three sides must by definition sum to the number being represented:

In[]:=

Total/@DotData

Out[]=

{1,2,3,4,5,6,7,8,9,10,11,12,13}

Now define a function to place dots in appropriate locations based on the dot data:

In[]:=

DotDiagram[n_,off_:{0,0}]:=Module{r,t,l,rad=0.09,dx=0.1},{r,t,l}=DotDatan;(*box*)LineTableoff+

Through[{Cos,Sin}[π/2i+π/4]]

,{i,0,4},(*right*)Ifr1&&t0,TableCircleoff+

-dx,

r+1

,rad,{i,r},(*Else*)TableCircleoff+

-dx,

-If[n11,.2,.0],rad,{i,r},(*top*)Ifr0||l>1,TableCircleoff+

If[n13,7,5]

t+1

If[n13,14,10]

-If[5<n<11,.05,0],

-dx,If[n13,.8,1]rad,{i,t},(*Else*)TableCircleoff+-

t+1

-If[5<n<11,.05,0],

-dx,0.8rad,{i,t},(*left*)TableCircleoff+-

+dx,

l+1

,rad,{i,l}

… and plot the patterns for all possible year 1–13 specifications:

In[]:=

GraphicsGrid[Partition[Table[Graphics[DotDiagram[i],PlotLabeli],{i,13}],UpTo[7]],ImageSizeLarge]

Out[]=

In order to compare this implementation with that depicted in the Codex, construct a function to return the necessary number of dots together with the glyph corresponding to a given calendar year:

In[]:=

AztecDate[y_]:={Mod[y-1428,13]+1,Mod[17+(y-1428)365,20]+1}

This implementation is based on the Codex Mendoza, which marks the reign of Itzcōhuātl, fourth ruler of Tenochtitlan, who ruled from 1 Flint (1428 CE) to 13 Flint (1440 CE). From the Codex, these years correspond to:

In[]:=

Row[{#,$AztecNames[[#2]]},Spacer[1]]&@@@Table[AztecDate[y],{y,1428,1440}]

Out[]=

Flint,2

House,3

Rabbit,4

Reed,5

Flint,6

House,7

Rabbit,8

Reed,9

Flint,10

House,11

Rabbit,12

Reed,13

Flint}

Now generate a full representation of the year designations and show its agreement with the Codex:

In[]:=

img=

ImageTake[

]

;Row[{Framed@Show[Module[{dots,glyph,pos},Table[{dots,glyph}=AztecDate[y];Graphics[{DotDiagram[dots,pos={0,-1.1(y-1428)}],Inset[$AztecGlyphs[[glyph]],pos,Automatic,.5]}],{y,1428,1440}]],ImageSize60],img}]

Out[]=

Additional Reading

Berdan, F. and Anawalt, P. R. The Essential Codex Mendoza. Berkeley, CA: University of California Press, 1997.
Bernal, I. and León-Portilla, M. Plate 5 in Historia de México, Vol. 2. Barcelona, Spain: Salvat Editores, p. 296, 1974.
Pappas, T. The Joy of Mathematics: Discovering Mathematics All Around You. San Carlos, CA: Wide World Publishing/Tetra, pp. 128–129, 1989.
Payne, S. E. and Closs, M. P. "A Survey of Aztec Numbers and Their Uses." Ch. 8 in Native American Mathematics (Ed. M. P. Closs). Austin, TX: University of Texas Press, pp. 213–235, 1996.

Additional Links

Image Credits

Wikimedia