The Calligraphic Zero

David Stuart1 Comment

One of the many interesting details revealed in the small glyphic notations discovered at Xultun (Saturno, Stuart, Aveni, and Rossi 2012) are the unusual simple forms of “zero” that appear in several LC records. In Classic-era texts, zeroes have traditionally been recognized in one of three forms (see Figure 1): (a) the common three-pedal form that resembles a darkened flower; (b) the “shell-hand,” or (c) the head-variant showing a human profile with a hand or bony snake as its lower jaw (Figure 1, a-c). All of these can be phonetically read as the syllable mi (see Grube and Nahm 1990) or perhaps as the logogram MIH. As word signs these would correspond to to root mih and its cognates, widespread in Mayan languages with the meaning of “nothing” (see Blume 2011 for a thorough overview of zero signs in Maya script).

Figure 1. Three variants of syllabic mi or logographic MIH, “zero.” (Sketch by D. Stuart)

The small notations on the walls of Structure 10K-2 at Xultun now show a lesser known form – a simple oblong oval sign with a small interior circle and borderline along its upper edge (six instances are shown in Figure 2).

Figure 2. Number array from north wall of Structure 10K-2 at Xultun, Guatemala. Note the oval “zero” signs in each column. (Preliminary drawing by D. Stuart, Proyecto Regional San Bartolo-Xultun)

This is a more informal zero probably reserved for mathematical computations and notations in manuscripts, and the sort of thing that was seldom known before the discovery of the Xultun mural. The only other example I know is from the string of numerals written on an AN or AHN logogram on Pomona Panel 7, now in the Dallas Museum of Art (see Figure 3).

Figure 3. a-AN glyph from Pomona Panel 7. Note the simplified zero at the lower right, in the sequence 1.7.0. (Photograph by David Stuart)

It’s likely that this oval form was simplified from the common three-pedal variant illustrated in Figure 1a, with the darkened outer areas omitted and the circular “core” preserved. This Classic calligraphic zero is presumably the origin for the oblong forms of used in the Post-Classic Dresden Codex, nearly always described as a representation shell. However, those zeroes in the Dresden are not quite identical, usually showing a more pointed outline on its two ends (see Figure 4).

Figure 4. Zero signs (in red) from page 43b of the Dresden Codex.

Regarding the forms of the Xultun zeroes, one alternative I’ve considered is that they are calligraphic variants of the very similar-looking logogram read PET, a sign deciphered many years ago by Nikolai Grube. This root, meaning “circular” in proto-Ch’olan, appears in a variety of settings in the inscriptions, such as PET-ne, peten, “island,” or the verb written PET-ta-ja, pet-aj, “become round” or “be encircled.” It is interesting to note that in the Calepino Motul dictionary of colonial Yucatec, pet is the basis of a derived noun petel meaning “totality” or “grouping.” Conceivably this might be an appropriate marker for a number position that has reached its “totality.” However, I prefer for now to see the Xultun zeroes simply as calligraphic forms, derived from the more complex and familiar sign of the stone inscriptions.

REFERENCES CITED:

Blume, Anna. 2011. Maya Concepts of Zero. Proceedings of the American Philosophical Society, vol. 155, no. 1., pp. 51-88. http://www.amphilsoc.org/sites/default/files/6BlumeRevised1550106%20(2).pdf

Grube, Nikolai, and Werner Nahm. A Sign for the Syllable mi. Research Reports on Ancient Maya Writing 33. Center for Maya Research, Washington, DC.

Saturno, William, David Stuart, Anthony Aveni and Franco Rossi. 2012. Ancient Maya Astronomical Tables from Xultun, Guatemala. Science, vol. 336 no. 6082, pp. 714-717.

Xultun Number A and the 819-Day Count

David Stuart8 Comments

by Barbara MacLeod and Hutch Kinsman

Within a few hours of the publication in the 11 May, 2012 issue of Science of “Ancient Maya Astronomical Tables from Xultun, Guatemala”, by William Saturno, David Stuart, Anthony Aveni and Franco Rossi, Hutch Kinsman contacted colleagues who regularly correspond by email, pointing out that Number A—1,195,740—is evenly divisible by 819. It is the only one of the four which contains this factor. He also noted that the coefficient of the tzolk’in day at the top of the column is 1. Since all tzolk’in dates which are stations in the 819-Day Count have a coefficient of 1, this was further evidence that the purpose of the interval was to commensurate the 819 (2.4.19)-Day Count with the Calendar Round.

Figure 1. Number array from north wall, Structure 10K-2, Xultun, Guatemala. (Preliminary drawing by David Stuart)

For anyone not familiar with this cycle, 819 is the product of 7, 13, and 9—numbers of ritual and calendric significance to the Maya. Following the Initial Series, the count appears as a short distance number leading to the previous station–one of four which are 819 days apart. These are associated with the cardinal directions and their corresponding colors. A verb meaning ‘stand still’ or ‘stop’ appears along with several regular protagonists. Yaxchilan and Palenque are noteworthy in having multiple monuments featuring the 819-Day Count. J. Eric S. Thompson (1950:214) and his contemporaries offered early suggestions about its purpose. Heinrich Berlin and David Kelley (1961) first described the structural similarity between the Dresden New Year pages and the color/direction symbolism of the 819-Day Count. Given the formula [4 x 819] = [9 x 364] one may add nine days to the latter to complete nine haabs. Michael Grofe (personal communication, May, 2012) suggests that it is an idealized system for tracking the sidereal position of eclipses.

Figure 2. Example of 819-day count record from Yaxchilan, Lintel 30. (Drawing by Ian Graham)

The interval of Xultun Number A—1,195,740 days– is [63 x 18,980] and [4 x 819 x 365]. It is also [9 x 365 x 364], which brings to mind the [9 x 364] = [4 x 819] formula mentioned above. The unit of 364 days is the Maya “computing year” discussed by Thompson (1950:256). The interval of Xultun Number A is also the smallest unit which commensurates the 819-Day Count with the Calendar Round.

Thompson (cited above) wrote: “as only once in every 63 times will a day with a coefficient of 1 also mark the start of the 819-day cycle, the fact that this first day before (13.0.0.0) 4 Ajaw 8 Cumku is a base in the 819-day cycle argues strongly for that count’s being primarily ritualistic”.

The day 1 Kaban before the Era Base 4 Ajaw 8 Kumk’u, per a discussion Carl Callaway and Barbara MacLeod had several years ago, is an 819-Day Count station in the east quadrant—the quadrant in which, for several reasons, we concluded that the count should begin. From this datum, counts both forward and back might reach other stations in the cycle; thus the pre-era date 12.19.19.17.17 1 Kaban 5 Kumk’u need not be the earliest documented station. The earliest station known, 12.9.19.14.5 1 Chikchan 18 Ch’en, recorded on the Palenque Temple XIX bench, is therefore not the base date but rather a distant-past station reached from it.

At the 1974 Segunda Mesa Redonda de Palenque, Floyd Lounsbury presented a meticulous analysis of the pre-era initial date of the Tablet of the Cross at Palenque. This paper is well worth reading and is available on Mesoweb:

http://www.mesoweb.com/pari/publications/RT03/Rationale.html

Per Lounsbury’s work, the Palenque interval is 1,359,540 days, or [4 x 819 x 415]. While it is not an even multiple of the 18,980-day Calendar Round, it is [5229 x 260] and [1734 x 780] and [3735 x 364]. It demonstrates the application to dynastic mythological narrative of large multiples of [4 x 819] by Maya scribes in deep-time calculations.

Saturno et. al. note that the tzolk’in day at the top of Column A is either 1 Kawak or 1 Kaban. We suggest that it is 1 Kaban—the tzolk’in position of the base date of the 819-Day Count. This in turn sheds light on the function of the other three tzolk’in dates. We tentatively suggest that the 9 K’an date atop Column B is that of the Dresden Codex Serpent Base 9 K’an 12 K’ayab. More will be said about the other three numbers in the near future.

References Cited

Berlin, Heinrich, and David H. Kelley. 1961. The 819-day Count and Color-direction Symbolism among the Classic Maya. Middle American Research Institute Publication 26.

Lounsbury, Floyd G. 1976 A Rationale for the Initial Date of the Temple of the Cross at Palenque. Second Palenque Roundtable, 1974. The Art, Iconography & Dynastic History of Palenque, Part III, edited by Merle Greene Robertson. Pebble Beach, California: Pre-Columbian Art Research, The Robert Louis Stevenson School.

Saturno, William, David Stuart, Anthony Aveni and Franco Rossi. 2012. Ancient Maya Astronomical Tables from Xultun, Guatemala. Science 336, 714.

Thompson, J. Eric S. 1950. Maya Hieroglyphic Writing: An Introduction. University of Oklahoma Press, Norman.

Xultun’s Astronomical Tables

David Stuart1 Comment

Our article has just published in the latest issue of Science (Vol. 336 no. 6082 pp. 714-717), co-authored by  William Saturno, David Stuart, Anthony Aveni and Franco Rossi.

Article Abstract

Maya astronomical tables are recognized in bark-paper books from the Late Postclassic period (1300 to 1521 C.E.), but Classic period (200 to 900 C.E.) precursors have not been found. In 2011, a small painted room was excavated at the extensive ancient Maya ruins of Xultun, Guatemala, dating to the early 9th century C.E. The walls and ceiling of the room are painted with several human figures. Two walls also display a large number of delicate black, red, and incised hieroglyphs. Many of these hieroglyphs are calendrical in nature and relate astronomical computations, including at least two tables concerning the movement of the Moon, and perhaps Mars and Venus. These apparently represent early astronomical tables and may shed light on the later books.

Full article can be accessed here

UPDATE: Mesoweb has posted a nice summary of the find and of our epigraphic work (click here). Thanks Marc and Joel.

Number table from the north wall of Structure 10K-2 at Xultun, Guatemala. (Preliminary drawing by D. Stuart)

A Liquid Passage to Manhood

mayoid6 Comments

by Stephen Houston

A few years ago I proposed that some of the most celebrated Maya vases were commissioned for young men in Classic society (Houston 2009: 166). My thoughts at the time: “pots with such labels could have been bestowed in the setting of age-grade rituals or promotions, a recognition of feasting and expensive drinks as markers of adult status, even trophies and material honours while in page service, ballplay or war.” The notion appealed to me on behalf of all those ungainly, ever-changing youth in past and present times—as a set, the boys and young men could be seen as an unexpected aesthetic locus, a target for what was arguably the summit of ceramic painting in the ancient New World. But this idea involved a second, very specific expectation. The painted vessels so-named and so-possessed would involve not only young men but youths at times of change, on transit through the rites of passage so familiar to comparative anthropology.

A fresh piece of evidence lends weight to this conjecture. An eroded and shattered cylindrical vessel in the Juan Antonio Valdés Museum in Uaxactun, Guatemala, contains the usual Primary Standard Sequence. The glyphs have a cadenced coloration of two red-painted glyphs followed by one left uncolored. This scheme recalls the Primary Standard Sequences on such luminous vessels as a bowl from the area of Tikal, now in the Museo Popol Vuh (Kerr #3395; the presence of Jasaw Chan K’awiil’s name on the bowl brackets it temporally to AD 682 ~ 734; see also K #595). The vessel in Uaxactun has the following sequence, somewhat occluded by the darkness of the photographs I have seen: ….u-tz’i ba-IL yu-k’i-bi ti-YAX-CH’AHB ch’o-ko AJ-?BAHLAM che-he-na SAK-MO’-‘o ?-?…., the final glyphs surely the name of the painter. Figure 1 reproduces a key passage, the name of the drinking vessel (yuk’ib), just before an expression ti yax ch’ahb, “for the first fast/penance(?).” Then, crucially, the name of the owner, a ch’ok or “youth.”

Figure 1. Passage on a vessel in the Juan Antonio Valdés Museum (drawing by author).

As Stuart and others have noted, the yax ch’ahb refers to a rite of passage for young males, perhaps most eloquently in a text from Caracol Stela 3 (Stuart 2008; also Houston et al. 2006: 131-132, fig. 3.30). There, a young prince, only a few months beyond 5 years of age, underwent this arduous rite. It formed part of his first bloodletting but probably involved much other pain besides, including the denial of food. The import is clear: the vessel at Uaxactun was intended specifically for an age-grade ritual, the first (presumably) of many sacrificial offerings from a noble youth. Did it offer a filling and restorative draft of liquid after penance? Was it a gift to others who might witness his ascent to adult duty? Of these matters we cannot be certain. But the likelihood is now stronger that most such vessels marked and materialized shifts of status: a liquid passage from boyhood to the obligations of elite men.

REFERENCES CITED

Houston, Stephen. 2009. A Splendid Predicament: Young Men in Classic Maya Society. Cambridge Archaeological Journal 19 (2): 149-178.

Houston, Stephen, David Stuart, and Karl Taube. 2006. The Memory of Bones: Body, Being, and Experience among the Classic Maya. Austin: University of Texas Press.

Stuart, David. 2008. A Childhood Ritual on The Hauberg Stela. Maya Decipherment weblog. http://mayadecipherment.com/2008/03/27/a-childhood-ritual-on-the-hauberg-stela/

The Misunderstanding of Maya Math

David Stuart13 Comments

A great many descriptions of ancient Maya mathematical notation read something like this:

The Maya made use of a base-20 (vigesimal) system with the units of 1, 20, 400, 8,000, 160,000, etc.. To write a number, a scribe would show multiples of these units in a set columnar order, moving down from highest to lowest, and add them accordingly. “32” for example would be written as single dot for 1, representing one unit of 20, above the two bars and two dots for 12, corresponding to the “ones” unit (1×20 + 12×1 = 32). A larger number such as 823 would be written in three places as two dots followed by one dot followed in turn by three dots, standing for the necessary multiples of 400, 20, and 1 respectively (2×400 + 1×20 + 3×1 = 823).

Similar descriptions of Maya math pervade the literature, textbooks and the internet. For example Michael Coe writes in the latest edition of The Maya (p. 232):

Unlike our system adopted from the Hindus, which is decimal and increasing in value from right to left, the Maya was vigesimal and increased from bottom to top in vertical columns. Thus, the first and lowest place has the value of one; the next above it the value of twenty; then 400; and so on. It is immediately apparent that “twenty” would be written with a nought in the lowest place and a dot in the second.

The illustration accompanying this text provides many examples of this purely vigesimal system:

Maya number notation from Coe’s The Maya (8th edition, p. 233)

Maya mathematical notation is described the same way in a number of other influential books widely read in classrooms and seminars, such as The Ancient Maya: New Perspectives (McKillop 2004:277) or the venerable The Ancient Maya (Sharer and Traxler 2006:101). In the latter work, two types of counts are represented (see below) – the purely vigesimal or base-20 count (with units of 1, 20, 400, and 8,000) alongside what’s called the “chronological count” (with units of 1, 20, 360, 7,200). The second is of course the basis for the familiar Long Count system.

Maya number notation as shown in The Ancient Maya (6th edition, p. 101).

A big problem exists with all of these seemingly straightforward descriptions of Maya mathematical notation. As far as I am aware no purely vigesemal place-notation system was ever written this way. It’s true that in Mayan languages numbers are base-20 in their overall structure, just as in most Mesoamerican languages. In Colonial Yukatek, for example, we have familiar terms for these units: k’al (20), bak’ (400), pik (8,000), and so on. However, ancient scribes never represented these units in a columnar place notation system, as is so commonly described in the textbooks. That format was instead always reserved for a for the count of time, in what we know as the Long Count. That system is mostly vigesimal, but it is skewed in one of its units (the Tun, of 360 days) in order to conform as much as possible to the number of days in the solar year (365). To reiterate: the columns of numbers we find in the pages of the Dresden Codex or painted on the walls of Xultun (stay tuned, folks…) are all day counts; the positional notation system was never used for reckoning anything else.

In the ancient inscriptions non-calendrical counts using large numbers are quite rare, mostly found in connection to tribute tallies, such as the counting of bundled cacao beans. But in those settings the scribes always seem to show nice rounded numbers (as in ho’ pik kakaw, “5×8,000 [40,000] cacao beans,” shown in the murals of Bonampak) without all the place units we know from the Long Count. In the Dresden and Madrid codices, counts of food offerings are given as groupings of WINIK (20) signs with accompanying bars and dots for 1-19. In this way a cluster of four such elements (4×20) with 19 writes 96 (See Love 1994:58-59; Stuart, in press).

There is a good deal we still don’t know about the ways the Maya wrote quantities, especially of non-calendrical things. The pattern nonetheless seems clear that the place notation system of the Long Count was restricted to time reckoning, and never applied to the purely vigesimal counting structure we see reflected in Mayan languages. The descriptions of written numbers found in the many texts about the ancient Maya therefore need to be corrected.

Sources Cited:

Coe, Michael. 2011. The Maya (8th edition). Thames and Hudson, New York.

Love, Bruce. 1994. The Paris Codex: Handbook for a Maya Priest. University of Texas Press, Austin.

McKillop, Heather. 2006. The Ancient Maya: New Perspectives. W.W. Norton, New York.

Sharer, Robert, and Loa Traxler. 2005. The Ancient Maya (6th edition). Stanford University Press, Stanford.

Stuart, David. In press. The Varieties of Ancient Maya Numeration and Value. To appear in The Construction of Value in the Ancient World, ed. by J. Papadopolous and G. Urton. Cotsen Institute of Archaeology, UCLA, Los Angeles.