Periodicity in the Dresden Codex Venus Table

Human Mosaic 36(1), 2006, pages # Periodicity in the Dresden Codex Venus Table Sonny Faulseit Tulane University The first page (24) is known as the introductory page, and it contains mathematical and calendrical information concerning the contents of the table itself. This information includes the base date and ring number, which are used to establish the entry or beginning date of the table. Starting with the Goodman–Martínez– Thompson (GMT) correlation, which places the origin of the Maya Long Count on the Julian day 584283, Floyd Lounsbury (1983) established the Venus Table entry date (1 Ahau) as the Gregorian calendar date of 23 November 934, and the results of this paper are based on that date.1 Other information found on the introductory page (24) includes a table of multiples that provides numeric information concerning the construction of the Venus Table. For example, multiples of the numerical sums in each row are tabulated, as well as multiples of the numeric sum of the entire table. Lastly, the table of multiples includes some “aberrant” multiples, or numbers that are not exact multiples of the table or any part within it. These numbers have been demonstrated to correct for the long-term drift that occurs because Venus’ synodic period does not equal a whole number of days (H. Bricker and V. Bricker, personal communication, 2002). All of the celestial bodies in our solar system have both sidereal and synodic periods. Even the Earth’s rotation has different sidereal and synodic periods, making the length of the day arguable. Basically, sidereal is suncentric and synodic is earth-centric, although either period can be observed from the earth’s surface. Without using astronomical definitions, it is best to think of the sidereal period as associated with the fixed background of stars and the synodic period as associated with the relative motion of the celestial bodies. For example, we can observe the synodic period of the moon as the number of days it takes to go from full moon to new and back to full again (approximately 29.5 days), while the sidereal period would consist of the number of days it takes for the moon to go through all the constellations of the zodiac (roughly 27 days). Venus’ synodic period involves the number of days between Venus’ first appearance as morning star. Like that of the moon, Venus’ sidereal period consists of the number of days the planet takes to traverse the zodiac. Personal Note It gives me great pleasure to include this paper as part of the Festschrift for Harvey and Victoria Bricker. Academically, both have contributed enormously to the discipline of Anthropology; their accomplishments are well-known and celebrated. That is why in this note I want to share with the reader something about these wonderful professors as I have come to know them personally. Vicki is one of the most dedicated and tireless scholars I have known, and she is equally devoted to the students and faculty of Tulane University. She always takes time out from her many duties to discuss issues about school, research, or any other problem a student may bring to her. She is a very caring and warm person, who has a selfless devotion to the Department of Anthropology that is only matched by Harvey’s devotion. Harvey has been my advisor since I came to Tulane. Any student will tell you that at first impression he is a very intimidating figure who expects a lot from graduate students. I remember how nervous I felt sitting in his office for the first time. Since then I have come to know Harvey and respect him greatly, for his criticism and candor have always been directed toward the development of my scholarship. His careful attention to detail and encyclopedic knowledge have left a permanent impression on me. Harvey is the type of person who will exhaust every means available to him to help a student, and I am greatly indebted to him for helping me. The Brickers will be sorely missed by both Tulane University and the Department of Anthropology. The 60s style dark-frame glasses and black beret will have to be retired to the Tulane Hall of Fame. ______________ Introduction The Dresden Codex is one of four pre-Columbian Maya documents. It is filled with hieroglyphic writing, almanacs, and numeric tables. Although the hieroglyphic texts have only been partially deciphered, some of the numerical information in the tables has been demonstrated to pertain to astronomical events (Aveni 2001). Pages 24, 46, 47, 48, 49, and 50 of this codex make up what has been identified as the Venus Table (Thompson 1972). This table more than likely was used by the Maya to predict the appearance of Venus as morning star, and it contains an astonishing amount of information related to the motions of this planet. 1. Susan Milbrath (1999:163–174) has argued for an alternate reading of the entry date, as supported by iconographic representations in the table itself; however, this is beyond the expertise of the author and does not pertain directly to this discussion. Either entry date would support the arguments presented in this paper, and for simplicity’s sake I have chosen Lounsbury’s. 5 6 A German librarian, Ernst Förstemann, was the first to recognize that the sum of the four numbers repeated in the bottom left corner of each of the pages of the actual table itself (46–50) added up to 584, a close approximation to the synodic Venus cycle of 583.92 days. He also noted that the individual intervals roughly related to the four events that take place throughout Venus’ synodic cycle. The construction of the table shows that the Maya were aware of the commensuration of the Venus synodic period with the tropical year (Aveni 2001), as the sum of five Venus cycles (5 x 583.92 = 2919.6) very closely approximates eight tropical years (365.2422 x 8 = 2921.94). Considering the fact that the Maya did not make any adjustments for the tropical year, but instead relied on a vague year of 365 days, it makes sense that they approximated the Venus synodic cycle as 584 days, thus making both periods of five Venus cycles and eight years equal to 2920 days. Indeed, the numeric intervals spanning the rows of the five pages of the Venus Table sum exactly to 2920 days. Each of the five pages of the Venus Table is made up of four columns representing the four events in the Venus synodic cycle: the heliacal rise event (MFIRST) corresponds to the time when Venus first becomes visible in the morning sky before the sun rises, and the heliacal set event (ELAST) corresponds to the last day Venus is visible in the evening sky after the sun sets. The other events (MLAST and EFIRST) correspond to the last visibility of Venus as morning star, and first visibility of Venus as evening star, respectively. These events are organized in groups of four, starting with MLAST, followed by EFIRST, then ELAST, and ending with MFIRST. This pattern results in twenty columns across the table. At the bottom of each page of the Venus Table the intervals between canonical events are listed. Table 1 shows the Maya canonical intervals between these events as well as the calculated means for the actual events. In this paper, I explore the meaning behind the canonical synodic Venus occurrences and their significance in the Dresden Venus Table. These events are referred to as canonical because they have not traditionally been conceived as coinciding with the actual periods of occurrence, and therefore must have had some canonical meaning to the Maya astrologers who used them. The data obtained from the thorough investigation of the seasonal and astronomical events surrounding the canonical intervals strongly support the notion that the Maya incorporated both the sidereal and synodic periods of Venus into the table format. Table Structure: Numerology and the Tzolkin The Dresden Codex Venus Table is brilliantly constructed so that its structure accommodates the 260-day tzolkin, or ritual calendar. This calendar works by rotating 20 named days, Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Human Mosaic Edznab, Cauac, and Ahau, through a 13 numeral pattern, so that the first day is 1 Imix, followed by 2 Ik, 3 Akbal, up to 13 Ix, and then the numerals repeat while the days proceed, so that the following day is 1 Men. Since the lowest multiple of both 13 and 20 is 260, no numeralnamed day combination will repeat until after 260 days, when once again the day 1 Imix will occur. The Venus Table in the Dresden Codex is apparently arranged to mirror this pattern, revealing the importance the Maya placed on commensuration with this unique calendar. To accommodate the tzolkin, the Maya constructed the table with 13 rows and 20 columns. As a result, the overall table covers 37,960 days (146 tzolkins and 104 haabs), and contains a total of 260 events. These events are further divided into five pages, each containing 52 events (4 rows and 13 columns). This pattern resembles Central Mexican representations of the 260-day calendar such as on pages 1 through 8 of the Borgia Codex (Díaz and Rodgers 1993), where the 260-day calendar is divided into five rows of 52. Another way in which the Dresden Codex Venus Table is divided in a similar pattern to the 260-day calendar involves the four synodic events. When looked at individually, each event has five columns of 13 rows for a total of 65 tzolkin dates recorded in the table. Thus, the table is structured so that it contains a total of 260 items, the number of days in the tzolkin, that are neatly organized into parts of 4, 5, 52, and 65, or all the multiples that could possibly be used to evenly separate the number 260, using the base numbers of 13 and 20. “The first annual morning or pre-dawn appearance of Venus is termed its heliacal rising (MFIRST). It was probably the most important single event in Maya astronomy” (Aveni 2001:83). The Venus Table in the Dresden Codex seems to verify Aveni’s statement, as it appears that the main function of the table was to predict occurrences of this phenomenon. The commensuration of the tzolkin and the synodic cycle of Venus reveals an interesting pattern. When 584 is divided by the number of named days in the tzolkin (20), there is a remainder of four. As a result, if the count of the synodic period begins with an MFIRST occurrence on an Ahau day in the tzolkin, the next MFIRST will occur four days later in the cycle on a Kan day. Since 20 is evenly divisible by four (five times), there are only five named days in which morning star appearances can occur using the canonical synodic interval. The resulting tzolkin day names for the MFIRST occurrences in the Venus Table are Ahau, Kan, Lamat, Eb, and Cib. In addition to this, 584 when divided by 13 yields a remainder of 12. Thus, in the cycle of 13 days, each successive MFIRST date will have a coefficient of one less than the previous date. Table 2 is a representation of the structure of the MFIRST occurrences in the Venus Table. The start date, 1 Ahau, appears in the bottom right-hand corner on the last page. This is a common practice found in many almanacs and tables from the Maya codices. Volume 36 7 Table 1. Canonical and actual intervals between synodic events. Events MFIRST to MLAST MLAST to EFIRST EFIRST to ELAST ELAST to MFIRST Period Visible as Morning Star Invisible at Superior Conjunction Visible as Evening Star Invisible at Inferior Conjunction Canonical Interval 236 days 90 days 250 days 8 days Actual Interval* 263 days 50 days 263 days 8 days *Numbers taken from Sharon Gibbs (1977). Table 2. Mfirst tzolkin dates with coefficients. Page 46 13 Kan 8 Kan 3 Kan 11 Kan 6 Kan 1 Kan 9 Kan 4 Kan 12 Kan 7 Kan 2 Kan 10 Kan 5 Kan Page 47 12 Lamat 7 Lamat 2 Lamat 10 Lamat 5 Lamat 13 Lamat 8 Lamat 3 Lamat 11 Lamat 6 Lamat 1 Lamat 9 Lamat 4 Lamat Page 48 11 Eb 6 Eb 1 Eb 9 Eb 4 Eb 12 Eb 7 Eb 2 Eb 10 Eb 5 Eb 13 Eb 8 Eb 3 Eb Notice that throughout the entire table, no numeral– day combination is repeated, but the coefficients progressively count down through the five named days from 13 Kan to 1 Ahau. Each individual occurrence of morning heliacal rise has its own name, a fact that would certainly help the astrologer–priest keep his place in the table. This pattern is very nearly replicated using the Central Mexican 260-day calendar, or tonalpohualli, on pages 53 and 54 of the Borgia Codex (Díaz and Rodgers 1993). Furthermore, because all of the synodic periods in the Venus Table consist of 584-day intervals (i.e. ELAST to ELAST = 584 days), any of the single events would fit neatly into the structure of Table 2, albeit it would have a different pattern of day names. For EFIRST the days are Cimi, Oc, Ix, Edznab, and Ik; and for MLAST and ELAST they are Cib, Ahau, Kan, Lamat, and Eb. The importance of Venus’ morning heliacal rise has been well-documented throughout Mesoamerica (Bierhorst 1992; Sahagún 1953:11–12). In the Codex Chimalpopoca (Bierhorst 1992:36), the story of Quetzalcoatl of Tollan includes a clue to the ritual significance of morning star appearances, and perhaps describes the events pictured on the pages of both the Dresden Codex Venus Table as well as the Borgia Codex. These pictures show beings armed with atlatls who are Page 49 10 Cib 5 Cib 13 Cib 8 Cib 3 Cib 11 Cib 6 Cib 1 Cib 9 Cib 4 Cib 12 Cib 7 Cib 2 Cib Page 50 9 Ahau 4 Ahau 12 Ahau 7 Ahau 2 Ahau 10 Ahau 5 Ahau 13 Ahau 8 Ahau 3 Ahau 11 Ahau 6 Ahau 1 Ahau shooting victims, which is precisely what the Codex Chimalpopoca attributes to Quetzalcoatl after he transforms into the morning star. More relevant to the topic of this paper is the reference to the number of days that Venus disappears around inferior conjunction, the period between ELAST and MFIRST. “He [Quetzalcoatl] died and disappeared for four days … and he spent four more days making darts for himself” (Bierhorst 1992:36). This sets the time interval to eight days, which closely approximates the astronomical mean interval, and is exactly the interval used in the Dresden Codex Venus Table. Floyd Lounsbury (1978) has demonstrated that the main function of the Venus table was to predict occurrences of MFIRST, and indeed throughout the table the canonical MFIRST day falls within –2 to +4 days of the actual event. Harvey and Victoria Bricker (2002) further demonstrated a method for using the table of multiples located on page 24 of the codex to adjust the dates for recycling the table. This correction is needed because the canonical Venus synodic period, 584 days, and the actual period, 583.92 days, differ enough to cause significantly faulty predictions (predictions that occur after the actual event) over the 104 years of the table. They successfully explain the “aberrant multiples” of 1.5.14.4.0 (185,120), 9.11.7.0 (68,900), and 4.12.8.0 (33,280) for this 6 function, leaving all but one of the numbers in the table of multiples associated with the prediction of heliacal rise as morning star. If the morning heliacal rising event were all the Maya were concerned with, it seems that the Venus Table would consist of only the information found in Table 2, as it does in the Borgia Codex. This is not the case, however, and there is evidence that the Maya were also concerned with accurately predicting the occurrence of at least one of the other three synodic Venus events, ELAST. The importance of Venus’ manifestation as evening star is supported by the work of Aveni and Hartung (1991), which was later refined by Ivan Šprajc (1993b), who documented an alignment at the site of Uxmal on the Yucatan Peninsula that appears to be associated with the extreme position of Venus along the horizon as evening star. The first canonical ELAST date in the table on page 46, 5 Cib, occurs on 21 June 935 of the Gregorian calendar, which is the exact date of the actual ELAST event, as well as the date of the summer solstice. Unfortunately, while the mean value of 584 days is accurate for the interval between consecutive ELAST dates, the actual period varies between 575 and 592 days. Anthony Aveni (1992b) first noted that the variance in the interval between ELAST and MFIRST has a seasonal dependence. In the next section, I will explore some of the seasonal patterns present in the Venus Table. Table Structure: Seasonality If one were to map Venus’ movement through the sky as morning or evening star by marking its position in both altitude and azimuth with respect to the horizon at the same time every day, one would notice an interesting pattern. There are five distinct paths that Venus makes that repeat approximately every eight years. Aveni (2001:184–196) proposed that these cycles account for the fact that the Venus table is divided into five pages, and indeed counting the number of days across any row of the table will yield 2920, which is exactly equal to eight vague years (365 x 8 = 2920). Without a doubt, the Maya recognized the coincidence of the five tzolkin day names and the five cycles of Venus in the evening/morning sky. The result of this phenomenon is that each of the five pages of the Venus table will have an associated seasonal pattern. Table 3 has been constructed in order to demonstrate this seasonal pattern. The canonical synodic events and their corresponding Gregorian calendar dates are listed for pages 46 through 50 of the Venus Table. Using the Planet Visibility program developed by Noel Swerdlow and Rainer Lange (2002 #1305), I documented all of the actual occurrences of the Venus synodic events that occurred between 23 November 934 (the entry date) and 29 October 1038, and compared them to the canonical dates listed in the Venus Table. The column located to the Human Mosaic right of the dates contains the difference in days between the canonical and actual event. The first row of page 46 shows that canonical MLAST occurred on 17 July, 30 days before the actual event; EFIRST canonical occurred on 15 October, 15 days after the actual event; canonical ELAST occurred on 21 June, the same day as the actual event; and finally MFIRST canonical occurred on 29 June, two days before the actual event. Moving down any column will give the same event 2920 days later, and it will reveal the drift associated with the difference between the canonical synodic periods of 584 days for the Venus cycle and 2920 days for the year and actual periods of 583.92 and 2921.92 days, respectively. On each of the five pages, MFIRST, ELAST, and MLAST dates all occur roughly in the same season. By season, I mean the division of the year into four roughly equal periods based on the summer solstice, fall and spring equinoxes, and the winter solstice. For example, anything that occurs between the Julian dates of June 21st (summer solstice) and September 21st (fall equinox) would be considered the same season. Since the canonical MLAST, ELAST, and MFIRST events in the first row on page 46 all occur in June or July, after the summer solstice and before the fall equinox, they are considered to be in the same season. In all cases, the canonical MLAST event occurs on the latest date, well into the season. This seasonal pattern is not precise as defined, and there are several notable cases where canonical events drift into the earlier seasons, but the fact does remain that MFIRST, ELAST, and MFIRST all occur within a 30-day window. On the other hand, the EFIRST canonical date on each page stands out, because it does not occur within the same season as the others. In addition to this, EFIRST also does not maintain the same tzolkin day pattern as the other three, Ahau, Kan, Lamat, Eb, Cib. Instead, EFIRST occurrences fall on the days Cimi, Oc, Ix, Edznab, and Ik. Due to the fact that the canonical dates for MFIRST and EFIRST are not intended to match the actual events—in fact, they differ by as many as 30 and 15 days, respectively (Table 3)—it seems that at least one of these canonical events is intended to establish or maintain the seasonal opposition of EFIRST with respect to the other events. Indeed, the canonical interval of 90 days between MLAST and EFIRST seems to belie some seasonal aspect, for 90 days is just short of the number of days between any solstice and equinox. This fact, coupled with the notion that MLAST is always the latest date of the three seasonsharing events, would ensure that the EFIRST canonical dates occur within the following season. Table 4 contrasts the actual and canonical dates for the events on page 46. From this list, it appears that the MLAST date was adjusted to be closer to the ELAST and MFIRST dates, while the EFIRST date was adjusted to move it into the following season. The right-hand columns in Table 3 reveal the Volume 36 7 Table 3. Seasonal pattern in the Dresden Codex Venus table. Page # 46 47 48 MLAST Greg *Can - ML CIB 17-Jul -30 15-Jul -30 13-Jul -29 11-Jul -29 9-Jul -29 7-Jul -29 5-Jul -28 3-Jul -28 1-Jul -28 30-Jun -25 28-Jun -25 26-Jun -25 24-Jun -25 AHAU 20-Feb -23 18-Feb -24 16-Feb -22 14-Feb -22 12-Feb -23 10-Feb -22 8-Feb -21 6-Feb -21 5-Feb -19 3-Feb -18 1-Feb -19 30-Jan -19 28-Jan -18 KAN 27-Sep -28 25-Sep -29 23-Sep -27 21-Sep -28 19-Sep -27 17-Sep -27 15-Sep -25 13-Sep -25 12-Sep -23 10-Sep -22 8-Sep -22 6-Sep -21 4-Sep -21 EFIRST Greg *Can - EF CIMI 15-Oct 15 13-Oct 14 11-Oct 16 9-Oct 16 7-Oct 17 5-Oct 16 3-Oct 18 1-Oct 18 29-Sep 19 28-Sep 21 26-Sep 21 24-Sep 22 22-Sep 22 OC 21-May 11 19-May 11 17-May 12 15-May 12 13-May 12 11-May 13 9-May 13 7-May 14 6-May 16 4-May 16 2-May 17 30-Apr 17 28-Apr 17 IX 26-Dec 13 24-Dec 13 22-Dec 14 20-Dec 14 18-Dec 14 16-Dec 15 14-Dec 16 12-Dec 16 11-Dec 19 9-Dec 19 7-Dec 20 5-Dec 20 3-Dec 21 ELAST Greg *Can - EL CIB 21-Jun 0 19-Jun 1 17-Jun 1 15-Jun 1 13-Jun 1 11-Jun 1 9-Jun 2 7-Jun 2 6-Jun 2 4-Jun 2 2-Jun 3 31-May 3 29-May 3 AHAU 26-Jan -4 24-Jan -4 22-Jan -3 20-Jan -3 18-Jan -3 16-Jan -2 14-Jan -2 12-Jan -2 11-Jan -1 9-Jan -1 7-Jan 0 5-Jan 0 3-Jan 1 KAN 2-Sep 7 31-Aug 7 29-Aug 7 27-Aug 8 25-Aug 8 23-Aug 8 21-Aug 8 19-Aug 8 18-Aug 9 16-Aug 9 14-Aug 9 12-Aug 9 10-Aug 9 MFIRST Greg *Can - MF KAN 29-Jun -2 27-Jun -2 25-Jun -2 23-Jun -1 21-Jun -1 19-Jun -1 17-Jun -1 15-Jun 0 14-Jun 0 12-Jun 1 10-Jun 1 8-Jun 1 6-Jun 2 LAMAT 3-Feb -1 1-Feb -1 30-Jan 0 28-Jan 0 26-Jan 0 24-Jan 0 22-Jan 1 20-Jan 1 19-Jan 1 17-Jan 2 15-Jan 2 13-Jan 2 11-Jan 3 EB 10-Sep -1 8-Sep -1 6-Sep 0 4-Sep 0 2-Sep 1 31-Aug 1 29-Aug 2 27-Aug 2 26-Aug 3 24-Aug 3 22-Aug 3 20-Aug 4 18-Aug 4 Error ! Not a valid link. 6 Human Mosaic Table 3. Seasonal pattern in the Dresden Codex Venus table (continued). Error! Not a valid link. MLAST Page # Greg *Can - ML EFIRST Greg *Can - EF interesting nature of the actual synodic periods of the particular Venus events and how they vary seasonally. For example, the difference between canonical and actual ELAST events varies from –4 to +10 days. It appears that when ELAST occurs before the summer solstice or after winter solstice, the synodic interval will be very close to the average 584 days. However, when ELAST occurs near the equinoxes, the synodic period becomes as many as 11 days shorter in the case of the autumnal equinox, or 10 days longer in the case of vernal equinox. This fact is most notable on page 50 of the Venus Table, where the period is roughly 580 days, when ELAST occurs in November on the first row. However, as the date of the event drifts toward the fall equinox, the actual event occurs earlier and earlier, making the distance between canonical ELAST and actual ELAST grow to as many as 10 days. In stark ELAST Greg *Can - EL Greg MFIRST *Can - MF contrast, the difference between canonical MFIRST and actual MFIRST is very stable from page to page. In fact, the variability seems to be common to Venus’ disappearance events, because the actual MLAST events also show greater seasonal variability than their EFIRST counterparts. Perhaps this is the reason why MLAST and ELAST seem to have been related by the Maya in their canonical association. Along with all the other numbers discussed above, the table of multiples on the introductory page (24) contains a number 1.5.5.0 in the Maya numeral system equal to 9100 in our decimal system. This number is located on the same line with the “aberrant” multiples and is associated with the tzolkin date 1 Ahau. As Michael Closs (1977) notes, this is the exact distance in days between two 1 Ahau dates in the Venus Table. On page 47, a 1 Ahau 6 Human Mosaic ELAST date occurs 9100 days after a 1 Ahau MLAST date. Indeed, 9100 is the exact interval in days between any MLAST tzolkin date and the identical tzolkin date in ELAST. The number 9100 is also an even Table 4. Actual and canonical Venus events for page 46. Greg CIB Canonical 17-Jul 15-Jul 13-Jul 11-Jul 9-Jul 7-Jul 5-Jul 3-Jul 1-Jul 30-Jun 28-Jun 26-Jun 24-Jun Greg CIB Canonical 21-Jun 19-Jun 17-Jun 15-Jun 13-Jun 11-Jun 9-Jun 7-Jun 6-Jun 4-Jun 2-Jun 31-May 29-May MLAST Greg Actual 16-Aug 13-Aug 11-Aug 9-Aug 7-Aug 5-Aug 3-Aug 1-Aug 30-Jul 27-Jul 25-Jul 23-Jul 21-Jul ELAST Greg Actual 21-Jun 18-Jun 16-Jun 14-Jun 12-Jun 10-Jun 7-Jun 5-Jun 4-Jun 2-Jun 30-May 28-May 26-May *Can - ML Greg CIMI Canonical Actual 15-Oct 30-Sep 13-Oct 28-Sep 11-Oct 25-Sep 9-Oct 23-Sep 7-Oct 20-Sep 5-Oct 18-Sep 3-Oct 15-Sep 1-Oct 13-Sep 29-Sep 10-Sep 28-Sep 8-Sep 26-Sep 6-Sep 24-Sep 3-Sep 22-Sep 1-Sep Interval (days) -30 -30 -29 -29 -29 -29 -28 -28 -28 -25 -25 -25 -25 *Can - ML Interval (days) 0 1 1 1 1 1 2 2 2 2 3 3 3 EFIRST Greg Greg MFIRST Greg KAN Canonical Actual 29-Jun 1-Jul 27-Jun 29-Jun 25-Jun 27-Jun 23-Jun 24-Jun 21-Jun 22-Jun 19-Jun 20-Jun 17-Jun 18-Jun 15-Jun 15-Jun 14-Jun 14-Jun 12-Jun 11-Jun 10-Jun 9-Jun 8-Jun 7-Jun 6-Jun 4-Jun multiple of 260 (35 x 260 = 9100). Since the two 1 Ahau dates (1 Feb [MLAST] and 1 Jan [ELAST]) are also in the same season, exactly 30 days apart, I thought it might be interesting to investigate the apparent seasonal structure in the tzolkin. Table 5 contains a cumulative count of 260day multiples, as well as the relative position with respect to the 365-day calendar written in months. As you can see, a repeating seasonal pattern of seven tzolkin periods equaling 1820 days does exist, and 9100 is an even multiple of this pattern. *Can - ML Interval (days) 15 14 16 16 17 16 18 18 19 21 21 22 22 *Can - ML Interval (days) -2 -2 -2 -1 -1 -1 -1 0 0 1 1 1 2 Michael Closs (1977:97) also noted that the number 9100 was equal to 15 canonical synodic Venus periods plus 340 days (15 x 584 + 340 = 9100). Perhaps this partly explains the choice the Maya made for the canonical position of MLAST in the Venus Table, as the number of days between MLAST and ELAST is equal to 340. This would certainly establish a seasonal connection; however, there may have been more to it than just earthly seasons. 6 Human Mosaic Table Structure: Canonical Intervals and Astronomy Because of the close approximation of the canonical intervals with the synodic period of the moon, Aveni (1992a) suggested that the two were possibly related. He noted that the canonical Venus intervals closely approximate multiples of the lunar synodic month, or the Table 5. Seasonal pattern in the tzolkin. 260 days (Sep) 520 days (Jun) 780 days (Feb) 1040 days (Nov) 1300 days (Jul) 1560 days (Apr) 1820 days (Dec) 2080 days (Sep) 2340 days (May) 2600 days (Feb) 2860 days (Nov) 3120 days (Jul) 3380 days (Apr) 3640 days (Dec) 3900 days (Sep) 4160 days (May) 4420 days (Feb) 4680 days (Oct) 4940 days (Jul) 5200 days (Mar) 5460 days (Dec) 5720 days (Sep) 5980 days (May) 6240 days (Feb) 6500 days (Oct) 6760 days (Jul) 7020 days (Mar) 7280 days (Dec) 7540 days (Aug) 7800 days (May) 8060 days (Jan) 8320 days (Oct) 8580 days (Jul) 8840 days (Mar) 9100 days (Dec) Table 6. Venus synodic events and the moon’s synodic period. Year 935 937 938 VENUS MLAST EFIRST 17 July 15 Oct 20 Feb 21 May 26 Sep 25 Dec Last 7 July 17 Feb 30 Sep interval between successive full moons (29.53 days). The canonical period of visibility as morning star, 236 days, is only six hours less than eight lunar synodic months. Also, the 90-day interval associated with the disappearance of Venus at superior conjunction is only a little more than a day greater than three lunar synodic months. While this is enticing, the synodic period of Venus (584 days) is almost six days earlier than the closest multiple of the lunar synodic month (20 x 29.53 = 590.6), which would cause a significant amount of drift in correlation between Venus synodic stations and the moon phases. To demonstrate this, I collected the days of first and last visibility of the moon for the years 935 to 938. Table 6 shows the closest matches from this search. While there seems to be a rather close relationship between Venus and moon events in the year 937, the pattern does not recur throughout the table. If the lunar synodic period were a factor in setting up the Venus intervals, the concept is not clearly demonstrated by this research. Pages 46–50 contain three drawings on the right-hand side of each page. The uppermost drawing depicts individuals seated on what appear to be skybands, while the middle and lower two drawings show figures hurling darts and victims being wounded, respectively. Captions written between the drawings reveal that the lower two are associated with MFIRST events (V. Bricker 2001). Aveni (1992a:90) suggested that the appearance of the lunar First 9 July 19 Feb 1 Oct MOON Last 6 Oct 16 May 30 Dec First 9 Oct 18 May 28 Dec goddess I in the upper right-hand side of page 49 could also signify some association of the moon with Venus. I loaded the date, 8 April 941, the date of the ELAST event that occurs in the first row on page 49, into the Starry Night software for 20.5˚ N latitude and 88.5˚ W longitude (corresponding to the Yucatan peninsula). From the point of view of an observer facing west at the sunset, the moon would have appeared as a bright crescent directly above Venus. Similarly, V. Bricker (2001) suggested that the figure seated on a sky-band in the upper right-hand side of page 46 is a representation of the summer solstice, and the ELAST date in the first row of the same page did occur on the June solstice in 936. These two similarities may suggest a correlation between the respective figures in the upper right-hand side of each page and the respective ELAST occurrences on the same page. There is certainly enough evidence to warrant further investigation. In an attempt to find an astronomical relationship that would bring meaning to the canonical Venus intervals, I followed many avenues, some of which include Mars/Venus conjunctions, Venus/Mercury conjunctions, Saturn/Venus conjunctions, last and first visibility of Jupiter and Mars, and the synodic intervals of Mercury. The results of some of these findings are summarized in Table 7. Although it is intriguing that the interval between Mercury’s MLAST and MFIRST is 80 days, a close approximation of the canonical 90-day period, none of these investigations led to anything. 6 Human Mosaic Aveni et al. (2003) have compiled information concerning the sidereal periods of the planets. The astronomical sidereal period of a planet generally relates to the length of its revolution around the sun. This is not directly measurable by a viewer on Earth; however, a terrestrial observer can mark the sidereal pattern of a particular planet by noting its celestial longitude. For example, an observer can mark when a planet such as Venus passes a certain star or constellation along the ecliptic, and begin counting days until once again Venus is in the same position. This would be the sidereal period of Venus as viewed from Earth. In my search, I found that ELAST and MLAST share a relationship with respect to Venus’ sidereal period. Table 7. Venus periods and astronomical events. Year Can ML 935 17-Jul 937 20-Feb 938 26-Sep 940 3-May 941-42 7-Dec 943 14-Jul 945 17-Feb 946 24-Sep 948 30-Apr Can EF 15-Oct 21-May 25-Dec 1-Aug 7-Mar 13-Oct 19-May 24-Dec 30-Jul Can EL 21-Jun 26-Jan 2-Sep 8-Apr 13-Nov 19-Jun 24-Jan 31-Aug 6-Apr Can MF 29-Jun 3-Feb 10-Sep 16-Apr 21-Nov 27-Jun 1-Feb 8-Sep 14-Apr Saturn-Venus Conj 13-May 29-Dec 1-Apr 17-Nov 4-Jul 24-May 14-Apr 30-Nov 18-Jul 7-Jun 23-Jan 10-Sep Year Can ML 935 17-Jul 937 20-Feb 938 26-Sep 940 3-May 941-42 7-Dec 943 14-Jul 945 17-Feb 946 24-Sep 948 30-Apr Can EF 15-Oct 21-May 25-Dec 1-Aug 7-Mar 13-Oct 19-May 24-Dec 30-Jul Can EL 21-Jun 26-Jan 2-Sep 8-Apr 13-Nov 19-Jun 24-Jan 31-Aug 6-Apr Can MF 29-Jun 3-Feb 10-Sep 16-Apr 21-Nov 27-Jun 1-Feb 8-Sep 14-Apr Mars-Venus Conj 9-May 4-Mar 1-Feb 30-Dec 25-Oct 2-Oct 22-Aug 2-Aug 1-Jun 16-Apr 12-Feb Venus Year Can ML 935 17-Jul 937 20-Feb 938 26-Sep 940 3-May 941-42 7-Dec 943 14-Jul 945 17-Feb Can EF 15-Oct 21-May 25-Dec 1-Aug 8-Mar 13-Oct 19-May MF 2-Jul 16-Jan 1-Sep 22-Mar Mercury-Venus Conj 14-Mar 30-Jul 5-Jan 14-Mar 19-Jul 19-Dec 22-May 25-Sep 10-May 28-Sep 29-Nov 24-Apr 28-Sep 2-Feb 5-Jul Mercury MARS Jupiter ML MF Last Vis First Vis Last Vis First Vis 27-Jul 20-Oct 12-Jul 5-Aug 4-Mar 25-May 12-Sep 6-Oct 24-Sep 15-Dec 25-Jan 13-Oct 5 Nov 4-May 28-Jul 15-Dec 3-Dec 12-Feb 15-Mar 2-Jul 19-Jan 14-Feb 4-Jul 27-Sep 8-Feb 25-Apr Using planetarium software,2 I found that Venus would appear to an observer on the Yucatan peninsula (20.5˚ N, 88.5˚ W) to be in the same position with respect to the stars Castor and Pollux in the constellation Gemini (Figure 1) on both the ELAST and MLAST canonical 2. I used EZCOSMOS version 3.0 software by Future Trends Software, Inc. for the initial study. The data were verified using Starry Night Pro version 4.5.2 by Imaginova. dates recorded on page 46. On page 47, where both MLAST and ELAST occur on Ahau days, Venus appears in the constellation Aquarius (Figure 2). The Kan MLAST and ELAST dates on page 48 place Venus in the constellation Virgo (Figure 3). The Lamat MLAST and ELAST dates on page 49 place Venus in the constellation Aries (Figure 4). Finally, the canonical Eb ELAST and MLAST dates occur when Venus is located in the same position with respect to the star Antares, located in the 6 Human Mosaic constellation of Scorpio (Figure 5). While the Maya most likely did not recognize the western zodiac, they surely could have used asterisms, or even prominent stars such as Antares, Castor, and Pollux, to mark the planet’s motion along the ecliptic. Thus, it is quite possible that the Maya used Venus’ sidereal position along the zodiac in selecting the canonical dates for the synodic event of MLAST. This sidereal relationship, coupled with the need to maintain the same tzolkin day for both events, as discussed above, would necessitate the 340-day interval between MLAST and ELAST, because any other multiple of 20 days, while maintaining the day name relationship, would have placed Venus quite distant from the same sidereal station. a 6 Human Mosaic b Figure 1. Venus’ position with respect to the stars Castor and Pollux (labeled) in the constellation Gemini for MLAST and ELAST on page 46 of the Dresden Codex. EZCOSMOS 3.0 software set at 20.5 N latitude and 88.5 W longitude: (a) MLAST, 17 July 935, 3 Cib; (b) ELAST, 21 June 936, 5 Cib. a Volume 36 7 b Figure 2. Venus’ position with respect to the constellation of Aquarius (labeled) for MLAST and ELAST on page 47 of the Dresden Codex. EZCOSMOS 3.0 software set at 20.5 N latitude and 88.5 W longitude: (a) MLAST, 20 February 937, 2 Ahau; (b) ELAST, 26 January 938, 4 Ahau. a 8 Human Mosaic b Figure 3. Venus’ position with respect to the constellation of Virgo (labeled) for MLAST and ELAST on page 48 of the Dresden Codex. EZCOSMOS 3.0 software set at 20.5 N latitude and 88.5 W longitude: (a) MLAST, 27 September 938, 1 Kan; (b) ELAST, 3 September 939, 3 Kan. a Volume 36 9 b Figure 4. Venus’ position with respect to the constellation of Aries (labeled) for MLAST and ELAST on page 49 of the Dresden Codex. EZCOSMOS 3.0 software set at 20.5 N latitude and 88.5 W longitude: (a) MLAST, 3 May 940, 13 Lamat; (b) ELAST, 8 April 941, 2 Lamat. a 10 Human Mosaic b Figure 5. Venus’ position with respect to the star Antares (circled) in the constellation of Scorpio for MLAST and ELAST on page 50 of the Dresden Codex. EZCOSMOS 3.0 software set at 20.5 N latitude and 88.5 W longitude: (a) MLAST, 8 December 941, 12 Eb; (b) ELAST, 13 November 942, 1 Eb. Discussion and Conclusions In this paper, I have tried to investigate the criteria for establishing the canonical periods for the synodic events in the Venus Table of the Dresden Codex. It appears that the canonical intervals were defined to maintain a seasonal relationship between MLAST, ELAST, and MFIRST on the one hand, and EFIRST on the other. While the assigned dates for both MFIRST and ELAST reveal a desire to predict accurately the actual events, the canonical assignments of MLAST and EFIRST quite obviously depart from reality. I have argued that the EFIRST assignment may have been made to maintain the seasonal structure of the table. The MLAST assignment may have been a consolidation of many factors, including: maintaining the seasonal relationship, the sidereal relationship, and the same tzolkin day as ELAST. Focusing on this MLAST/ELAST relationship, I would like to investigate some aspects of the cultural significance of the structure of the Venus Table. The Popol Vuh, a Quiche Maya mythological work that contains obvious influences from Central Mexico (Tedlock 1996), may provide a cosmological perspective for what is depicted on the pages of the Dresden Codex Venus Table. The central part of the story involves the adventures of a pair of Hero Twins, Hunahpu and Xbalanque, who could quite possibly be mythical representations of Venus as morning and evening star. Similarly, the Central Mexican deity Quetzalcoatl, who has definitively been associated with Venus as morning star (Bierhorst 1992:28–37), also had a twin Xolotl, who could correspond to Venus as evening star (Šprajc 1993a:30). Ivan Šprajc (1993a) pointed out that the name of the father of the Hero Twins, Hun (One) Hunahpu, is the equivalent of the lowand Maya date 1 Ahau, which is interesting because the Venus Table starts with the tzolkin date 1 (Hun) Ahau. Dennis Tedlock (1996:206–207) suggested that the characters One and Seven Hunahpu (Ahau), the Hero Twins’ father and uncle, respectively, represent different appearances of Venus as morning star, whereas the characters One and Seven Death (Cimi) represent Venus’ appearance as evening star. This notion is based on an association of the characters with MFIRST and EFIRST, which seems plausible because these names (Ahau and Cimi) are tzolkin day names that appear in the Venus Table on MFIRST and EFIRST events. In Tedlock’s assessment, the coefficients 1 and 7 before the names symbolize the entire span of the 13 numerals in the tzolkin. Since none of these numerals repeats in any column of the Venus Table, 1 and 7 represent a column of the table. Thus, One and Seven Death represent the 6 entire EFIRST column on page 46, and One and Seven Hunahpu the entire MFIRST column on page 50. What about the disappearance events, MLAST and ELAST? It is these events in the table that are matched by tzolkin day names, and I suggest they may represent the Hero Twins themselves. Some of the characters in the Popol Vuh are thought to represent stellar constellations. For example, 7 Macaw has been associated with the asterism of the Big Dipper (Tedlock 1996:240). If the Maya were indeed paying attention to the sidereal motion of the celestial bodies, constellations would have served as a useful means of memorizing the regions of the sky. If constellations were associated with mythical characters, such as the Four Hundred Boys, Zipacna, etc., then it is possible that Venus’ motion in the sky with respect to the background of stars could be described as interaction with these mythical figures. This was a common practice of many ancient astronomers from Greece to China, and I believe the story of the Hero Twins’ adventures is a Mesoamerican example of such a myth. Furthermore, it may be the mnemonic device that matches the pages of the Venus Table in the Dresden Codex, where Venus’ sidereal position during its last visibility as morning star and evening star relates to the Hero Twins’ adventures in the upper and lower worlds. If in some way the characters that the twins encounter in the Popol Vuh in the upper or lower worlds can be associated with stellar constellations, perhaps then the story can be fitted onto the pages of the Dresden Codex Venus Table. In any case, the fact that the canonical MLAST date is adjusted so that it shares the same tzolkin day name, season, and sidereal position with the corresponding ELAST date on any given page of the Venus Table certainly supports the idea that the Maya desired to connect the similarities between Venus as morning and evening star. Perhaps these are the Hero Twins of the story. Acknowledgments I am greatly indebted to Anthony Aveni, Gabrielle Vail, Tim Knowlton, and the students of Seminar in Mayan Archaeoastronomy (Spring 2003, Tulane University) for their comments and suggestions on this paper. In addition to this, I am grateful for the guidance I received from Harvey and Vicki Bricker on Mayan hieroglyphs, astronomy, and codices throughout my time at Tulane University. References Cited Aveni, Anthony F. 1992a The Moon and the Venus Table: An Example of Commensuration in the Maya Calendar. In The Sky in Mayan Literature. Anthony F. Aveni, ed. Pp. 87–101. New York: Oxford University Press. 1992b The Sky in Mayan Literature. New York: Oxford University Press. 2001 Skywatchers: A Revised and Updated Version of Human Mosaic Skywatchers of Ancient Mexico. Austin: University of Texas Press. Aveni, Anthony F., Harvey M. Bricker, and Victoria R. Bricker 2003 Seeking the Sidereal: Observable Planetary Stations and the Ancient Maya Record. Journal for the History of Astronomy 34:145–161. Aveni, Anthony F., and Horst Hartung 1991 Archaeoastronomy and the Puuc Sites. In Arqueoastronomía and Etnoastronomía en Mesoamerica. J. Broda, S. Iwaniszewski, and L. Maupomé, eds. Pp. 65–95. Mexico: Universidad Nacional Autónoma de México. Bierhorst, John 1992 History and Mythology of the Aztecs: The Codex Chimalpopoca. Tucson: The University of Arizona Press. Bricker, Harvey M., and Victoria R. Bricker 2002 Astronomical Tables in the Maya Codices. Workshop presented at the First Annual Tulane Maya Symposium: Archaeology, Astronomy, and Texts from the Northern Maya Lowlands. Tulane University, New Orleans. Bricker, Victoria R. 2001 A Method for Dating Venus Almanacs in the Borgia Codex. Archaeoastronomy 26 (Supplement to the Journal for the History of Astronomy 32):S21–S43. Closs, Michael P. 1977 The Date–Reaching Mechanism in the Venus Table of the Dresden Codex. In Native American Astronomy. Anthony F. Aveni, ed. Pp. 89–99. Austin: University of Texas Press. Díaz, Gisele and Alan Rodgers 1993 The Codex Borgia: A Full-Color Restoration of the Ancient Mexican Manuscript. New York: Dover. Gibbs, Sharon L. 1977 Mesoamerican Calendrics as Evidence of Astro- Volume 36 nomical Activity. In Native American Astronomy. Anthony F. Aveni, ed. Pp. 21–36. Austin: University of Texas Press. Milbrath, Susan 1999 Star Gods of the Maya: Astronomy in Art, Folklore, and Calendars. The Linda Schele Series in Maya and PreColumbian Studies. Austin: University of Texas Press. Sahagún, Fr. Bernardino de 1953 Florentine Codex: General History of the Things of New Spain. Book 7: The Sun, Moon, and Stars, and the Binding of the Years. Arthur J. O. Anderson and Charles E. Dibble, trans. Monographs of The School of American Research, no. 14, pt. VIII. Santa Fe: School of American Research, and the University of Utah. Šprajc, Ivan 1993a The Venus–Rain–Maize Complex in the Meso- 7 american Worldview, part I. Journal for the History of Astronomy 24:17–70. 1993b The Venus–Rain–Maize Complex in the Mesoamerican Worldview, part II. Archaeoastronomy 18 (Supplement to the Journal for the History of Astronomy 24):S27–S53. Tedlock, Dennis 1996 Popol Vuh: The Mayan Book of the Dawn of Life. New York: Simon and Schuster. Thompson, J. Eric 1972 A Commentary on the Dresden Codex: A Maya Hieroglyphic Book. Memoirs of the American Philosophical Society, 93. Philadelphia.