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The civilization of the Maya, which has existed since about 1800 BC on the Yucatan Peninsula, was at its peak during the Classic Period between 250 and 800 AD.

Classical Maya civilization was highly advanced and developed many areas of science. Perhaps the most significant one was astronomy, which was important for agriculture. Based on astronomical observations, the Maya invented an elaborate system of calendars. There are four calendars: Tzolkin Calendar, Haab Calendar, Round Calendar, and Long Count. In this article we discuss some relationships between the calendars from the mathematical standpoint. We illustrate our discussion with actual examples of dates found on a stela in Coba and on the Leyden Plaque, in which we show how to convert from Long Count dates into Tzolkin and Haab dates.

**Figure 1: **Millersville University students at Chichen Itza, Mexico, 2007 (Photo by Ximena Catepillán.)

**Tzolkin Calendar**

The Tzolkin Calendar, also called the Tonalamatl or Sacred Calendar, uses 20 names of gods with 13 days attached. The names of the gods are: Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etznab, Cauac, and Ahau.

The consecutive dates are formed as follows: 1 Imix, 2 Ik, 3 Akbal, 4 Kan, …, 12 Eb, 13 Ben, 1 Ix, 2 Men, …, etc.

As a result 260 dates are formed, which constitute 1 Tzolkin year. The year begins with 1 Imix and ends with 13 Ahau.

**Haab Calendar**

The *Haab* Calendar, also called the *Vague, Solar,* or *Civil *Calendar, was used by farmers. It uses 18 month names (gods) of 20 days each to make a year of 360 days plus five “unlucky days” or Uayeb.

The names of the gods are: Pop, Uo, Zip, Zotz, Tzec, Xul, Yaxkin, Mol, Chen, Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, Cumku, and Uayeb (month of 5 unlucky days).

To each god the numbers 0, …, 19 are attached as in our Gregorian calendar. The first date of the year is 0 Pop followed by 1 Pop, 2 Pop, …, 19 Pop, 0 Uo, 1 Uo, …, 18 Cumku, 19 Cumku, 0 Uayeb, 1 Uayeb, 2 Uayeb, 3 Uayeb, 4 Uayeb.

As a result 365 dates are formed, which constitute 1 Haab year.

**Round Calendar**

The Maya merged the Tzolkin and Haab calendars into a superior cycle known as the Calendar Round. The least common multiple of 260 and 365 is 18,980. Thus, this combined calendar consists of 18,980 days, which is 73 Tzolkin years (73 x 260), or 52 Haab years (52 x 365). An example of a Round Calendar date is 4 Ahau 8 Cumku, in which 4 Ahau is the Tzolkin date and 8 Cumku is the Haab date.

**Long Count Dates**

The Round Calendar date gives us the exact day within one cycle of 18,980 days, but we cannot locate the date in time as the Round Calendar cycles are not numbered (like our years, e.g. 2009). To locate dates over long periods of time, the Maya used Long Count dates.

The shortest period in the Long Count was the kin (day). Then:

1 uinal = 20 kins

1 tun = 18 uinals = 18 x 20 kins = 360 kins

1 katun = 20 tuns = 20 x 18 x 20 kins = 7200 kins, and

1 baktun = 20 katuns = 20 x 20 x 18 x 20 kins = 144,000 kins.

For example, the Long Count date 9. 12. 2. 0. 16 represents 9 baktuns, 12 katuns, 2 tuns, 0 uinals, and 16 kins. This means there have been 9 x 144,000 + 12 x 7200 + 2 x 360 + 0 x 20 + 16 = 1,383,136 kins since the beginning of the current era. An era or creation of 13 baktuns has 13 x 144,000 = 1,872,000 kins, which is approximately 5,128 years.

According to the *Popol Vuh**,* a book recording the creation story from Maya mythology, the beginning date, or the day of the creation of the world, is the combination 4 Ahau 8 Cumku (Round Calendar date) or 0. 0. 0. 0. 0 (Long Count date), which corresponds most recently to August 11, 3114 BC. We are now living in the fourth creation. The previous creations have ended at the conclusion of the 13^{th} baktun, or in Long Count dates, the day after 12. 19. 19. 17. 19. Another 12. 19. 19. 17. 19 will occur on December 20, 2012, followed by 13. 0. 0. 0. 0, or 4 Ahau 3 Kankin**,** on December 21, 2012, the day of the winter solstice in the northern hemisphere.

Ximena Catepillan (Millersville University of Pennsylvania) and Waclaw Szymanski (West Chester University of Pennsylvania), "Maya Calendar Conversions - The Maya and their Calendars," *Loci* (October 2010), DOI:10.4169/loci003536