# Maya Calendar Conversions - In Summary: Calendar Conversion Methods

Author(s):
Ximena Catepillan (Millersville University of Pennsylvania) and Waclaw Szymanski (West Chester University of Pennsylvania)

The method described in Examples 1 and 2 can be expressed in general terms as follows:

1. Compute the number of kins K.
2. Divide K by 18,980; the remainder is R.
3. Tzolkin date: Divide R by 13 and by 20 to obtain respective remainders A and B. To get the Tzolkin number, we imagine the numbers 1 through 13 on a circle, locate 4 and move A places ahead.  To get the Tzolkin god, we look at the list of Tzolkin gods and, imagining them on a circle, locate Ahau and move B places ahead.
4. Haab date: Divide R by 365; the remainder is H.
5. If 0 ≤ H ≤ 11, the Haab date is 8 + H Cumku.
6. If 12 ≤ H ≤ 16, the Haab date is H – 12 Uayeb.
7. Finally, if 17 ≤ H ≤ 364, the Haab date falls between 0 Pop and 7 Cumku (inclusive), and can be found as follows.
8. Compute H’ = H – 17.
9. Divide H’ by 20; the quotient is Q, the remainder is W.
10. The Haab number is W and the Haab month (god) is Pop + Q, so that the Haab date is W Pop + Q.

Finally, the Round Calendar date is the Tzolkin date followed by the Haab date.

More sophisticated mathematics students could approach these examples using more formal modular arithmetic notation. They would consider the dates in both the Tzolkin and the Haab calendars as ordered pairs (Number, God). In the Tzolkin Calendar, 1 ≤ Number ≤ 13 and God is taken from the Tzolkin 20-cycle of gods. The following day would have date given by (Number + 1 (mod 13), God + 1 (mod 20)) and, in fact, for any positive integer k, the day k days later would have date given by (Number + k (mod 13), God + k (mod 20)). In the Haab Calendar, 0 ≤ Number ≤ 19, and God is taken from the Haab 18-cycle of gods together with Uayeb. Here, the day after (Number, God) would have date (Number+1, God) if either God ≠ Uayeb and Number < 19, or God = Uayeb and Number < 4. Otherwise, the following day has date (0, God + 1 (mod 19)). Since the Haab date k days later, for any positive integer k, depends on how close the original date is to the five days of Uayeb, Haab dates for k > 1 are even more complicated to describe using modular notation.

Ximena Catepillan (Millersville University of Pennsylvania) and Waclaw Szymanski (West Chester University of Pennsylvania), "Maya Calendar Conversions - In Summary: Calendar Conversion Methods," Convergence (October 2010), DOI:10.4169/loci003536