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The Calendars of the Maya

Dr. Ximena Catepillán Millersville University Dr. Waclaw Szymanski West Chester University

MILLERSVILLE UNIVERSITY OF PENNSYLVANIA MATHEMATICS IN NON-EUROPEAN CULTURES MATH 102 A survey of Mathematical ideas developed by nonEuropean cultures including, but not limited to, those of Africans, Asians, and native North, Central and South Americans. Includes culture, specific examples from the following areas of mathematics: number theory, topology, probability, group theory and logic.

Number Theory 1. Magic squares (African, Chinese). 2. Number systems (Inuit and Ojibway of Canada, Chumash of the Santa Barbara Channel). 3. Number words (Mayan, Nahuatl of Central Mexico, Yoruba of Africa). 4. Chinese triangle. 5. Mayan and Aztec calendars. 6. Javanese-Balinese calendar.

Topology 1. Graphs (Bushoong and Tshokwe of Africa). 2. Mazes (Malekula of the South Pacific Islands). 3. String Art (African Batwa Pygmies, Native American). Logical Structures 1. Kinship (Warlpiri of Australia, Tongans of the South pacific Islands). 2. Quipus (Inca coding methods).

Group Theory 1. Symmetry (African, Inca, Maori of New Zealand) 2. Geometric Patterns (African, Inca, Native American) 3. Groups (Warlpiri of Australia) Probability 1. Games of chance (Cayuga, Hupa, Cherokee) 2. Games of strategy (African, Native American, Maori of New Zealand) 3. Puzzles (Kpelle puzzles of Africa, African river crossing puzzles).

COURSE OBJECTIVES: Develop written and oral communication skills Develop analytical reasoning and problem solving skills Gain a broader understanding of the mathematical aspects of art Become more aware of the cultural diversity of mathematics Be able to relate western and non-western mathematical ideas

MILLERSVILLE UNIVERSITY OF PENNSYLVANIA STUDY ABROAD

"MATHEMATICS IN NON-EUROPEAN CULTURES" MATH 102 Summer 2010

Join Dr. Ximena Catepillán (mathematician) and Dr. Chris Powell (archaeologist) on this study abroad course and learn about ancient Maya History, Culture, and Mathematics as we travel through the ruins of the Yucatan Peninsula in Mexico

Credits

3-credits G2 D-label This course counts towards the required G2 mathematics course for nonmathematics and science majors, D-diversity course.

Dates

May 17 21 classes at Millersville University A survey of mathematical ideas developed by non-European cultures May 22 28 Yucatan Peninsula, Mexico A study of ancient Maya history, culture, and mathematics

Cost

$ 1,100 plus air and tuition

Contact

Dr. Ximena Catepillán Office: Room 211-C1, Wickersham Hall Phone: 717-872-3993 E-mail: [email protected] Web site: http://www.millersville.edu/~xcatepil/#travelsabroad For more information visit the Maya Exploration Center at http://mayaexploration.org/ View from a student's perspective (Mike West) http://mayanexploration.blip.tv The highest pyramid in Mexico, Coba Mysterious ancient roads The White Reading House Temple with tunnels inside leading to tombs at Ek-Balam El Castillo, a time temple in Chichen Itza El Caracol, a Maya observatory Sacred wells Ancient ball courts Lectures in Playa del Carmen a beautiful beach resort

Trip Highlights

Dr. Chris Powell, Maya Exploration Center Dr. Ximena Catepillán, Millersville University Summer 2009 August 22, Saturday Arrival Land in Cancun, meet MEC transport at the airport and travel to Playa Del Carmen Orientation meeting and course's first lecture, A Brief History of Ancient Maya Civilization. August 23, Sunday Seminars in Playa del Carmen 8-9am Breakfast 9-11am Free time to explore Playa del Carmen 11-12pm Lecture: Ancient Maya History 12-2pm Lunch break 2-3pm Lecture: Maya Numbers and Mathematics 4-5pm Lecture: Maya Calendars: Part 1 Overnight in Playa de Carmen

August 24, Monday Coba 7-8am Breakfast and hotel check out 8-1pm Transport to and tour of Coba ruins 1-4pm Lunch and travel to Chichen Itza 6-7pm Lecture: Ancient Maya Astronomy Overnight in Chichen Itza August 25, Tuesday Chichen Itza 7-8am Breakfast 8-1pm Tour of Chichen Itza 1-3pm Lunch and swimming at Cenote Ik'il 6-7pm Lecture: Maya Geometry in Art and Architecture Overnight in Chichen Itza

August 26, Wednesday Ek Balam 7-8am 8-1pm Breakfast and hotel check out Tour of Ek Balam, Lecture at the "White Reading House" Temple 1-3pm Lunch in Valladolid 3-6pm Transport to and check in at Playa del Carmen Overnight in Playa de Carmen

August 27, Thursday Playa del Carmen 8-9am Breakfast 9-3pm Free time 3-4pm Lecture: Maya Calendar Systems: Part II 5-6pm Lecture: 2012 and the 13th Baktun 6-8pm Closing Ceremonies and final dinner Overnight in Playa de Carmen

Millersville University Math 102 students in Chichen Itza, May 2007

Madrid Codex

17+8x20=177 8+ 7x20=148

THE CALENDARS

The Round Calendar uses a 52-year cycle composed of two different calendars: The Tzolkin Calendar and the Haab calendar. The Tzolkin Calendar is also called Tonalamatl or Sacred calendar. Uses 20 day names (Gods) with 13 days each

TZOLKIN CALENDAR

20 Gods (numbers 1 through 13)

Imix Ik Akbal Kan Chicchan Cimi Manik Lamat Muluc Oc Chuen Eb Ben Ix Men Cib Caban Eznab Cauac Ahau Sea Air Night Corn Serpent Death Deer Rabbit Rain Dog Monkey Broom Reed Jaguar Bird Owl Force Flint Storm Lord, Sun

These Gods' names had numbers 1,..., 13 attached. 4 Ahau, 7 Ben, 11 Oc Thus 13 numbers combined with 20 Gods' names gives us 260 days because lcm(13,20)=260 It begins with 1 Imix and ends with 13 Ahau

The Haab Calendar also called Vague, Solar or Civil was used by the farmers. Uses 18 months names (Gods) (uinals) of 20 days each to make a year of 360 days plus five "unlucky days" or Uayab.

HAAB CALENDAR

18 Gods (numbers 0 through 19)

Pop Uo Zip Zotz Tzec Xul Yaxkin Mol Chen Yax Zav Ceh Mac Kankin Muan Pax Kayak Cumku Uayab (month of 5 unlucky days)

The Maya merged both calendars into a superior cycle known as the Calendar Round. This combined calendar consists of 18,980 days 73 Tzolkin years 73x260 or 52 Haab years 52x365 Each Calendar Round date reoccurs every 18,980 days. Only every 18,980 days, one of the 260 Tzolkin days coincides with another of the 365 days of the Haab again.

260 = 2²·5·13 365 = 73·5 lcm = 2²·5·13·73 = 18,980

Long Count or Era Calendar

The Maya used the Long Count dates to mark events which occurred over long periods of time. The calendar comprised a period of 5,128 years. The shortest period was the kin (day) uinal = 20 kins tun = 18 uinals = 18x20 kins = 360 kins katun = 20 tuns = 20x18x20 kins = 7200 kins baktun=20 katuns=20x20x18x20kins=144000 kins Long count date 9.12.2.0.16 In this example 9 baktuns, 12 katuns, 2 tuns, 0 uinals , and 16 kins have gone since the beginning of the current era, i.e., 1,383,136 kins

1 1 1 1

Math 102 student in Cobá, Mexico, 2005

Long Count Date

12.19.15.6.6

The mythical beginning, according to the Maya, the day of the creation of the world was the combination 4Ahau 8 Cumku 0.0.0.0.0 August 11, 3114 BC According to the Maya the end of time will occur when the 13th baktun begins, i.e., 13.0.0.0.0, that is December 21, 2012 AD winter solstice in the northern hemisphere. 4Ahau 3 Kankin 13.0.0.0.0 December 21, 2012 AD An Era of 13 baktuns has 13x144,000=1,872,000 kins, which is approximately 5,128 years.

The following example illustrates how to convert the Long Count date 9.3.10.5.3 (9 baktuns, 3 katuns, 10 tuns, 5 uinals, 3 kins) to Round Calendar date. The date is carved in Stela D in Cobá, Mexico.

Stela D in Cobá, Mexico, 2009 (picture taken by

Brian Viera Math 102 Abroad student)

First we need to convert the long count date to number of kins. Baktun 9 X 144,000 = 1,296,000 Katun 3 X 7200 = 21600 Tun 10 X 360 = 3600 Uinal 5 X 20 = 100 Kin 3X1=3 Total 1,321,303 Kins

Now we divide the number by the number of days in the Round Calendar 1,321,303/ 18,980 = 69 + 11,683/18980 the remainder is 11,683 Next we divide the remainder by 13, the number of days attached to each God of the Tzolkin. 11,683/13 = 898 + 9/13 the remainder is 9.

Now we divide 11,683 by 20 (the number of Gods) and obtain a remainder of 3. Thus we have 9 for number of days and 3 for Gods. The mythical beginning of the Maya world was the combination 4 Ahau 8 Cumku 4 Ahau is the Tzolkin date and 8 Cumku the Haab date. Therefore we add 9 to 4, and 3 (Gods) to Ahau obtaining the Tzolkin date 13 Akbal

Now for the Haab date we divide 11,683 by 365 the number of Haab days and obtain a remainder 3. 11,683 /365 = 32 + 3/365 Next we divide the remainder 3 by 20 the number of days per God in the calendar. This is 0 months and 3 days. We add these numbers to 8 Cumku obtaining 11 Cumku.

The Round Calendar date is 13 Akbal 11 Cumku We used a slight different arithmetic among the calendar conversions because the Tzolkin calendar combines first God with number 1, second God with 2, etc. whereas in the Haab calendar we have first God with first number, first God with second number, etc.

Given the long count date

9.3.10.5.3

Stela D in Cobá, Mexico

Baktun Katun Tun Uinal Kin Total 9 X 144,000 = 1,296,000 3 X 7200 = 21600 10 X 360 = 3600 5 X 20 = 100 3X1 =3 1,321,303Kins

Total/ 18,980 = 69 + 11,683/18980 The remainder is 11,683 TZOLKIN 11,683/13 11,683/20

= 898 + 9/13 = 584 + 3/20

the remainder is 9 the remainder is 3

HAAB 11,683 /365 = 32 + 3/365 3/20 = 0 + 3/20 4 + 9 AHAU 3

the remainder is 3 This is 0 months and 3 days 8 3 CUMKU 0 months and 3 days

13 AKBAL

11

CUMKU

http://users.hartwick.edu/hartleyc/mayacalendar/mayacalendar.html

The Leyden Plaque contains the first text with a date in the standard notation of the Classic Period

Conversion Long Count date to Round Calendar date

Given the long count date 8.14.3.1.12 This is in the famous Leyden Plaque Baktun Katun Tun Uinal Kin Total 8 X 144,000 14 X 7200 3 X 360 1 X 20 12 X 1 = 1,152,000 = 100,800 = 1,080 = 20 = 12_________ 1,253,912 Kins The remainder is 1,232

Total/ 18,980 = 66 + 1,232/18980 1,232/13 = 94 + 10/13 1,232/20 = 61 + 12/20 1,232/365 = 3 + 137/365 137/20 = 6 + 17/20 4 +10 14 AHAU 12 EB 8 12 20

remainder is 10 remainder is 12 remainder is 137 This is 6 months and 17 days CUMKU 6 months &12 days (17 - 5 unlucky) YAXKIN (7th month)

1

EB

0

YAXKIN

The Dates of Stela A at Copan, Honduras

9.15.0.0.0 4 Ahau 13 Yax

9.15.19.5.0 4 Ahau 18 Muan

Computations with Long Count dates Adding dates: 5. 3. 9. 15. 1 + 6. 12. 14. 10. 3 __________________________________________ 11. 15. 23. 25. 4 11. 11. 15. 16. 24. 4. 7. 7. 4 4

1 uinal=20 kins 1 tun=18 uinals 1 katun=20 tuns 1 baktun = 20 katuns

Subtracting dates: 7. 6. 12. 14. 10. 3 __________________________________________ 1. -4. -11. 4. -1 1. 1. 0. -4. -5. 15. -11. 9. 9. 3. 3. 3. 19 19 19 8. 3. 14. 2

1 uinal=20 kins 1 tun=18 uinals 1 katun=20 tuns 1baktun=20 katuns

Zenith, May 2007 Chichen Itza

FIN

References Maya Exploration Center MEC http://www.mayaexploration.org

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