THE MATHEMATICS OF THE MAYANM. Alejandra Sorto & Aaron Wilson

SMMG University of Texas

Austin March 31, 2012

MAYAN NUMERICAL SYSTEM

BASE-10 AND BASE-20 COUNTING

Base-10

1s = 0-9 10s = 10 100s = 10x10 1000s = 10x10x10 2012= (2x1000) (0x100) (1x10) (2x1)

Base-20

1s = 0-19 20s = 20 400s = 20x20 8000s = 20x20x20 2012= (0x8000) (5x400) (0x20) (12x1)

THE MAYAN CALENDARS

THE RITUAL CALENDAR OR TZOLKIN

Cycle of 20 days in combination with…

Cycle of 13 months to form…

260 uniquely named days of the year

THE SOLAR CALENDAR OR HAAB

18 “months” each with…

20 days (0 - 19) to form…

A cycle of 360 days, plus 5 (0-4) additional days

1 1

22

3

4

3

45

How many turns of each wheel does it take before you’re back to the starting position- with the same tooth 1 and space 1 meshing together again?

You found that five turns of the 4-gear (five groups of 4) will bring you the same place as four turns of the 5-gear (four groups of 5). So if the gears represent two different calendars, we can say that there is a 20-day cycle in the system using both calendars. Once every 20 days, it will be New Year’s Day on both calendars4 x 5 = 205 x 4 = 20

What about the two Mayan calendars, with 365 and 260 days? Will it take 94,900 days (365 x 260) for the two New Year’s Day to happen together again? That’s only once every 260 astronomical years!

How many turns of each wheel does it take before you’re back to the starting position- with the same tooth 1 and space 1 meshing together again?

1 1

2 3

4

56

2

34

You found that tooth 1 and space 1 line up again after only two turns of the 6-gear and three turns of the 4-gear.

Why is this so?

THIS CORRESPONDS TO THE MATHEMATICAL IDEA OF THE LEAST COMMON MULTIPLE (LCM)

Source: “The Mayan Calendar Round Keeping Time” by Bazin and Tamez.

WHEN WOULD THE TWO MAYAN CALENDAR COINCIDE?

THE CYCLE OF 18,980 DAYS – A CALENDAR ROUND

The combination of both calendars create a major cycle of 18,980 days (the LCM of 260 and 365: 5 x 52 x 73)

They will come together again after 52 astronomical years of 365 days each. The 52-year cycle is called “Calendar Round”

COUNTING FOR A LONG, LONG TIME“LONG COUNT” CALENDAR

Tun: 360-day “year” Katun: A period of 20 tuns (7, 200 days) Baktun: A period of 20 katuns (144, 000

days)

“Great Cycle” of the Long Count: A period of 13 baktuns = 5, 200 years long (in 360-day years).

The Great Cycle will be completed on December 21, 2012