THE MATHEMATICS OF THE
MAYAN
M. 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

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?
2
3
2
1
3
1
4
4
5
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 20day cycle in the system using both
calendars. Once every 20 days, it will be
New Year’s Day on both calendars
4 x 5 = 20
5 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 positionwith the same tooth 1 and space 1 meshing
together again?
2
2
3
1
4
3
1
4
6
5
You found that tooth 1 and space 1 line
up again after only two turns of the 6gear 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