Mayan religion: the ceiba tree a central metaphor as a tree of life.  It existed in an underworld (roots) in the earth (trunk) and in the heavens (branches).  It exhaled the breath of Hunab K’U, the creative force.  Without trees, no life.  Human sacrifice was practiced to appease hungry gods, though probably not on Aztec scales (sometimes ca. 1000s in one festival for the Aztecs).   


Vigesimal Number System: The Maya also developed perhaps the most sophisticated number system before the decimal (base-10) system we use now (originating from the 15th century and utilizing Hindu-Arabic symbols and place-holding zeros).  The Mayan system is based on 20s (vigesimal = 20-based) and also used zeros to hold places and denote the somewhat abstract idea of nothing.  This numerical concept was not usable in the Roman numeral system (e.g. VII or MCMLIX), although the idea of nothing was certainly in the Latin language (nihil or nil).  It is often stated that the concept of “zero” was unknown to the Greeks and Romans and Medieval Europeans.  This is inaccurate.  The number “zero” generally did not exist in their computational systems, but the concept was clearly available to them.  [Ptolemy, an Egyptian from the 2nd c. AD in Roman–ruled Egypt, actually used a zero in his hybrid computational system, but this is generally disregarded by historians who usually don’t understand enough math to say anything coherent.  This is a good example of historians not knowing what they are talking about.]  None-the-less, the Maya had a zero, and it was utilized in their general system and was not just an anomalous occurrence.


Example of vigesimal numbers compared to our decimal system.


Decimal mimicking Vigesimal style

Decimal = Vigesimal

Vigesimal (std. notation)

1 = 1 x 100        (100 = 1)


1 = 1 x 200 = 1 x 1      (200 = 1)


3 = 3 x 100


3 = 3 x 200 = 3 x 1


10 = 1 x 101  (101 = 10)


10 = 10 x 200 = 10 x 1


20 = 2 x 101


20 = 1 x 201                 (201 = 20)


30 = 3 x 101


30 = (1 x 201) + (10 x 200)


100 = 1 x 102 = 1 x 100


100 = 5 x 201


125 = (1 x 102) + (2 x 101) + (5 x 100)


125 = (6 x 201) + (5 x 200)


135 = (1 x 102) + (3 x 101) + (5 x 100)


135 = (6 x 201) + (15 x 200)



This makes computation much, much easier to do on paper.  All you have to do is add or subtract corresponding powers. 



Vigesimal (same idea)


-   104

=  031

=    31


-  5|4

= 1|11

(= 31 in decimal notation)



Notes from Mann’s book (pp. 403-407) are in blue.
Mayan Calendar: The Mayan Calendar was arguably the most sophisticated calendrical system produced before the modern age.


The Maya had two time divisions (three if you count the combined Long Count) of note:


-One was 260 days long, called Tzolk'in.  It is in evidence as early as 200 BC.  It was divided into 13 20-day “months.”  It is 260 days long because this is the lowest common multiple of 13 and 20. 

            Mann describes this somewhat differently, calling the 13-day cycle the month because it is numbered, and the 20-day cycle as a week because its days are named like we name Tuesday and Wednesday. This system produces a unique combination which only repeats after 260-day cycle has been completed.


Me: example: first place is base 13 and the second place is base 20.  First day of the year would be 1.1 and the last would be 13.20.  If our months were consistently 30 days long, our system would cycle every 210 days.


The number 20, as already noted, is the basis for the Mayan number system and most literature likes to suggest that this number may have developed from the number of fingers and toes.  This is sheer speculation, but not unreasonable speculation in my estimation.  But it does make you wonder if shoe wearing is a significant factor in the development of number systems?

The number 13 is the number of layers in the Mayan heaven.


Why 260?

Perhaps…260 closely approximates the number of days in a typical pregnancy (calculated to be 266) and is roughly the number of days in a typical southern Mexican growing season.


Mann speculated that the visible times of Venus as morning star, may have inspired this system.  It’s visible in the morning for 263 days, then behind the sun for 50, then visible as the evening star for 263 days.


-The other was 365-days long (approximately a solar year) (called Haab') and was divided into 18 20-day “months” with 5 additional days considered to be unlucky tacked on at the end. When performing calendrical calculations, the Mayan number system was modified to incorporate this 18-“month” year.  Instead of the strict vigesimal (base-20) system, the 202 place (20 x 20) was changed to 18 x 20.  Thus one dot in this part of a number represented 360 [1 x 18 x 20], two dots equaled 720 [2 x 18 x 20], and so on… This is similar to our way of writing dates as 10/26/10.  Each place is not based on 10, but is based on the number of months, number of days, and number of years.  Try adding in this whacky system like we add our decimal number system or the vigesimal system.  Not very easy.  Of course adding or subtracting dates in our system is not so easy either.  You cannot simply add 15 days [0/15/0] to 10/26/2010.  You'd get 10/41/2010 and then have to convert the "41" to the next month with a remainder of days, taking into account that not all months have an equal number of days.  It's messy.


From Mann, p. 405.


The Long Count: Mann describes this quite differently.

The combination of these two cycles (the Tzolk'in and the Haab’) results in what is referred to as the “Long Count.”  If both systems start at the same time, it will take 52 Haab’ (years) for both systems to synch up again.  It will take 73 Tzolk'in (73 x 260 days). 

52 x 365 days = 18,980 days

73 x 260 days= 18,980 days


This cycle was of considerable importance.  It is coincidentally(?) about the lifespan of a long-lived Mayan (ca. 52 years).


As you can see, none of these systems include leap years of any sort, so this system simply accumulated “error” over the centuries and it cycled around completely every 1460 years. 


The sophistication of the Mayan calendar has captured the imagination of historians for many years.  The Mayan solar year has been calculated by some to be approximately 365.2422 days long, although the .2422 was not incorporated into the calendar.  (One should treat this number with much skepticism, but I’ll accept it for fun.) The modern value is generally given as approx. 365.24219 for what is referred to as the mean tropical year or approx. 365.2424 days for the vernal equinox year.  These are just slightly different ways to measure the solar year. The difference between these years is measured by the 1/10,000 of a day. Even if we use the vernal equinox year which differs from the Mayan year more significantly, the difference between the two years is only 18 seconds. It would take 200 years for the two systems to differ by an hour, and 4800 years to differ by a full day.  But it should be noted that the Maya did not treat fractional days in any way and their calendars did not reflect this level of accuracy in the way that we have leap years and leap seconds and the like, even though such accuracy appears to be inferred from their astronomical observations. 


b'ak'tun = 144,000 days  (18 X 203)

There is yet another system of time lasting approximately 394 years called b'ak'tun, a word which more or less refers to the calendrical number system which differed from the regular number system in that instead of the 202 place it used 18 x 20 (referred to above).  This 360-day grouping was called a tun  (18 x 201).  The next vigesimal place was called the k'atun  (18 x 202), and the next on the b'ak'tun (18 x 203).  One b'ak'tun, lasts 144,000 days, or about 394 years.


Using this system, the Maya established a beginning year: ca. 3114 BC.  (Compare with the beginning for the Jewish calendar: 3760 BC.)  Although there are references to events before this date in some inscriptions.  It was prophesized in some writings that at the 13th b'ak'tun the world would enter a new cycle of existence, whatever that means.  This new cycle corresponds to our year 2012 AD.  Did you see the movie?


The Maya wrote dates like so: # b'ak'tun – # ?, #?, # ???  ….Mann is horribly vague.  But they wrote the dates in 5 places from greatest to least division.  I’m thinking:   [X x 18 x 203], [Y x 18 x 202], [Z x 18 x 20], [# of 20 day cycle], [day in 365 system]


Me: the system seems to be based on this: 18 X 20n. 


An even longer division than the b’ak’tun was called the alautun, which is 23,040,000,000 days long (ca. 63 million years).  [18 X 207]


The counts in order of duration:

13 days, 20 days, 260 days, 365 days, 7,200 days, 18,980 days (= 52 years.  the lowest common multiple of 260 and 365), 144,000 days (ca. 394 years). 




Similar to the Maya, the Aztecs believed that a series of cosmic cycles that preceded the present time.



This is about 12 feet in diameter and is in Mexico City.


This shows the previous 4 ages of the world. 

The first age (upper right quadrant) tells of giants created by the god.  But they did not till the earth (no agriculture) and the gods sent jaguars to eat them. 

The second age (upper left quadrant) tells of another race created by the gods that didn’t work out.  They became apes and escape a windy destruction by clinging to the world tightly.

The third age (lower left quadrant) tells of another creation resulting in birds who escape a volcanic apocalypse. 

The fourth age (lower right quadrant) tells of another aborted creation which resulted in fish who escaped a cataclysmic flood.

In the next age (presumable the 5th epoch) we are what the gods created and we must work to appease the gods who will otherwise destroy the world with an earthquake. 


Because the gods made sacrifices in order that we may exist, it stands to reason that we should return the favor and sacrifice to the gods.  Hence, human sacrifice. 



Stonehenge – England                                                       A view of the Summer Solstice.


-Built in 3 phases spanning 3,100 to 1500 BC

-Think of it as a large clock.

-Summer solstice (June 21) sunrises exactly from the “Heel Stone” (see above)

-Winter Solstice and the two equinoxes are also indicated by stones as well as other aspects of lunar motions. 



Gregorian Calendar: Implamented by Pope Gregory XIII in 1582

Leap years are all years divisible by 4, with the exception of those divisible by 100, but not by 400. These 366-day years add a 29th day to February, which normally has 28 days. (So, in the last millennium, 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not. In this millennium, 2100, 2200, 2300 and 2500 will not be leap years, but 2400 will be.)

Easter established at first council of Nicaea in 325.

The problem was, people used a calendar to tell them when the vernal equinox was, rather than observation.


A new calendrical system was developed to replace the Julian system which was simply not working. When the new calendar was put in use, the error accumulated in the 13 centuries since the Council of Nicaea was corrected by a deletion of ten days. The last day of the Julian calendar was Thursday 4 October 1582 and this was followed by the first day of the Gregorian calendar, Friday 15 October 1582 (the cycle of weekdays was not affected). Nevertheless, the dates "5 October 1582" to "14 October 1582" (inclusive) are still valid in virtually all countries because even most Roman Catholic countries did not adopt the new calendar on the date specified by the bull, but months or even years later (the last in 1587).  Non-Catholic countries like England resisted this change but eventually gave in to its functionality.


Alaska adopted it in 1867 and Greece in 1923, China in 1912, Japan in 1873,