Its amazing how ancient civilizations, had the creativity to develop custom numerical systems, that aproximated complex natural systems, with an accuracy that was surpassed only in the last centuries by our occidental culture
For example the predictive maya table of Solar and Lunar Eclipses (http://tzolkinhaab.googlepages.com/Tabla_Eclipses (broken link) _Mayas.pdf) that its totally integrated with his numerical astronomical time system...
Even more wonderfull, its the simplicity how were defined the time units from the visible planets trought (in equivalent modern concepts) sampling from a (clock signal) basic serie derived of the combination of two counting numeric systems (base 13 and base 20)
From the pdf, it looks like the theory i first saw in a book called "Los libros del tiempo" (buyed in the Monte Albán, Oaxaca bookstore) wich proposes that the mesoamerican calendar was based in numerical relations, derived from the observation of the visible planets and through the analisis of their astronomic periods
[URL="http://in.solit.us/archives/download/124955"]in.solit.us: Los Libros del Tiempo download page[/URL]
[URL="http://www.4shared.com/file/36499619/33662a16"]4shared.com - document sharing - download Los Libros del Tiempo.pdf[/URL]
[URL="http://www.mediafire.com/?cg7f11zm3iy"]Los Libros del Tiempo.pdf - ArqueoastronomÃ*a, maya, mesoamerica[/URL]
The mayas to designate a day used 20 names, counted with a numeric system base 13 (from 1 to 13), grouped in 5 days "weeks", and 20 days "months", to form 260 unique combinations like designations for days (number:name) in a repetitive calendar we call "tzolkin" divided in 65 days "seasons" (of course i am making a simil, werent called like that) Examples of the names are: Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etznab, Cauac, Ahau. An example of a tzolkin date could be "1 Imix"
The list of names had intercalated orientation (orient, nort, ponient, sout) so every "week" (5 days) started and ended with the same orientation... and every of the 260 designations of days informs elapsed days and orientation. They also divided the calendar in periods of 52 days (unique combinations of 13 numbers with 4 orientations)
This system its then based on the relation of two sets: the base 13 numbers and the 20 names. The consecutive asociation was done trought simple counting, making pairs... there were not needed any operations with positional notation.
All this counts (asociations) are cyclic... repetitive, but not identical because every repetition starts at a diferent point in the cycle. I think of them, like time units (for example [day
rientation] to name a "week") The name of the repetition its given by is starting day (for example "week" 1-nort) The relation of two sets with a difference in cardinality of 1 its very common in the maya "time units" (for example 5 days:4 orientations) and its called "movement efect" because the maya didnt have a word for "time", to talk about the elapsing of time they used words related to movement
The maya also used a calendar of 365 days we call haab (divided in 73 "weeks" of 5 days or 18 "months" of 20 days plus 1 week... of bad luck) The haab is an alternate form of the tzolkin that relates (trought consecutive asociation) the number of day (from 0 to 19) and name of the month. Examples of the names of the month are: Pop, Uo, Zip, Zotz, Tzec, Xul, Yaxkin, Mol, Chen, Yax, Zac, Ceb, Mac, Kankin, Muan... An example of a haab date could be "4 Ahau - 18 Pop". Therefore, the day 0 Pop was the maya new year, and the haab date (for example "1 Imix - 0 Pop") designed the name of the year. The tzolkin-haab dates gave the mayas 94,900 unique combinations (of 260 tzolkin dates with 365 days of haab months) to name his days.
Here comes the really interesting part... To measure longer periods of time, the maya keeped lists of dates "sampled" from the tzolkin-haab succession of days at some fixed interval. Doing a mathematical analysis of the numeric properties of this calendaric system, was found that some astronomical intervals produced lists of "sampled" dates sorted (as a whole or in part) in ascending, descending order or with a fixed marker. The astronomical intervals where aproximated because the mayas used only whole numbers.
For example here are some lists of "sampled" dates at fixed intervals (using for simplicity 1Imix-0Pop as origin, not the historical 4Ahau-8Cumhu)
365 days : 1Imix-0Pop, 2Cimi-0Pop, 3Chuen-0Pop, 4Cib-0Pop, 5Imix-0Pop, 6Cimi-0Pop, 7Chuen-0Pop, 8Cib-0Pop...
Here, obviously se fixed marker its the "0 Pop" or new year day
584 days : 13Chicchan, 12Muluc, 11Ben, 10Caban, 9Imix, 8Chicchan, 7Muluc, 6Ben, 5Caban, 4Imix, 3Chichan...
Aproximated Venus sinodic revolution
116 days : 13Caban, 12Ben, 11Muluc, 10Chicchan, 9Imix, 8Caban, 7Ben, 6Muluc, 5Chicchan, 4Imix, 3Caban...
Aproximated Mercury sinodic revolution
378 days : 2Cauac, 3Caban, 4Men, 5Ben, 6Chuen, 7Muluc, 8Manik, 9Chicchan, 10Akbal, 11Imix, 12Cauac...
Aproximated Saturn orbital sinodic period
780 days : 1Imix-10Zip, 1Imix-0Xul, 1Imix-10Mol, 1Imix-0Zac, 1Imix-10Mac, 1Imix-0Pax...
Aproximated maximun Mars oposition to Sun
399 days : 10Ahau, 6Cahuac, 2Etznab, 11Caban, 7Cib, 3Men, 12Ix, 8Ben, 4Eb, 13Chuen, 9Oc, 5Muluc...
Aproximated maximun Jupiter oposition to Sun (this succession its ordered in the names)
The mayas used what now we call "astronomical" days (that start/end on midday) in order to keep all the astronomic observations of one night on the same date. Divided the sky in sections, and for the record of some visible planet, used like reference point the presence of close deep space objects and changed sections and objects over time.
There is much more to this theory.... but i am late already, and dont have more time to write about it.
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