# Mayan math

INTRODUCTION Two mathematical systems, one linear and the second modular, encoded Mayan astronomical texts. The linear system was positional with respect to the Olmec long count bases 18 and 20. The long count recorded day numbers in 1-18 and 1-20, respectively.

Two examples 1.1.1.1 and 1.1.0.0.0.0 disclose distance numbers that are nearly equal to 19 sidereal cycles of Jupiter and 400 sideral cycles of Saturn, respectively, thereby extending visual retrogrades discussions of the planets (Brickers). Mayans therefore nominally selected 399 for Jupiter and 378 for Saturn within a methodology that commenserated actual and nominal cycles of Mercury, Venus, earth, and Mars in paired planetary almanacs, measured against the paired Saturn and Jupiter long count.

The modular system also recorded base 13 remainder arithmetic in a manner that defined least common multiple (LCM) almanacs. Almanac remainders were written in red and black, and often double and triple checked, recorded multiples of calendar rounds 18,980 days. Several LCMs 117, 260, 360, 584, 585, 780 and higher scaled planetary cycles in base 13 remainders. Calendars were recorded 1-13 remainders and distance numbers were recorded in 0-12 remainders.

The long count can be seen as a numeration system predated the Maya positional system that allowed Mayans to encode 260 day, 360 day, 364 dates followed by four-part lunar months and days and solar months and days. The linear system followed the sun across the window of the annual migration of solstices equinoxes. The mid-year summer solstice (SS) marked the beginning of the primary Mayan solar time-keeping method.

Mayan astronomy linked lunar and solar modular calendars that followed aspects of the Chinese ’string of pearls’ visual approach in families of almanacs (i.e Dresden Codex) that discussed nominal and actual planetary cycles.

The linear aspect, written from right to left by Mayans, is written here left to right to conform with modern conventions , was positional, with exponent n = 0, 1, 2, …:

The long count calculated four distance numbers on a 419 AD wall (near Tikal. Guatemala) with four super-numbers. The first is divisible by 117, 260, 360, 365, 584, 585 and 780: Mercury, lunar, earth, Venus, Mars and 18 periods of the Mayan 52 year calendar round period of 18980 days; the second divisible by 117, 260, 360, 364, 365, 780 and 63 (18980); the third by 117, 260, 365, 780 and 91(18980), and the fourth by 117, 260, 365, 780 and 129(18980) as parsed by:

1. 341640 = (8)(365)(117) = (2)(3)(3)(73)(260) = (13)(73)(360) = (6)(156)(365)= (5)(9)(13)(584) = (6)(73)(780)= (3)(6)(18980) = 18-CR. An independent analysis of 120 LCM combinations of Mayan nominal planetary cycles 260, 360, 364, 365, 584, 585 and 780 yields 18-CR 28 times. The larger LCM CR data base is ”implied” and thus not on the wall.

In 2012 Aveni, et al, data reported the remaining super-numbers:

2. 1195740 = (4)(7)(365)(117) = (3)(3)(7)(73)(260) = (3)(3)(5)(73)(364)= (21)(156)(365)= (4)(7)(73)(585)= (21)(73)(780)= (3)(21)(18940) = 21(56940) meant LCM (260, 364, 365, 585) = LCM (364, 365, 585,780)= LCM (260, 364, 365, 585, 780) = LCM(364, 365, 584, 585).

3. 1765140 = (31)(219)(260) = (31)(156)(365) = (31)(73)(780)= (3)(31)(18980) = 31(56940)= LCM (260, 365, 780, 2263)

4. 2448420 = (43)(219)(260) = (43)(156)(365) = (43)(73)(780) = (3)(43)(18980)= 43(56940) = LCM (260, 365, 780, 3139)

Quotients of 260, 365, 780 and 56940 validates a Mars focus per 73(160) = 11680 rather than 11679 cited by Aveni and the Brickers. Note 20 Venus synodic cycles approximates 11680, off by one day. Powell ”New View of Mayan Astronomy” detail Mayan LCM methods that expose Mars, Jupiter and Saturn nominal cycles and methods that coincide with the LCM premise of this paper.

The planets Mercury, Venus, Mars, Saturn, and Jupiter, plus the moon lined up in ”string of pearls” combinations that aligned Chinese calendars to Feb. 2, 1951 BCE (facts seen on Stellarum and other astronomical programs), related evnets that Mayans placed at the center of their mythic and scientific worlds that double checked Mayan calendars.

Mayan rational numbers scaled super-number distance numbers in exacting ways. Planetary almanacs divided calendar round periods that defined rational number quotients and day remainders.

Mayan calendars were recorded in day quotient and day remainders. Solar calendars used in China, India, and the Hellene world only used remainder arithmetic. Ancient Near East lunar calendars and weights and measures were also written in quotient and exact remainder arithmetic.

Joseph Needham that wrote of the CRT indeterminate equation solution method reached the Hellene and medieval worlds via the Silk road no later than 100 AD. Fibonacci used the CRT in the Liber Abaci that solve several non-astronomical indeterminate problems.

A 405-moon lunar calendar decodes Mayan arithmetic texts within a practical 3 x 4 (base 4 x base 5) abacus,Oxford Press, 1992.

*"The Question of Jupiter and Saturn, Astronomy in the Maya Codices"*, American Philosophical Society, 2011.

*"Elementary Number Theory*, Allyn and Bacon, 1976.

*"Nepohualtzintzin. Computador Prehispanico en Vigencia" [The Nepohualtzintzin: a pre-Hispanic computer in use]*, Mexico City, Mexico: Editorial Diana, 1977.

*"A Solution for the Number 1.5.5.0 of the Mayan Venus Table", THE SKY IN MAYAN LITERATURE, ed. A. Aveni*, Oxford Press, 1992.

*"Number Theory and its History"*, McGraw-Hill, 1948.

*"Arithmetic in Maya"*, Austin-Texas, 1961.