|Ivars Peterson's MathTrek|
March 12, 2001
In 1601, Johannes Kepler (1571-1630) undertook the challenge of deciphering the orbit of Mars and developing a mathematical theory of its motion to fit observations of the planet's changing position in the sky. In assuming that Earth itself traveled around the sun, Kepler's immediate hurdle was to find a way to disentangle Mars' motion from that of Earth. He then faced the daunting task of choosing an appropriate geometry for the two planetary orbits so that a line joining Mars and Earth and projected to the stars would correctly mark the position of Mars relative to the stars as seen from Earth.
What made Kepler's task feasible is the cyclic nature of planetary movements. For example, astronomers could determine the position of Mars on a certain date in the calendar for several years in succession. Because Mars travels more slowly in its orbit than Earth, its position in the sky, as viewed annually from Earth, differs from year to year. Astronomers could use such observations, carried out over a sufficiently long period of time, to plot the orbit of Mars.
Similarly, astronomers could deduce the shape of Earth's orbit by using the fact that Mars takes 687 days to travel around the sun relative to the stars. By collecting and plotting observations of Mars every 687 days, they would obtain the same data as they would by looking from a stationary Mars toward a moving Earth.
Kepler eventually found the answer that he sought. To do so, he had to propose that Mars and Earth followed elliptical orbits around the sun.
Remarkably, several centuries earlier in Central America, Mayan astronomers had developed their own model to describe the motion of Mars with uncanny accuracy. Anthropologists Harvey M. Bricker and Victoria R. Bricker of Tulane University in New Orleans and astronomer Anthony F. Aveni of Colgate University in Hamilton, N.Y., describe the evidence supporting the Mayan model in the Feb. 13 Proceedings of the National Academy of Sciences.
"While Kepler solved the sidereal problem of Mars by proposing an elliptical heliocentric orbit, a daring leap for its time, equally ingenious Maya astronomers, operating in a less abstract, earthbound frame of reference, managed to discover a pair of time cycles that not only accurately described the planet's motion but also married it to other cosmic and terrestrial concerns," the researchers remark.
The evidence resides within the so-called Dresden Codex, one of four written documents that have survived from pre-Columbian Mayan civilization. Scholars have long suspected that a 780-day table in the Dresden Codex has something to do with Mars. The table apparently concerns the times when the planet becomes visible again in the morning after its period of invisibility (termed heliacal risings) and the times when the planet's motion reverses its direction relative to the stars (retrograde motion).
The Brickers and Aveni argue that another complex table in the same document also concerns Mars. In this case, Mayan astronomers apparently focused on two cycles related to the number of days that elapse between consecutive passages of Mars through a given celestial longitude while traveling across the sky. One cycle lasts about 700 days and includes a 75-day period when the planet is in retrograde motion. The other cycle lasts about 540 days and does not include a retrograde loop.
It turns out that these periods alternate rhythmically in an easily discoverable manner, with one short period following seven or eight long periods, the researchers say. Moreover, the starting and ending points of these cycles follow a pattern related to seasons on Earth. Use of these two particular intervals allowed Mayan astronomers to track the motion of Mars across the zodiac and to relate its movements to the terrestrial seasons.
Both of these directly observable Martian cycles have hitherto gone unrecognized in western astronomy, the researchers note.
Interestingly, the Dresden Codex has a table based on 702-day intervals. "A 702-day value is much more relevant than the Western value of [about] 687 days for a terrestrial observer keeping track of Mars' position against the background of the stars," Bricker and his colleagues suggest.
A value of 707 rather than 702 for the length of the Mars period would have been astronomically more accurate, however. The researchers speculate that Mayan astronomers chose 702 because of this value's commensurability with the 780-day period representing the time it takes Mars to make one orbit of the sun relative to Earth and the 260 days of the sacred Mayan calendar: (702 x 10) = 7,020 = (780 x 9) = (260 x 27).
Moreover, Mayan astronomers found a way to link the time that it takes Mars to orbit the sun relative to Earth (its synodic period) with the time that it takes Mars to orbit the sun relative to the stars (its sidereal period). The synodic period of 780 days = 10 x 78, the long period of 702 days = 9 x 78, and the short period of about 543 days is close to 7 x 78.
"One of the great benefits of studying the astronomies of other cultures lies in the possibility of appreciating alternative ways of understanding the cosmos," the researchers conclude. "The pages of the Dresden Codex dealing with Mars provide specific examples of such alternative views."
Copyright 2001 by Ivars Peterson
Bricker, H.M., A.F. Aveni, and V.R. Bricker. 2001. Ancient Maya documents concerning the movements of Mars. Proceedings of the National Academy of Sciences 98(Feb. 13):2107. Abstract available at http://www.pnas.org/cgi/content/abstract/98/4/2107.
Peterson, I. 1997. Fractions, cycles, and time. MAA Online (Oct. 13).
______. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.
An introduction to Mayan astronomy and calendar systems is available at http://www.civilization.ca/membrs/civiliz/maya/mmc07eng.html.
Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He also writes for the children's magazine Muse (http://www.musemag.com) and is working on a book about math and art.
NEW! NEW! NEW!
Math Trek 2: A Mathematical Space Odyssey by Ivars Peterson and Nancy Henderson. For children ages 10 and up. New York: Wiley, 2001. ISBN 0-471-31571-0. $12.95 USA (paper).