Calculations and Calendrics

Now, as you know, the Gregorian Calendar solution is to add a leap day. Apparently, the Mesoamericans did the opposite, they subtracted. Recognizing that the 1508 Vague (365-day) Years of the solar era actually make up only 1507 real tropical years of 365.2422 days, they set out to erase the extra year by subtracting one 20-day month every 83 years. Actually, this was not in fact performed as a real correction, because any Mesoamerican calendar was allowed to run its course without change, but every time that a new calendar was founded, it was timed to the ongoing awareness of when the era was due to begin and end. By the way, regarding this subtraction of 20 days every 83 years, try this: Take the Fibonacci number 1597, subtract 20 and you get an exact multiple of 83.

1597 - 20 = 1577 = 83 x 19.

The Mesoamericans used sequences of additive numbers in their astronomical calculations, as is apparent in their five-part Long Count dating system, composed as follows. (Because 20 and 13 are involved, the great cycle can also be expressed as 7200 tzolkin, so that the great cycle begins and ends on the same name day, 4-Ahau, also called Four Flower.)

1 uinal = 20 days

 1 tun = 18 uinal = 360 days

 1 katun = 20 tuns = 7200 days

 1 baktun = 20 katuns = 144,000 days

 1 great cycle = 13 baktuns = 1,872,000 days (about 5125 years)
 

Note the use of the term "sequences of additive numbers," rather than "multiples." There is a subtle difference here that may assist us in getting into a Mesoamerican "headset." Multiplication is really a special form of addition. "There is no evidence showing that the Maya ever used their notation for multiplying numbers, or even that they ever multiplied two numbers, with or without their notation."[16] Many important numbers of astronomical significance can be created from additive functions. For example, if you add the numbers one through twenty-seven, you get 378, the synodic period of Saturn (in days). Adding one through 39 equals 780, the synodic period of Mars. As we have seen, adding the numbers one through thirteen gives 91, the number of days in a season.

Thinking in terms of consecutive, repetitive cycles is a slightly different approach than straight multiplying, using fractions, or "reducing to common denominators." For example, if you alternate the addition of 260 and 364 (the Maya calculational year in days), you get:

260 + 364 + 260 + 364 + 260 = 1508

We could instead think in a Western mathematical fashion and express this as an algebraic equation, ( 3 x 260 ) + ( 2 x 364 ) = 1508. Since 364 can also be expressed as 7 cycles of 52, or 28 cycles of 13, and 260 is 20 x 13, we can rewrite this as 1508 = ( 3 x 13 x 20 ) + ( 2 x 13 x 28), and find the factor 13.

1508 = 13 x 116

And 116 days just happens to be the synodic cycle of Mercury -- another eerie coincidence of resonance in the solar system that is revealed through the all-important Mesoamerican number 13 (a Fibonacci number). The word "coincidence" is used facetiously, of course, for this is no accident. The Mesoamericans held an astro-numerological key to the structure of astronomical cycles, and their culture was destroyed "coincidentally" during the rise of the Western European Scientific Revolution. Johannes Kepler attempted to reveal a similar mathemagical structure, and his work along these lines has been put down as "mystical" by the modern offspring of that same Scientific Revolution, the academic mafia. This latter term was coined by the Mesoamerican archaeologist Michael Coe in his critical assessment of his colleagues and a system that mitigates against academic heresy:

"Most anthropologists are so fuddy-duddy. They're not willing to let their minds roam ahead, speculate....There used to be more freedom of thought and expression, less worry about what peers said. Today there's sort of an academic mafia that runs things....If you say the wrong thing, you're bad and you don't get in... There are reputable publications that won't accept papers written based on anything but stuff dug up by archaeologists."[17] Needless to say, the paper you are reading right now falls into this category.

Modern academic prejudice, the debunking of astrology, or burning sixteenth century works of "superstition and falsities of the devil" -- what's the difference? Independent thinkers are not hampered by such restrictions to the growth of knowledge, so let us continue...

It turns out that 1508 days is a key value for determining other astronomical cycles.

1508 + 52 = 1560 = 2 x 780 (Mars cycle) = 9 x 173.31 (eclipse cycle)

 = 6 x 260 (sacred calendar)

Also, taking the amount 1508 years (Edmunson's "solar era") and multiplying by 17, we get 25,636 years, a good approximation of the precessional cycle of the equinoxes.

Other calculational values were used repeatedly by the Mesoamericans. In the Dresden Codex, we find indications for another significant period of time, 1820 days.
 

1820 = 5 x 364 (calculational year) = 7 x 260

1820 = 1508 + six cycles of 52

One particularly important calculational value was the 819-day-count.[18] In Classic Maya inscriptions, it was one of the ongoing cycles specified by a Distance Number that was counted backwards to a particular tzolkin name day that had a coefficient of one. Because 819 is divisible by 13, this coefficient always remained the same. In Mayan hieroglyphics, there are "verb" glyphs that usually accompanied such a count, such as the God K who was commonly associated with the 819 count. According to some sources, God K was a rain god, possibly related to Mercury. Another connection with the 819-day-count was the period of forty days, symbolized in the texts by images of footprints. In the mythological passage from the Chilam Balam of Chumayel called "The Creation of the Uinal," there are images of footprints used to measure the world,[19] and this period of 40 days (twice the uinal) is still called by some modern Maya "one foot of the year."[20]

The 819-day-count is an interesting computational formula. It is the sum of the numbers 2 through 40, and can also be expressed as:
 
 

819 = 9¹ + 9² + 9³.

819 is divisible by 13, 9, and 7, and can therefore be linked to several other cycles, such as the computational year:
 

4 x 819 = 9 x 364

There is much evidence that multiples of 364 days, particularly 20 x 364, were used by the Maya. The quantity 20 x 819 (or 45 x 364 = 16,380 days) was also used. The 819 days is equivalent to three tzolkins (39 veintenas) plus 39 days (or 20 x 41, minus one), which means it could function as a "footstep." In 819 days, the trecena coefficient and that of the Lord of the Night (a multiple of nine) remains the same, but the day name (and the kin coefficient in the Long Count) drops by one.

One researcher[21] has noted that the Maya could have used the 819-day-count as a means of tracking planetary positions, because the synodic periods of the visible planets can be integrated thus:

819 = 780 (one Mars cycle) + 39 (or 3 x 13)

 = 2 x 377 (one Saturn cycle) + 65 (or 5 x 13, or 1/4 of a tzolkin)

 = 7 x 116 (one Mercury cycle) + 7

 = 2 x 399 (one Jupiter cycle) + 21 (or 3 x 7)

Adding together these four equivalents of 819, we get several useful calculational formulae:
 

819 x 4 = 364 x 9 = 63 x 13 = 117 x 7 = 21 x 39 = 13 x 7 x 9

We also note that 819 days can be linked to the cycle of Venus:
 

819 x 5 = 7 x 584 (one Venus cycle) + 7
 

And thus we see the value of the quantity 4 x 5 x 819 (or 20 x 819), which brings the cycle of Venus into alignment with the other planetary cycles. If we use 585, the Mayan calculational cycle of Venus, instead of 584, we see even more exact connections. (584 days is technically the mean synodic period; the cycle actually varies between 581 and 588)[22]

819 x 5 = 585 x 7 = 45 x 13 = 65 x 9

And by linking the 819-count with the 260-day tzolkin, we get a precise earth-Venus relationship:
 

819 x 260 = 364 x 585
 

The quantity 819 forms other interesting celestial relationships:
 

819 x 115.95 (Mercury cycle) = 260 x 365.2425 (tropical year)

 819 x 40 (one "footstep of the year") = 360 (one tun) x 91 (one season)
 

The following fractions are all equivalent and suggest the Maya had an understanding of the intricate relationships between different cycles:
 

4     20    28     52   116   364
9     45   63   117  261   819

819 can also be expressed as the sum of various Fibonacci numbers, for example:

819 = 610 + 144 + 34 + 21 + 8 + 2 (or other combinations)

In fact, there are several numbers of astronomical significance that can be created from the Fibonacci series. For example, 2 + 3 + 5 + 8 +13 + 21 = 52, one-fifth of a tzolkin. We have mentioned that 1 + 1 + 2 + 3 + 5 + 8 = 20, and 20 times the next number in the series (13) equals 260. Take seven numbers in the series (8, 13, 21, 34, 55, 89, 144), add them together to get 364, divided by seven is 52. Or take alternating numbers in the series, adding two and skipping one:
 

0 + 1.....+ 2 + 3.....+ 8 + 13......+ 34 + 55......+ 144 = 260

        (1)            (5)              (21)                (89)
 

Using another sequence from Fibonacci numbers (55, 89, .... 1597), we add the first six:

55 + 89 + 144 + 233 + 377 + 610 = 1508

Continue forward two more in the series (987, 1597), subtract the second number (89), and you have 1597 - 89 = 1508. As previously mentioned, the reason this works is because of the general property of Fibonacci numbers: With any sequence of numbers in the Fibonacci series, you get the sum of all of them by going forward two more numbers in the sequence and subtracting the second in the sequence.

The synodic periods of the planets can also be created from Fibonacci numbers. The Mercury cycle of 116 days is 1508 divided by 13; the Jupiter cycle of 399 days is 34 more than one year, 365 days; the Saturn cycle of 378 days is 13 more than 365, or 377 plus one. The number of days in the Venus cycle, 584, is created from the sequence:

8 + 13 + 21 + 34 + 55 + 89 + 144 + 233 = 584 minus 13

The synodic period of Venus can also be expressed via the Fibonacci number 8:

584 = 8¹ + 8² + 8³

What we are talking about here, as we jump back and forth between Maya calculations and European Fibonacci numbers, from the astrologically-oriented Mesoamerican culture to Western Science, is nothing less than the grandest quest of the human mind: to understand man's place within the universe. To perceive the pattern which connects man and nature has been the goal of many great thinkers. For example, it is the subject of Mind and Nature: A Necessary Unity, an extraordinary book by Gregory Bateson, the highly respected modern anthropologist. Posing the question, "what is the pattern which connects all living creatures," Bateson recognized that spirals, such as the one produced through the Fibonacci series, the Golden Proportion, are created by living things. The key, Bateson remarked, is "never quantities, always shapes, forms, and relations."[23]

Form, order, and pattern are terms that continually appear in Bateson's writings. He does not classify human beings according to category (such as religion, family, gender), but in terms of their relationships. "Patterns," Bateson said, "are the necessary outward and visible sign of the system being organized."[24] The numerological understanding of such patterns was also the goal of Johannes Kepler, of astrology in general, and Mesoamerican "calendrics" in particular. While the Western mind has investigated the principle of the Divine Proportion, the Mesoamerican mind observed the Divine Calendar.
 

Divine Proportion

The Divine Proportion, also called the Golden Ratio, and has been designated by the 21st letter of the Greek alphabet, PHI j (21 is, of course, a Fibonacci number). PHI is an irrational number created by the successive convergents of numerators and denominators following the Fibonacci sequence. To demonstrate this, take any Fibonacci number and divide it by the previous Fibonacci number. For example, 377 / 233 » 1.61803. The further you go in the Fibonacci series with this method, the closer you approach the Divine Proportion, which is equal to

PHI (j) =( Ö5 + 1 ) / 2

Being an irrational number, it extends far beyond 1.618033988749894...but is often approximated at 1.618. The ratios of successive Fibonacci numbers tend toward PHI, so PHI is the limit of the sequence 2/3, 3/5, 5/8, 8/13, etc. These ratios approach, but never equal PHI. Mathematicians measure how "irrational" a number is by seeing how quickly the differences between these fractions and PHI shrink toward zero. It so happens that they shrink more slowly for PHI than for any other irrational number. This is why number theorists say that PHI is the "most irrational number." Two fascinating properties of PHI are seen in its reciprocal and its square:
 

1 = j - 1 or .618...
j
 
 

j² = j + 1 or 2.618...
 
 

Like j², the higher powers of j can all be expressed very simply in terms of j:
 

j² = j + 1

j³ = 2j + 1

j⁴ = 3j + 2

j⁵ = 5j + 3

j⁶ = 8j + 5

j⁷ = 13j + 8

... where each power is the sum of the two previous powers, and the coefficients of j form the Fibonacci sequence over again, as do the integer parts of the powers.

The history of the Western understanding of the Divine Proportion is worth mentioning. There is some evidence that the ratio was important to the Egyptians, because the Rhind papyrus refers to a "sacred ratio," and it is also prominent in the dimensions of all the pyramids at Giza.[25] The ancient Greeks came close to worshipping its aesthetic relationship in their architecture and sculpture. Indeed, it was the the name of the Greek sculptor Phidias which inspired the modern invention of the term PHI (j).

Leonardo da Vinci was probably the first to refer to the ratio as secto aurea, the Golden Section. The Italian Renaissance mathematician Luca Pacioli published a book called De Divina Proportione, illustrated by his friend da Vinci. "Coincidentally," Pacioli wrote this work in the year 1508 (remember that number?). He was also more than a little interested in probability and in what came to be known eventually as Pascal's Triangle, which is closely related to the Fibonacci series. Also "coincidentally," the last pre-Conquest Aztec ceremony was held the previous year, in 1507.[26]
 

Pacioli presented thirteen remarkable properties of the Divine Proportion. His "Ninth Most Excellent Effect" is that two diagonals of a regular pentagon divide each other in the Divine Proportion. If you tie an ordinary knot in a strip of paper and flatten it, the same figure appears. In the pentagram, which the Pythagoreans considered a symbol of health, the ratio is the Golden Ratio. Euclid in his Elements called this division "in the extreme and mean ratio," and used it to construct a regular pentagon, as well as the dodecahedron and icosahedron.

Kepler called the Divine Proportion one of the two great treasures of Geometry. Renaissance artists and architects regularly used it to create pleasing proportions, as did the modern architect Le Corbusier and the painter Georges Seurat. In fact, the Golden Proportion is the basis for many common objects such as playing cards, book covers, and windows, and psychologists have demonstrated that it is apparently preferred unconsciously by most people. It has been noted by one Fibonacci-aficionado, "Greek columns have golden proportions; so do a pack of Marlboros."[27]
 

Fibonacci numbers and the Divine Proportion also show up repeatedly in music. For example, the 8-note octave is produced on the piano keyboard as five black keys and eight white keys; thus the sequence 5, 8, 13. It has been suggested that the major 6th chord is "the one our ears like best," because the note E vibrates at a ratio of 0.625000 to note C, very close to the golden mean.[28] These notes produce pleasing vibrations in our inner ear's cochlea, a spiral-shaped organ, a fact that brings us to the next application of Fibonacci numbers.
 
 

PHI, Fibonacci, Phyllotaxis -- plus some Philosophizing

The Fibonacci sequence is also linked with the growth of plants. Certain statements by Kepler indicate that he understood this,[29] and Goethe wrote a paper about the subject, "Metamorphose der Pflanzen." The continuous application of PHI produces a logarithmic spiral which is the only curve in math that enlarges without changing its shape and which occurs frequently in nature.

The Divine Ratio/Fibonacci sequence is not accepted as a universal law in nature, but it is a fascinating tendency that appears far too often to be discounted as mere chance. Tendrils coil spirally; protoplasm streams in a spiral course; spiral threads occur in cytoplasm; molecules of DNA are spiral; and spiral grain has been found almost invariably in tree trunks.[30] Surfers shooting the gap underneath a breaking wave are covered by a logarithmic spiral of ocean spray. The shoreline of Cape Cod is a logarithmic spiral. Most of nature's horns, claws, and teeth exhibit the spiral, as in a ram's horn or parrot's beak, an elephant tusk or lion's claw, and the tusk of the now-extinct mammoth and saber-toothed tiger. And any dentist knows it is easier to pull a tooth in the direction of its curve.[31] The Fibonacci sequence reveals the breeding patterns of rabbits and the ratio of males to females among honey bees. We see the logarithmic spiral in the foetus of humans and animals, in the multiple reflections of light through mirrors, and the arrangement of spiral shells.

It is most apparent in plants in the distribution of leaves on branches, the seeds of sunflowers and pinecones, the scales of pineapples, and the spines on certain cacti. Such patterned arrangements of growth are called phyllotaxis. Some of the most common arrangements are in ratios of alternating Fibonacci numbers: 2/5, found in roses and fruit trees, or 3/8, which is found in plantains, or 5/13, found in leeks, almonds, and pussy willows.[32] If you examine the stalk of almost any green plant, you can observe this in action. Start at the bottom with a given leaf; move up the stalk, counting the leaves until you reach a leaf that is directly above the first one (do not count the first one), and you will have a Fibonacci number. In addition, the number of times you have circled the stalk will be another Fibonacci number. For example, there are generally thirteen buds arranged between two vertical lines on a pussy willow stem, and to count the buds you must circle the stalk five times.

In one recent survey of nearly 13,000 observations of 650 species, 96.5% conformed to classic, Fibonacci-type phyllotaxes.[33] Fibonacci numbers are found in the number of petals in a majority of common flowers: iris (3), primrose and buttercups (5), dahlia (8), ragwort (13), marigold and aster (21), daisy (34), Michaelmas daisy (55 and 89).[34]

Similar ratios appear in the scales of a fir cone or the florets of a sunflower. Here the packing is regular, forming sets of spiral rows, or parastichies. A pineapple usually has 5, 8, and sometimes 13 parastichies; a sunflower may have 21/34, 34/55, 55/89, or 89/144. In a good specimen of a sunflower, this remarkable feature will be seen: two sets of equiangular spirals superimposed or intertwined, one spiral turning clockwise and the other counterclockwise, with each floret filling a dual role by belonging to both spirals. Comparable arrangements of opposing spirals associated with Fibonacci numbers are found in the pine cone (5 and 8).[35]

If you look at the tip of the shoot of a growing plant, you can detect the parts from which all the main features of the plant form -- leaves, petals, sepals, florets, etc. At the center of the tip is the apex, and around it, the primordia emerge. Each primordium moves away from the apex and eventually develops into a leaf or petal. Following their order of appearance, the primordia form a generative spiral. If you measure the angles between successive primordia, you find they are equal; their common value is called the divergence angle, and it measures about 137.5 degrees, which is 360 minus 222.5 degrees. It so happens that this angle is 360 degrees multiplied by j, the ratio formed by successive Fibonacci numbers.

While many scientists have considered the subject of phyllotaxis too susceptible to mysticism and ("God-forbid!") nature worship, there are in fact labyrinthine interconnections among several phenomena studied by mathematicians and biologists: Fibonacci numbers, the Divine Proportion, parastichies, plant growth and shoot size, branching processes, primordia, divergence angle, spatial packing, and fractal geometry.

Regarding the Fibonacci number 5, there is a particular preponderance of pentagonal symmetry in living organisms, especially in botany and among marine animals such as starfish, jellyfish, and sea-urchins. Noted among the five-petalled flowers (or multiples of five) are all fruit blossoms, water-lilies, roses, honeysuckle, geraniums, marshmallows, campanulas. Pentadactylism (five fingers, or corresponding bones) is common in the animal kingdom. According to one authority on sacred geometry, five is the number of petals on all flowers of the edible fruit-bearing plants:

"Thus, five signals to man his proper foods. Five is dominant in the substructure of living forms, while 6 and 8 are most characteristic of the geometry of mineral, inanimate structures. The plants displaying a sixfold structure, such as the tulip, the lily and the poppy, are very often poisonous or only medicinal for man. Traditional medicine considered seven-petalled plants to be poisonous. Among these are the tomato and other plants of the belladonna or nightshade family. The very exotic flowers, on the other hand, the flowers of love such as the orchid, the azalea and the passion flower, are all governed by pentagonal symmetry."[36] This is why the Pythagoreans considered five to be a number of life and health. The awareness of sacred time, the harmonious concordance between heaven and earth, has formed the basis of agricultural cycles in all traditional societies. We moderns have lost this sense of this sacred connection, which is not surprising since the average modern person can now acquire all manner of "fresh" produce out-of-season. The question is, can you trust a tomato in January?[37]

Dr. J. William Littler wrote a paper entitled "On the Adaptability of Man's Hand," where he suggests that the human hand is composed of finger bones that represent the Fibonacci sequence. Measured in centimeters and starting with the metacarpal and working out to the finger tip, the bones average 8.8, 5.5, 3.3, 2.2. This predominance is a definite characteristic of living forms, and pentagonal forms do not and cannot appear among inorganic, crystalline systems.[38]

Why do living things grow this way, in mathematical patterns? Scientists have often rephrased this question, as Dick Teresi did in The God Particle, "Why does nature choose math as its language? Why is it that the overarching principles of the universe can be broken down into equations?" Such wording can only arise from a distorted anthropocentric approach which puts the cart before the horse. But Western Science often gets things (literally) ass-backwards, like a current editor of The Scientist who wrote that "the tendrils of some vines mimic the coil springs of a car."[39] -- Vines mimic cars? Obviously, nature was here first, and mathematics was a later invention, evolved and refined through the human attempt to perceive nature.

Botanists do not agree on a scientific explanation for phyllotaxis, despite the mathematicians' fascination for numbers involved. There is no doubt, however, that people down through the ages have observed and worshipped the spiral growth in nature. The beauty we admire in ancient Minoan art, for example, is based almost entirely on organic spiral forms.

We find a great deal of evidence for nature worship in etymology. Words like heliotrope (sunflower) derive from Greek words meaning "sun turning." From the same root comes the word helix, meaning "a whorl, curl, or spiral," which has been adopted as the name of the genus of land-snails. We are also reminded of double helix, the DNA term that holds the secrets of identity for living things. Just as the flower symbol was used by Mesoamericans to denote their calendrical "first day," we note that primrose (one of the 5-petaled flowers and one of the earliest to appear) is derived from the Latin prima, meaning "first." The daisy, another Fibonacci flower, is a word derived from the Saxon words for "day's eye." The cosmos, an 8-petaled flower, is named for a word meaning "order."[40]

Regarding the worship of nature, there is certainly something to be said for the act of meditating upon "the most irrational number." We might follow the example of the 17th century poet Richard Lovelace and admire the "Sage snail, within thine own self curled." Or, as one 17th century scholar put it, "From the contemplation of Plants, men might first be invited to Mathematical Enquirys."[41]

In the spirit of this advice, we can compare the development of Mesoamerican and Western European mathematics, and note a general philosophical difference. Mesoamerican mathematics evidently paid greater attention to numbers associated with organic growth, while Western mathematics became rather more fixated on the numbers associated with crystalline growth -- "crystalline numbers" which have been subsequently used by scientists to describe an organic world, rather like forcing a square peg into a round hole. The Divine Proportion and the progression of the Fibonacci series are better suited to describing organic growth because they have the property of producing simple addition, the accretion of identical parts, or a succession of similar shapes, what has been called "gnomonic growth."[42]

The respected naturalist Sir D'Arcy W. Thompson pointed out that "the shell, like the creature within it, grows in size but does not change its shape. The existence of this constant relativity of growth, or constant similarity of form, is of the essence." In other words, the very shape of the logarithmic spiral enables growth to occur without any change in form.[43] The spiral formed by living things is genetic or developmental, and crystallographic symmetry is absent in spiral phyllotaxy.

The gnomonic type of growth, from the inside outwards, is associated with living organisms and allows greater movement potential, whereas in crystals the growth is by agglutination, or simple addition from outside inwards, and the final distribution of energy in the system being such as to cause no further motion. This comparison implies that immobility is created through the crystalline number system of Western mathematics and could therefore be potentially the final product of Western culture. Perhaps this is what Ian Stewart meant when he called for the development of a new kind of mathematics that can explain the patterns in nature, complaining that our mathematical schemes are "too inflexible, geared to the constraints of pencil and paper."[44]

Modern science is equipped with the kind of thinking which rests on analytical mathematics and has produced the so-called "miracles" of technology. But the successes resulting from the use of analytical mathematics have eclipsed other ways of mathematical thinking. To transform one's thinking beyond the inflexible world of measurement requires the development of an awareness that is independent of the laws manifest in the material world. To quote Rudolf Steiner, "The highest level of individual life is that of conceptual thinking without reference to a definite perceptual content."[45] The act of contemplating Infinity is one way to transform thinking, and this can be performed by meditating upon the logarithmic spiral.

Remember that multiplication is really a special form of addition, and PHI represents a coinciding of the processes of addition and multiplication. What was a linear accumulation suddenly becomes a square ( j + 1 = j² ), and there is a leap of growth. In the plant, the simple additive growth in the stem suddenly erupts into a flower. When such a moment occurs in the context of spiritual development, it is called enlightenment. In our brains, the additive accumulation of data suddenly blossoms into a genuine understanding. There is a development from points to lines to planes, and finally, to volume. This is demonstrated by simultaneously adding and multiplying in the mathematical equation:[46]

j + j² = j³ = j x j²

Euclidean deductive geometry is directed towards measurement between points and is thus closed to the concept of infinity, whereas Projective Geometry is based on the idea of relations between ideal planes and thus opens the thinking. For example, compare two types of spiral. In the first case there is a spiral staircase, which is built on the principle of the Archimedian screw (useful for technology but never found in living forms). The other is, of course, the logarithmic spiral based on the Fibonacci sequence or PHI. Ideally this spiral is posed between two infinitudes; it curves on and on towards inner and outer points that it will never actually reach. The preponderance of Fibonacci/PHI spirals in nature offers a multitude of opportunities to discover infinity, and thus, divinity.[47] In this regard, we quote from William Blake, with italics to emphasize the Fibonacci significance:
 

To see the world in a grain of sand,
and a heaven in a wild flower;
Hold infinity in the palm of your hand,
and eternity in an hour.

We have travelled far from Fibonacci and PHI to phyllotaxis and finally, to philosophy. Plato described this same journey in the following way. "The sight of day and night, the months and the returning years, the equinoxes and solstices, has caused the invention of number, given us the notion of time, and made us enquire into the nature of the universe; thence we have derived philosophy."[48] We have seen that organic, developmental numbers are commonly associated with living, moving forms in nature. It is logical in our progression of thought to suggest that moving forms such as astronomical cycles may also be expressed by the organic numbers of the Fibonacci sequence. What Luca Pacioli revealed in De Divina Proportione can be applied to astronomy, and without crystalline numbers.
 

Luca in the Sky, without Diamonds

Let us look at the dynamics of the solar system, which is full of resonances or patterns of relationship created by interlocking cycles, whereby moving bodies take up the same relative positions at regular intervals. The sidereal periods of all the major planets are approximately in resonance, which means the relationship between their motions can be expressed as a ratio of whole numbers, usually in the Fibonacci sequence. For example, it takes Saturn 29.5 years to circle the sun, while Jupiter takes almost 11.9 years. Twice the Saturn cycle is about equal to five times the Jupiter cycle, so the ratio of these two cycles is 2:5. Numerous other planetary ratios are formed with the Fibonacci numbers -- Pluto/Neptune is 2:3, Earth/Venus is 3:5, Mars/Mercury is 1:8, and so on.[49]

The different kinds of motion of the planets can also be in resonance with each other. For example, Mercury rotates once every 58.65 days and revolves around the sun every 87.97 days, forming a resonance of 2:3 (2 x 87.97 = 3 x 58.65). There is a similar relationship between the earth's tropical year and the rotation of Venus. Venus rotates with a sidereal period of 243 days, forming a close 2:3 ratio with the earth's revolution (243 x 3 is close to 2 x 365).[50] And the daily cycle on Venus is 117 earth days, almost equal to one synodic period of Mercury.

There is also the principle of Bode's Law, which can be understood in terms of the Divine Proportion. Bode's Law states that the mean distances of the planets from the sun are proportional to the squares of simple integral numbers, and is usually expressed as

0.4 + (0.3 x 2ⁿ)

where n is the number signifying a planet's position from the sun. But substituting j will also produce close approximations where n is planets beyond earth:
 

0.4 + (0.618 x 2ⁿ)
 

Modified (Fibonacci) Version of Bode's Law

While astronomers complain that Bode's Law is "not of precise physical significance," they have still been goaded by its challenge, to question whether the distribution of planetary orbits is due to chance, or to some "real" physical principle.[51] Bode's Law is, like the equation of PHI, a formula that cannot be "explained" scientifically; it offends many scientists because it is numerological (mystical) and yet it works, for it was responsible for at least three valid predictions. The discovery of both Uranus and Ceres conformed to the formula, and recognition of a similar law led to the discovery of Saturn's moon Hyperion. The validity of Bode's Law has been supported by computer simulations of the N-body problem which have shown that arbitrary planetary configurations, started with purely random initial positions and velocities, tend to "relax" after thousands of years into a Bode's Law-type of resonant configuration.[52]

Paul Jacques Grillo, author of Form, Function, and Design, has also found evidence of the golden mean in the distribution of planets. He suggests that such a "coincidence" (he uses quotation marks) may offer a clue to the evolution of the solar system, that current elliptical paths evolved from the original form of a log spiral. Swedish astronomer Carl-Gustav Danver studied the characteristic shape of the great galaxies of outer space. He reported that the predominant appearance is that of logarithmic spirals.[53]

The subject of orbital resonance is indeed directly related to the initial formation and composition of the solar system, the question of origin for the known planets and asteroids. Astronomers have been slow to recognize that the solar system is not orbitally stable, but instead is "in process." They have thus denied Immanuel Velikovsky's theories about Venus being a newcomer to the solar system and have similarly rejected postulations about "missing planets" or a hypothetical planet beyond Pluto.[54] Appropriately for our discussion of Mesoamerican astronomy, one astronomer has proposed a now-missing planet, originally located in the asteroid belt, whose explosion produced the current configuration. He called the planet Aztex.⁵⁵

Astronomers are reluctant to recognize Bode's Law as a "real" principle, and botanists are uncomfortable with the "magic" of Fibonacci-related phyllotaxis. This is because Western Science cannot be satisfied that something works without explaining the material cause in scientifically-acceptable terminology. The Mesoamericans, on the other hand, were quite content to make monumental achievements without obsessing over causative principles.

Kepler expressed this in his rejection of atomism: "The cause is not to be looked for in the material, but in an agent." Instead of seeking the causes of material phenomena deep down in the structure of matter in ever-smaller sub-units, Kepler, like Gregory Bateson, looked for patterns and principles. Modern science, on the other hand, continues to ask why plants or planets do what they do, and continues to break down the world into more and more microscopic levels as molecular, atomic, subatomic, cellular, nuclear, and DNA, ad infinitum. We suggest that all this academically-funded busywork must eventually reduce to a final cause called "because that's the way it is." Or better yet, and even more offensive to scientists, "because the gods made it that way."

Kepler's terminology for this final agent was the Facultas Formatrix, the formative faculty, a morphogenic principle responsible for all organic and inorganic shapes in nature, from snowflakes and planetary orbits to plants and animals. As Kepler explained it, this was due to the Creator's design; things were shaped a certain way because "it is in their nature to do so." Modern scientists are violently opposed this kind of thinking, claiming there is no predictive power in this conclusion. There is an inconsistency involved here about the predictive value of mathematics, for common sense dictates that you don't need to understand underlying principles in order to use them, as we have seen with Bode's Law. If a portion of a pattern has been observed and the symmetry is either known or suspected, the whole can be predicted. Accepting such a precept and rejecting atomism would not only put a lot of scientists out of work, it would suggest that time might be better spent in applying what obviously works and in worshipping nature; i.e., doing what every known culture but our own modern one has done.
 

Fibonacci-Related Images in Maya Culture and Hieroglyphics

Let's look closer at how the Mesoamericans applied and worshipped the spiral principle. The Maya inscriptions include a great number of images of spiral growth patterns from nature, including the shell, which was used in inscriptions to indicate "zero."

The cipher (nought, zero) and place numerations are such an integral part of our cultural heritage and are such obvious conveniences that it is difficult to imagine mathematics without them. Yet they were a late development in European mathematics; the ancient Greeks and Romans had no knowledge of "zero" or place numeration, but the Maya did. This is one of the reasons they could excel at long calculations. Once you have a cipher and a system of place notation, long problems in simple arithmetic become infinitely easier, and it doesn't matter whether the system is decimal, vigesimal, or whatever. The Maya system was not decimal but vigesimal, counting by 20s.

Maya numbers were placed in vertical lines, with space fillers somewhat analogous to our "zero," but which did not mean "nothing." Rather, the space filler-cipher meant "completion." And the completion symbol most commonly used was a shell. This is rather interesting, considering the fact that many shells are formulated according to the divine proportion, or Fibonacci sequence. Among those that build their tiny bodies in a logarithmic spiral are the globigerinae, planorbis votex, terebra, turritellae, trochida, and the common snail.[56] The chambered nautilus is a spectacular example and the one most commonly used to illustrate how the Divine Proportion appears in nature. It belongs, incidentally, to a group of mollusks that include the 8-tentacled octopus. Outwardly it resembles the univalves, but the animal inside is more like an octopus.

While several Mesoamerican scholars (Seler, Forstemann, Tozzer) have recognized the snail shell as a symbol of birth, David Kelley believes that it means "emergence."[57] He notes a repetitive representation at Palenque of a snail shell with God K and the Corn God emerging from the shell. The three appear together with a 1-Ahau date, and Kelley further notes that One-Flower is associated with the Corn God and is also associated elsewhere with the birth date of God K.

The day Kan (the snake) is represented by a glyph which has been identified as a shell. The Yucatec word kan can be translated as "shells or stones used as money." Kelley points out that the Kan glyph shows a decided similarity to the "shell variant" of the kin glyph. Mesoamerican scholar Thompson suggested that the kin (day, sun) glyph represents a five-petaled flower converted to four petals, because four is the number of the Sun God (Four-Ahau). His first opinion was that it was the five-petaled tobacco flower, but he later decided it was the five-petaled Plumiera, a flower of great religious importance in Mesoamerica, which in Maya is known as nicte. Interestingly, this is close to the word nicotine, which is believed to be adapted from the name of an ambassador, Jean Nicot, who introduced tobacco into France in 1560.[58] We have to wonder about the true origin of many New World plant names, however, which is often found in the native languages.

One particular example of word origin is very revealing for our discussion here. We have the word pineapple because the Spanish recognized that this New World plant resembled the pine cone; they called it piña de Indias. But how does a pineapple resemble a pine cone? The only visible similarity is the identical Fibonacci whorls of 8:13.

Spirals associated with Fibonacci numbers are found in some groups of cacti, where the 3:5 ratio appears as bijugate spirals in paired parastichies of six and ten.[59] The prickly pear cactus, incidentally, has been cultivated for its fruit in Mexico for over six thousand years, and is called tuna.⁶⁰ Is it coincidental that the Maya glyph for tun was used to designate the last day of any period? In Yucatec, tun means "stone," possibly referring to stones used to count intervals in a cycle.[61] The connection is not unlike our own word calculus or calculate, which derives from the Latin word for pebble, used by the ancients for counting.

Although the study of Maya inscriptions is quite complex and not easily understood, we suggest that the scholars who attempt to decipher inscriptions should take the Fibonacci theory into account. Is it really just coincidence that there are so many Maya glyphs for Fibonacci objects in nature that are associated with numbers and calendrical cycles? One Mayan glyph is known to represent a shell and has been connected with the bar used for the number five, which one scholar has compared with hub, meaning "large sea shell," and is not unrelated to the word haab which refers to the year. Another glyph, naab, representing the five-fingered human hand, was used in the Venus table of the Dresden Codex, and we know that five is an important number for Venus and divides 360 days into 5 x 72. One meaning of the word naab is "to measure," much like we use the word "hand" as a measure of height for horses. A different hand glyph has been read as ka, which we know as part of the word katun, a measurement of 7200 days. The hand glyph is also read as lah, which has been associated with an inverted ahau glyph, and has been read as "end, die" in the compound word for numbers above ten,[62] as well as "completion" in counting cycles.[63] There is also a glyph called the "shell star" which appears often in Mesoamerican inscriptions where planetary cycles are interconnected. This topic of astronomical glyphs has been studied in great detail by David Kelley.[64]

If we examine Maya inscriptions, we find other evidence that implies an understanding of the Fibonacci numbers. The ancient Mesoamericans viewed the cosmos as a layered universe. The heavens were conceived of as thirteen in number, with the earth included as the first layer. There was also a nine-layered underworld, beginning with the earth again as the first layer. This is depicted on page 2 of the Codex Vaticanus A.⁶⁵ This is one of the examples of the use of the Fibonacci numbers 8, 13 and 21. There are 21 "worlds," when the earth is "counted" twice as the first world. (We note that the number one appears twice at the beginning of the Fibonacci series). There is evidence that the Maya also expressed the day as having 13 hours of daylight (the Thirteen Birds of the Day) and 9 hours of darkness (The Nine Lords of the Night).[66]

Flowers are one of the most consistent expressions of the Fibonacci sequence in nature, and they also form some of the most important symbols in Mesoamerican writing, mythology, and calendrics. The sunflower held a particularly important place in ancient Mexican mythology. When the Spaniards arrived in the New World, they found the sunflower serving in the temples as a sign and a decoration, and the sun-god's officiating handmaidens wearing upon their breasts representations of the sacred flower in beaten gold.[67]

As described earlier, the special name-day which began the sacred calendar was 1-Ahau, also called One-Flower, while the day that began and ended the great cycle was Four-Flower. The word Ahau means Lord and refers to the Sun, while the glyph used to represent it was a flower.

That the Mesoamericans would link flowers with astronomical cycles should not seem so surprising to us, since using flowers to mark time has had a long, though obscure, history in Europe. One 18th-century astronomer, Jean-Jacques De Mairan, noted that the opening and closing of certain "sensitive plants" followed a specific daily rhythm, and he linked this to the sun's influence. A similar observation had been made by a general in the travelling army of Alexander the Great. The great Swedish naturalist Carolus Linnaeus planted a timekeeping garden which he called "Flora's Clocks." The flower beds were laid out to form a clock face, with each bed representing a different hour of the day and planted with flowers that were known to open or close at certain times.

It is not only the circadian rhythm that is evident in plants, but also longer cycles. For example, there is a species of bamboo in Argentina which has a very precise life pattern -- exactly 30 years from seed to seed. And botanists have discovered that it is not only daylight which regulates the rhythms of plants; a number of experiments have shown that the length of night is just as important as the length of day. The chrysanthemum, for example, requires an uninterrupted period of about 13 hours of darkness (an autumn night) before it will flower. Ragweed cannot begin its flowering process until the summer nights have lengthened to about nine and a half hours.

Finally, and most relevant for our discussion here, there is the 260-day calendar, which we have shown to be based on Fibonacci numbers. There have been many theories that attempt to explain the origins of the 260-day cycle, and most of them show the connection with natural cycles. For example, the cycle of seed-to-harvest for certain types of corn grown in Guatemala is about 260 days. In the Dresden Codex, there are 258 days between Venus rising as an evening star and its emergence as a morning star, a period which correlates with the Aztec myth of Quetzacoatl's journey to the underworld. Perhaps the most significant correspondence is the period of human gestation. Nine lunar months are approximately equal to 260 days. One Mayan scholar discovered that some modern Mayan people explicitly state the human gestation period as the explanation for the origin of the tzolkin.⁶⁸

Conclusions

When we study the "facts" of Mesoamerican mathematics-calendrics, we cannot avoid realizing the mystical nature of life. To those who would deny the synchronicity, the great Mayan scholar J. Eric S. Thompson wrote, "One must appreciate the impact of the divinatory aspects of the 260-day Sacred Almanac, and never lose sight of the fact that the ends of Mayan astronomy were not scientific, but astrological....Mayan astronomy is too important to be left to the astronomers."
 
 

We have seen that the numbers produced by the Fibonacci sequence offer a numerological key to the inter-relationships between man and the cosmos. The ancient Mesoamericans evidently recognized the correlations between biological processes, agricultural cycles, and timing of astronomical events, and they created a mathematical system that could express these connections -- a system that emphasized what we call Fibonacci numbers.

References
 
 

[1]Vincent Malmstrom, "Origin of the Mesoamerican 260-Day Calendar," Science, Vol. 181 (Sept. 7, 1973), pp. 939-940; and "A Reconstruction of the Chronology of Mesoamerican Calendrical Systems," Journal of the History of Astronomy, Vol. 9 (1978), pp. 105-116.

[2]J. Eric S. Thompson, The Rise and Fall of Maya Civilization (Univ. of Oklahoma Press), pp. 148-149.

[3]John Teeple. Maya Astronomy. Carnegie Institution of Washington, Publication 403, Contribution 2 (1930), pp. 94-98. This was confirmed by J.E.S. Thompson in Maya Hieroglyphic Writing: An Introduction. Carnegie Institution of Washington, Publication 589 (1950), pp. 226-227.

[4]A.M. Tozzer. Landa's relacion de las cosas de Yucatan. Papers of the Peabody Museum, Harvard Univ., no. 18 (1941), p. 133.

[5]Sharon Gibbs, "Mesoamerican Calendrics as Evidence of Astronomical Activity," in Native American Astronomy (ed. Anthony Aveni), Univ. of Texas Press (1977), pp.30-31.

[6]J. Eric S. Thompson, The Rise and Fall of Maya Civilization (Univ. of Oklahoma Press), pp. 152 ff..

[7]Michael Coe, "Native Astronomy in Mesoamerica," in Archaeoastronomy in Pre-Columbian America (ed. Anthony Aveni), Univ. of Texas Press (1975), pp.11-12.

[8]David Kelley and K.A. Kerr. "Mayan Astronomy and Astronomical Glyphs," in Mesoamerican Writing Systems (ed. E. P. Benson), Washington, D.C.:Dumbarton Oaks (1974), pp. 179-215.

[9]Astronomical Almanac for the Year 1989. Washington, D.C.: Government Printing Office (1988).

[10]For suggestions on the Gregorian leap year determination, see Noel Swerdlow, "The Origin of the Gregorian Calendar," Journal for the History of Astronomy (1974), pp. 48-49; Jeffrey Shallit, "Pierce Expansions and Rules for the Determination of Leap Years," Fibonacci Quarterly (Nov. 1994), pp. 416-423; Jacques Dutka, "On the Gregorian Revision of the Julian Calendar," The Mathematical Intelligencer, Vol. 10, No. 1 (1988), pp. 56-64; and V. Frederick Rickey, "Mathematics of the Gregorian Calendar," The Mathematical Intelligencer, Vol. 7, No. 1 (1985), pp. 53-56.

[11]Gordon Moyer, "The Gregorian Calendar," Scientific American (May, 1982), p. 151. Such extrapolations are uncertain. As one calendar scholar remarked, "For one thing, the way in which Earth's rotation varies is not well understood. For another, the computations depend on Simon Newcomb's value of the length of the tropical year, which may not be correct." (Charles Kluepfel, "How Accurate Is the Gregorian Calendar?" Sky and Telescope, Nov. 1982, p. 418)

[12]The process used here is similar to a proven method for creating pi ( [pi] ):

 [pi] = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 .....

[13]John Major Jenkins, Tzolkin: Visionary Perspectives and Calendar Studies, Borderland Science (1994).

[14]Munro Edmunson, The Book of the Year. Univ. of Utah Press (1988), p. 112.

[15]Ibid., p. 114.

[16]A. Seidenberg, "The Zero in the Mayan Numerical Notation," in Native American Mathematics, ed. Michael Closs, Univ. of Texas Press (1986), p. 380.

[17]Caleb Bach, "Michael Coe: "A Question for Every Answer," Americas (Jan/Feb. 1996, pp. 15-21.

[18]Barbara MacLeod, "The 819-Day-Count: A Soulful Mechanism," in Word and Image in Maya Culture: Explorations in Language, Writing, and Representation (ed. William Hanks and Don Rice), University of Utah Press (1989), pp. 112-126.

[19]R. L Roys, The Book of Chilam Balam of Chumayel, Univ. of Oklahoma Press (1967).

[20]J. E. S. Thompson, Maya Hieroglyphic Writing: An Introduction, Univ. of Oklahoma Press (1950), p.248

[21]MacLeod, p. 116.

[22]Bruce Scofield, Signs of Time: An Introduction to Mesoamerican Astrology, One Reed Press (1994), p. 220; and David Kelley, "Mayan Astronomy and Astronomical Cycles," in Mesoamerican Writing Systems, ed. Elizabeth Benson, Dumbarton Oaks (1973).

[23]Gregory Bateson, Mind and Nature: A Necessary Unity. Bantam (1988), p. 9.

[24]Rollo May, "Gregory Bateson and Humanistic Psychology," Journal of Humanistic Psychology, Vol. 16, No. 4 (Fall 1976), p. 40.

[25]William Hoffer, "A Magic Ratio Recurs throughout Art & Nature," Orion (Winter 1985), p. 31.

[26]Bruce Scofield, Signs of Time: An Introduction to Mesoamerican Astrology, One Reed Publications (1994), p. 207.

[27]The Sand Dollar and the Slide Rule, p. 99.

[28]Randy Moore, The Numbers of Life.

[29]See Kepler's The Six-Cornered Snowflake and Harmonices Mundi Libri V.

[30]Edmund Sinnott, Plant Morphogenesis, Krieger (1979), p. 165.

[31]Hoffer, "A Magic Ratio," Orion (Winter 1985), pp. 28-38.

[32]H.E. Huntley, The Divine Proportion: A Study in Mathematical Beauty, Dover (1970), p.162. See also Peter Stevens, Patterns in Nature, Atlantic Monthly (1974).

[33]Roger V. Jean, Phyllotaxis: A Systemic Study in Plant Morphogenesis, Cambridge Univ. Press (1994).

[34]Huntley, p. 163; Ian Stewart, "Daisy, Daisy, Give Me Your Answer, Do," Scientific American (Jan. 1995), pp. 96-99; D'Arcy Wentworth Thompson, On Growth and Form, Vol. II, Cambridge Univ. Press (1968), p. 930. Thompson's chapter "On Leaf-Arrangement, or Phyllotaxis" is particularly informative, pp. 912-933. See also Jay Kappraff, Connections: The Geometric Bridge between Art and Science, McGraw-Hill (1991).

[35]Huntley, pp. 164-165.

[36]Robert Lawlor, Sacred Geometry: Philosophy and Practice, Crossroad (1982), p. 58.

[37]Vince Staten, Can You Trust a Tomato in January: Everything You Wanted to Know (And a Few Things You Didn't) about Food in the Grocery Store, Simon & Schuster (1993).

[38]Matila Ghyka, The Geometry of Art and Life, Dover (1977), p. 18.

[39]John Carey, "Spiral Effect," National Wildlife (April/May 1989), p. 56.

[40]Stephen Potter, Pedigree: The Origins of Words from Nature, Taplinger (1974).

[41]Nehemiah Grew, The Anatomy of Plants (1682), p. 152, quoted in D'Arcy Wentworth Thompson, On Growth and Form, Vol. II, Cambridge Univ. Press (1968), p. 912.

[42]D'Arcy Thompson, On Growth and Form.

[43]Hoffer, "A Magic Ratio," p. 38.

[44]Ian Stewart, Nature's Numbers: The Unreal reality of Mathematical Imagination, BasicBooks (1995), p. 148.

[45]Rudolf Steiner, The Philosophy of Spiritual Activity, Anthroposophical Press.

[46]Robert Lawlor, Sacred Geometry: Philosophy and Practice, Crossroad (1982), p. 56.

[47]For more on this topic, see George Adams and Olive Whicher, The Plant Between Sun and Earth, Rudolf Steiner Press (1980).

[48]Plato, The Timaeus and the Critias, translated by Thomas Taylor, Pantheon (1952), p. 164.

[49]See A.E. Roy and M.W. Ovenden, "On the Occurrence of Commensurable Mean Motions in the Solar System," Monthly Notices of the Royal Astronomical Society, 114 (1954), pp. 232-241, and "The Mirror Theorem, Part II," MNRAS, 115 (1955), pp. 296-309.

[50]Fred Whipple, Orbiting the Sun: Planets and Satellites of the Solar System, Harvard Univ. Press (1981), p. 184.

[51]Valerie Vaughan, Persephone Is Transpluto: The Scientific, Mythological and Astrological Discovery of the Planet Beyond Pluto, One Reed Publications (1994), pp. 91-92.

[52]J.G. Hillis, "Dynamic Relaxation of Planetary Systems and Bode's Law," Nature, 225 (1970), pp. 840-842. This phenomenon was also foreseen by Ernest W. Brown (see his "Observation and Gravitational Theory in the Solar System," Publications of the Astronomical Society of the Pacific, 44 (1932), pp. 21-40).

[53]Hoffer, "A Magic Ratio," p. 36.

[54]See Valerie Vaughan, Persephone Is Transpluto.

⁵⁵Michael W. Ovenden, "Planetary Distances and the Missing Planet," Recent Advances in Dynamical Astronomy, Reidel (1973), pp. 319-332.

[56]Hoffer, "A Magic Ratio," p. 34.

[57]David Kelley, Deciphering the Maya Script, Univ. of Texas (1976).

[58]Eric Partridge, Origins: A Short Etymological Dictionary of Modern English, Macmillan (1958).

[59]Edmund Sinnott, Plant Morphogenesis, Krieger (1979), p. 163.

⁶⁰Robb Walsh, "Prickly Paradox," Natural History (June 1996), p. 62-65.

[61]David Kelley, Deciphering the Maya Script, Univ. of Texas Press (1976), p. 135.

[62]Kelley, Deciphering, p. 139.

[63]Kelley, Deciphering, p. 35.

[64]David Kelley, "Mayan Astronomy and Astronomical Glyphs," in Mesoamerican Writing Systems, ed. Elizabeth Benson, Dumbarton Oaks (1973).

⁶⁵Michael Coe, "Native Astronomy in Mesoamerica," in Archaeoastronomy in Pre-Columbian America (ed. Anthony Aveni), Univ. of Texas Press (1975), pp. 7-8.

[66]Coe, "Native Astronomy," p.14.

[67]Mrs. William Starr Dana, How to Know the Wild Flowers, Dover (1963), p. 212.

⁶⁸Barbara Tedlock, Time and the Highland Maya.

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