Letter to the Editor, Borderlands Journal, Fall 1994
PHI, OR 'SORT OF PHI'
I just got the new Borderlands and after diving straight through Peter's column, went back to the start. It wasn't long before I was into the Jenkins article. The note in the first column that the Mayan calendar, "has been followed unbroken for over 2700 years," fits right in with Velikovsky's discovery of a worldwide calendaric change from a year of 360 days in the eighth century BC.
I was pretty excited about this article and I knew by the third page that I would have to send a copy of this to my astrologist friend who has been interested in Arguelles. It was on the next page that I bumped my head. Perhaps I'm not the one who should be saying this since I am far from being a mathematician. But, I have a fascination with the Golden Proportion and spirals in general. If, as Mr. Jenkins indicates, Arguelles and the others have played fast and loose with the calendaric correlations, he has played fast and loose with the Golden Mean.
1.618 X 13 = 21.034, A figure closer to 22 than 20, and an error of 5%.
.618 X 20 = 12.36, Closer to 12 than 13, and, again, an error of 5%.
Is 1.618 +/- 5%, good enough for pyramid building? What would you say? On page four is the statement, "5:8, like 13:20, is a PHI ratio." Wrong! 13:20 is a 1.538 ratio. Am I over the line? Is this too small a nit to pick? In the box, Tzolkin Chart with Spiral, is written: PHI 13:20. Wrong again, but a tile-like grid image is presented with a 1.618 spiral overlaid. The perfect fit is immediately discernible. The grid shown is 13 x 21, not 13 x 20. We could not say that 13:20 even would look like 13:21, hence, what is pictured is 13:21.
I would have to say this about the 13:20 portion of this otherwise intriguing article, "close, but no cigar." One part in twenty seems too much to give as a benefit of the doubt. If Mr. Jenkins has the goods on the calendar's correct alignment, great. This would be most important. As I'm sure he would readily admit, this particular endeavor would not easily tolerate a much greater than zero error.
R. L. Richards in Rosburg, Washington
Mr. Jenkins' Reply:
I'll agree that the "Tzolkin Chart with Spiral" diagram on page 4 is misleading. However, this diagram is not central to my primary argument. In the end, we still find that the Golden Proportion is the core principle of the Mayan tzollan calendar. The Maya derived their calendric cosmology by observing nature. Organic spirals in plants, leaves, conch shells as well as the 5:8 ratio of Sun and Venus all contributed to their suspicion that one principle was involved. 5:8 points us to two numbers in the Fibonacci series, and by virtue of this the 5:8 ratio is considered to be a reference to the Golden Proportion, and yet it only approximates it (8/5 = 1.6 rather than 1.618). By definition, the Golden Proportion is irrational and, like PHI, can never be precisely known. Nature itself does not use the Golden Proportion to a degree of perfect accuracy all the time. In fact, I see references in nature to the Golden Proportion very loosely. If I actually counted all the right and left facing seeds in a sunflower and it didn't come up 89/55 (the expected PHI ratio), I wouldn't say that PHI wasn't involved.
13:20 is a loose reference to the Golden Proportion. The 13 numbers and 20 daysigns combine to create the 260 day tzolkin, and 260 days was seen as a metaphorical reference to the human gestation period. In addition, the 260-day cycle serves as the key to a larger calendric system which can predict eclipses and planetary cycles. The number 260 is, philosophically, referring to a principle in nature which bridges the microcosm and macrocosm. In other systems such as Egyptian Sacred Science, the principle with this characteristic is identified as the Golden Proportion. I should point out here my primary observation which connects the Golden Proportion with the number 260, which Mr. Richards didn't mention in his letter. This is the formula 100 x PHI squared = 260. Precisely, it equals 261.8, but again we are dealing with the ancient Maya's attempt to model a principle they were observing in nature. The percentage of discrepancy here, if we need io state it, is 1.8 units out of 260, which is less than 7/10ths of a percent is small indeed. My proposed 13:20 reference to PHI is quite separate from this primary observation. When I use the "Tzolkin Chart with Spiral" diagram, I usually state that I have added an extra column lengthwise to provide a graphic example of what, I feel, the Maya were intending to model by their use of 13, 20 and especially the number 260. I apologize for the diagram being somewhat misleading. I would like to respond also to the question posed: "Is 1.618 ± 5% good enough for pyramid building?" I never say that the Golden Proportion was used by the Maya for pyramid building. It was used for cosmology building.
So, this is an important aspect of Mayan myth-making to clarify, and thank you to Mr. Richards for calling it to my attention. Overall, I feel that, for the Maya, perfect accuracy is less important than the realization that the Golden Proportion manifests in both the biological and astronomical realms. The ancient skywatchers who created the tzolkin calendar did not use 1.618 as a constant for PHI, they used 260. And all models of reality are to some degree inaccurate. Since PHI is essentially irrational, best expressed as an approximate ratio, we cannot, by definition, expect a "zero error" with any cosmology that tries to model nature in terms of the Golden Proportion. In this regard the Mayan concern was apparently not directed towards accuracy, but comprehensiveness; unlike the scientific paradigm, Mayan model-making was meant to encompass all aspects of existence. As such, human beings can not be viewed as some chance-created epi-phenomena, as Darwinian evolution would have us believe. Instead, our own unfolding is strangely reflected in vast celestial processes. In the Mayan view, each time cycle was seen as a sub-unit of larger time cycles to which it was related by harmonics. In other words, they had realized the intrinsic syncretism between earth, sky, and humanity. Furthermore, the unifying principle running through these different levels was seen to be the principle of "regeneration via self-same similarity", the Golden Proportion, which they conventionalized into their sophisticated cosmo-conception by way of the 260-day tzolkin calendar.
John Major Jenkins