The waxing and waning functions of phi that reflect successive twentieths of pi: Angle(180^{0} = pi) pi / 20 = 9^{0} pi / 10 = 18^{0} 3 pi / 20 = 27^{0} pi / 5 = 36^{0} pi / 4 = 45^{0} 3 pi / 10 = 54^{0} 7 pi / 20 = 63^{0} 2 pi / 5 = 72^{0} 9 pi / 20 = 81^{0} | ( 2 sin)^{2} of that angle 2 - R (phi+ 2) = 0.097887 1 - 1/ phi = 0.381966 2 -R (2- 1/phi) = 0.824429 2 - 1/ phi = 1.381966 phi - 1/ phi = 1.000000 phi + 1 = 2.618034 2+R (2- 1/phi) = 3.175571 phi + 2 = 3.618034 2 +R (phi+ 2) = 3.902113 | ( 2 cos)^{2} of that angle 2 +R (phi+ 2) = 3.902113 phi + 2 = 3.618034 2 +R (2- 1/phi) = 3.175571 phi + 1 = 2.618034 phi - 1/ phi = 1.000000 2 - 1/ phi = 1.381966 2 -R (2- 1/phi) = 0.824429 1 - 1/ phi = 0.381966 2 -R (phi+ 2) = 0.097887 |
If any ancients explored such a system of measuring angles by the periphery they enclosed, then this reflecting of pi the sun advancing along its circuit by the waxing and waning and self- mirroring of the corresponding phi- functions would probably also have impressed them as clear similarities with the behavior of the moon. The 360 degree system of measuring angles The 360 degree system produces its own connection between phi and the moon because an angle of 1.618... degrees has a cotangens of 35.4013. Ten times this cotangens matches the 354 calendar days in the lunar year which was and still is used widely in the Near East. We may dismiss such matches as accidental because degrees are an artificial division, but the ancients would have seen that division as part of the natural order decreed by the gods since 360 was a holy number, six times the “Great One” of sixty which formed a new unit in the Sumerian sexagesimal system. The 360- degree system for dividing the circle is not attested in writing until the last few centuries BCE on some Mesopotamian tablets^{2} . However, the Assyrian cultic year had long divided the annual cycle as well as the circumference of the universe into 360 days or parts^{3}. And long before them, the Sumerians believed that the number world was ordered by the sexagesimal system on which this division is based, so chances are they would have applied the same order to their surroundings. On the other side of the Fertile Crescent, the Egyptian astronomers also tracked circles the way we still do. It is true that their builders measured angles only by means of the seked which was the cotangens, or horizontal distance from the top of a vertical cubit rod, just as roof builders today express their slopes as the tangens, in inches of rise per foot of run. The Egyptian skywatchers, however, had organized a band of stars along a complete turn of the sky into 36 sections, the “decans”, that succeeded each other in ten- day intervals and so matched the 36 ten- day weeks in their 360- day civil year. This system is attested already from Old Kingdom times by the mention of “night hours” in the Pyramid Texts^{4} from about 2,300 BCE, and later coffin lids and tomb ceilings^{5} were sometimes decorated with elaborate pictures of those star- clocks. Like the Assyrian cultic year, the Egyptian civil year matched no observable cycle but was an artifical construct with a round and religiously significant number that also happened to come close to the 359.805 day average between the solar and lunar years and so paid equal respect to both. The Egyptians made up most of the difference with the actual solar year by tacking on an extra five “birthdays of the gods” that were not part of any year and did not count in the flow of time. These were considered unlucky days, but that was a small price to pay for maintaining the numerical harmony of the yearly cycle. The idea of dividing cycles and circles into 360 equal parts is thus much older than the first written documentation of the 360- degree system. It seems therefore reasonable to assume that many ancient number investigators may well have been aware of it, and that they could also have noticed the numerical link it produced between the cotangens of phi and the lunar year. We may consider such patterns at most as curious coincidences, but the ancient number magicians did not believe in random chance. They would have seen these perceived similarities and symbolic identities as a sign of deep connections between the moon and the mysterious golden ratio which mirrored that celestial body’s behavior and produced its numbers. In addition, they would also have taken the mirror reflections in those digit sequences as a symbol for water, just as the tide- and fluid- controlling moon was in many beliefs closely associated with water. |