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## Converting Phi and Pi into a year calendar | |

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Anonymous Coward (OP)User ID: 171724 United States 04/06/2007 07:26 PM Report Abusive Post Report Copyright Violation | Some properties of the golden ratio phi The geometric definition of the “golden section” is that it divides a length into a larger part and a smaller part so that the smaller part forms the same ratio with the larger part as the larger part does with the whole. Raster picture .gif = 31 Kb view vector version 25 Kb info on vector viewing plug-in This definition is fairly simple, but the magic starts when you add that larger part to the previous whole because the new whole is again in the same phi ratio to the initial whole, and that new one to the next, and so on. Each time the new length grows out of the old in unchanged proportion. The same is also true for the triangles and rectangles formed from this golden ratio, as well as for their logarithmic spirals which these form, as shown in the picture above. It also applies to the construction of the regular pentagon since this figure is entirely formed from golden triangles and produces phi over and over again with the ratios between its component lengths. The study of phi reveals many such unusual geometrical and arithmetical properties, and many mathematicians have written with great enthusiasm about this extraordinary number. Some of these surprising features of phi are illustrated in the picture above or listed in the Tables below. A small sampling from the many unique properties of the “golden section” Phi and its reciprocal 1 / phi are the two solutions: phi = (Root 5 + 1 ) / 2 and 1 / phi = (Root 5 – 1) / 2 of the equation x2 - x - 1 = 0. These solutions are perfect mirrors that reflect each other’s digit sequence after the decimal point, all the way to infinity: phi = 1.618033988749894848204586834365.... 1 / phi = 0.618033988749894848204586834365.... Phi is its own reciprocal added to 1, and adding 1 to phi is the same as squaring it. Adding phi to its square yields its cube; this adding of sequential powers always produces the next one. 1 / phi + 1 = phi phi + 1 = phi2 phi + phi2 = phi3 phi2 + phi3 = phi4 phin + phin+1 = phin+2 The same holds true for negative powers and continues the same way, on and on, again as in a mirror. 1 = 1/phi + 1/phi2 1/phi = 1/phi2 + 1/phi3 1/phi2 = 1/phi3 + 1/phi4 1/phi3 = 1/phi4 + 1/phi5 Phi has also a special relationship with the square root of 5 which is the irrational number Root 5 = 2.236068..., derived from the fingers of one hand that so creates this equally versatile number. phi + 1/phi = Root 5 1/phi = (Root 5 – 1) /2 phi = (Root 5 + 1) /2 phi2 = (Root 5 + 3) /2 phi3 = Root 5 + 2 phi4 = Root 5 + 3 + phi Phi and its mirroring functions create together the first five integers, and phi itself is the sum of all its reciprocal powers: 1/ phi + 1/phi2 = 1 phi + 1/phi2 = 2 phi2 + 1 /phi2 = 3 (phi + 1/phi2)2 = 4 (phi + 1/phi )2 = 5 1/phi + 1/phi2 + 1/phi3 + 1/phi4 + 1/phi5 + 1/phi6 + ... = phi Each next number in the Fibonacci sequence below is formed by adding the second last entry to the last one. As the entries grow larger, the ratios between successive pairs of Fibonacci numbers, shown in the last row, converge ever closer towards phi. 0+1 1 1/1 1.00 1+1 2 2/1 2.00 1+2 3 3/2 1.500 2+3 5 5/3 1.666 3+5 8 8/5 1.600 5+8 13 13/8 1.625 8+13 21 21/13 1.6154 13+21 34 34/21 1.6190 21+34 55 55/34 1.6176 34+55 89 89/55 1.6182 55+89 144 144/89 1.6180 Many modern writers have expressed amazement and delight about these and other unique properties of this “golden” ratio. If any ancient number researchers were aware of some among these astonishing properties, they would surely have been impressed. They would also have looked for symbolic analogies of that number’s behavior outside the numerical domain. These analogies are easy to find. For instance, the cycle of the moon resembles that of the digit sequences in phi behind the decimal point as you multiply phi with itself. These digit sequences are the same for the odd powers of phi and their reciprocals in the bold rows below, but not for the even powers between them. The decimal expansion of the negative even powers is one less the decimal digits of the corresponding positive power, so the expansion of the even powers looks different, and the mirroring of the digits seems to disappear in them. The successive powers of phi produce therefore a pattern in which they alternately display that reflection and then hide it again, just as the moon alternately reflects the light of the sun and then does not. This striking pattern is easy to find, and any curious ancients who explored the powers of this constant could easily have discovered it. Digit sequence symmetries in positive and negative powers of phi: phin phi1 = 1.61803398875 phi2 = 2.61803398875 phi3 = 4.23606797750 phi4 = 6.85410196625 phi5 = 11.09016994375 phi6 = 17.94427191000 phi7 = 29.03444185375 phi8 = 46.97871376376 phi9 = 76.01315561752 phi10 = 122.9918693812 phi11 = 199.0050249987 phi12 = 321.9968943800 phi13 = 521.0019193789 1 / phin 1/phi1 = 0.61803339887 1/phi2 = 0.38196601125 1/phi3 = 0.23606797750 1/phi4 = 0.14589803375 1/phi5 = 0.09016994375 1/phi6 = 0.05572809000 1/phi7 = 0.03444185375 1/phi8 = 0.02128623624 1/phi9 = 0.01315561752 1/phi10 = 0.00813061875 1/phi11 = 0.00502499874 1/phi12 = 0.00310562001 1/phi13 = 0.00191937894 Phi suggests also another similarity with the moon in the radians system of expressing angles. This system is less arbitrary than the division of a circle into 360 degrees, and some early mathematicians could easily have used it although we have no surviving written record of their having done so. In that radians system, you simply measure the rim length of the circle section that corresponds to an angle, and you divide this length by the radius of the circle. The 180 degrees of a half circle produce then a rim length of pi, and a right angle is pi / 2. In that system, the squares of the doubled sinus and cosinus for each successive twentieth of pi are all reflected by simple functions of phi that grow and decrease according to the progress of those pi fractions along the circle. Together, these form the well- ordered display of quasi- mirrored symmetries in the list below1 . (“R” means here “square root”) The waxing and waning functions of phi that reflect successive twentieths of pi: Angle(1800 = pi) pi / 20 = 90 pi / 10 = 180 3 pi / 20 = 270 pi / 5 = 360 pi / 4 = 450 3 pi / 10 = 540 7 pi / 20 = 630 2 pi / 5 = 720 9 pi / 20 = 810 ( 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 tablets2 . However, the Assyrian cultic year had long divided the annual cycle as well as the circumference of the universe into 360 days or parts3. 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 Texts4 from about 2,300 BCE, and later coffin lids and tomb ceilings5 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. Source: [link to www.recoveredscience.com] |

Anonymous Coward (OP)User ID: 171724 United States 04/06/2007 08:44 PM Report Abusive Post Report Copyright Violation | This puts the following timeline: 2012 CE 1640.419 BCE 5292.838 BCE 8945.257 BCE 12597.676 BCE 16250.095 BCE 19902.514 BCE 23554.933 BCE 27207.352 BCE 30859.771 BCE 34512.19 BCE 38164.609 BCE What events happened in 1640 BCE? Around 1640 BCE, towards the end of the "Middle Kingdom", central authority in lower Egypt seemed suddenly to perish and the rulers of the land fleed to the southern city of Thebes leaving the capital in Memphis, while some nomadic tribes supposedly coming from Asia were plundering the towns around the Nile delta. Source: [link to www.planetxinbound.com] Thera (Santorini) exploded circa 1640 BCE. Source: [link to www.waterencyclopedia.com] Exodus of the Hyksos/Hebrews occurred circa 1640 BCE. Source: [link to www.heardworld.com] Building of pyramids ended circa 1640 BCE. Source: [link to www.livius.org] [link to www.google.com] What events happened in 5293 BCE? Sahara desert became a dessert in circa 5300 BCE after a 3000+ year wet period. Source: [link to antigreen.blogspot.com] The first such a farmers settled in the Danube Valley inabout 5300 BCE. Within the seven centuries their descendants had slashed and burned their way from the Balkans to the Netherlands and east into the Ukraine. Source: [link to 64.233.167.104] Newgrange in Ireland dates to about 5,300 BCE. Stories have been handed down orally for a very long time, but it is possible that formal histories and storytellers become more important at this time. However, they have always been important if not vital. Maybe this is how Shamans began millenia ago, as protectors of stories and the knowledge of the group or tribe? We can only wonder just how far back some of these things actually go. Around 5300 BCE pottery entered Finland. The earliest representatives belong to the Comb Ceramic Cultures, known for their distinctive decorating patterns. This marks the beginning of the neolithic for Finland, although the subsistence was still based on hunting and fishing. Extensive networks of exchange existed across Finland and Northeastern Europe during 5th millennium BCE. "By 5300 BCE Bandkeramik settlement had spread along the major river valleys of central Europe, and communities of Bandkeramik farmers were established in eastern France and the Low Countries." [link to www.google.com] What happened 8950 BCE? At about 9,000 BCE (i.e. 11,000 years ago), was the end of the last ice age. The abruptness of the termination is startling. Agriculture, and all of our civilization, developed since this termination. The enormous glacier, several kilometers thick, covering much of North America and Eurasia, rapidly melted. Only small parts of this glacier survived, in Greenland and Antarctica, where they exist to this day. The melting caused a series of worldwide floods unlike anything previously experienced by Homo sapiens. (There had been a previous flood at about 120 kyr, but that was before Homo sapiens had moved to Europe or North America.) The flood dumped enough water into the oceans to cause the average sea level to rise 110 meters, enough to inundate the coastal areas, and to cover the Bering Isthmus, and turn it into the Bering Strait. The water from melting ice probably flooded down over land in pulses, as ice-dammed lakes formed and then catastrophically released their water. These floods left many records, including remnant puddles now known as the Great Lakes, and possibly gave rise to legends that persisted for many years. As the glacier retreated, it left a piles of debris at its extremum. One such pile is now known as New York’s Long Island. [link to www.google.com] What about 12600 BCE? Prior to 12500 BCE there is no evidence of any human habitation in North America. About 12,500 B.C.E. the Irmnistic religion (freely invented by Wiligut) arose, quickly becoming the dominant religion in all of the Germanic area. Later, a competing religion appeared, the Wotanism (worshippers of Odin) and challenged the Irminists. A violent religious war ravaged Germania, with the crucifixion of Baldur-Chrestos, one of the most holy prophets of the Irmnists, by the Wotanists in Goslar in the year 9,600 B.C.E. Source: [link to www.skepticreport.com] Werner von Bülow translates "Legend tells, that two Suns, two wholesome in change-rule UR and SUN, alike to the hourglass which turned upside down ever gives one of these the victory / The meaning of the divine errant wandering way / dross star in fire's sphere became in fire-tongue revealed to the Earth-I-course of the race of Paradise / godwilling leaders lead to the weal through their care in universal course, what is visible and soon hidden, whence they led the imagination of mankind / polar in change-play, from UR to SUN in sacrifical service of waxing and waning, in holy fire Santur is ambiguously spent in sparks, but turns victorious to blessing" Santur is interpreted as a burnt-out Sun that was still visible at the time of Homer. Rüdiger speculates that this was the center of the solar system hundreds of millennia ago, and he imagines a fight between the new and the old Suns that as decided 330,000 years ago. Santur is seen as the source of power of the Hyperboreans. Source: [link to www.answers.com] 12,000 BCE Rapid global warming (most likely an interstadial) which lasted a few thousand years and was followed by a return of cold known as the Younger Dryas Data suggest a periodicity of just over 3600 years to history's catastrophes (3652.419 years). This puts the next one due on or about 2012. Period for sun's binary companion to make 1 cycle is 3652.419 years (based on formula): (phi*(360(*10))*pi)/5.01025. This puts the following timeline: 2012 CE 1640.419 BCE 5292.838 BCE 8945.257 BCE 12597.676 BCE 16250.095 BCE 19902.514 BCE 23554.933 BCE 27207.352 BCE 30859.771 BCE 34512.19 BCE 38164.609 BCE Check it out yourself... WOW! All of this is coming from the formula: Period for sun's binary companion to make 1 cycle is 3652.419 years (based on formula): (phi*(360(*10))*pi)/5.01025. It appears the Sumerians were well aware of Phi and Pi; Plato was well aware of the importance of the number 5 -- but I'll bet he did not know why. It is the orbital revolution period of Tiamat. |

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