1 Table adapted from H.E. Huntley: “The Divine Proportion -- a Study in Mathematical Beauty”, Dover Publications, New York, 1970,., page 40.

2 O. Neugebauer: “The Exact Sciences in Antiquity”, 1957, edition consulted Dover, New York, 1969, see note 9, page 25.

3 Simo Parpola: “The Assyrian Tree of Life: Tracing the Origins of Jewish Monotheism and Greek Philosophy”, Journal of Near Eastern Studies, Volume 52, July 1993, Number 3, pages 161-208, see note 103 on pages 188 and 189.

4 Marshall Clagett: “Ancient Egyptian Science”, Volume 2: “Calendars, Clocks, and Astronomy”, American Philosophical Society, Philadelphia, 1995, page 49, citing Utterances 251 and 320 in which the word for “hours” is determined both times by three stars.

5 R. Böker and F. Schmeidler: “Über Namen und Identifizierung der ägyptischen Dekane”, Centaurus 1984, Institute of History of Science, Aarhus, Denmark, Vol. 27, pp. 189 to 217.

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.

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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....

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:

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:

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.