recoveredscience.com              We offer surprises about      and numerals and their ancient religious uses     in our e-bookAncient Creation Stories told by the Numbersby H. Peter Aleff

Footnotes :

[1] This quasi-equation follows from the second and third among the three combinations of phi and pi shown earlier, but that would not necessarily have diminished the symbolic importance of the so obtained ratio for the mace designer.

[2] George Hart: “Pharaohs and Pyramids: A Guide through Old Kingdom Egypt”, The Herbert Press, London , 1991, page 55

[3] Aidan Dodson: “On the Threshold of Glory: The Third Dynasty”, KMT Summer 1998, pp. 30 & 39.

Numerals and constants

tell the creations of numbers and world

Pi for sun and phi as moon

The likely symbolic meanings of the circle number pi in this postulated ancient Egyptian numerology are easy to retrieve because the most prominent circle in all of nature is the sun, both for its shape and for its motion through the sky, and the Egyptian hieroglyph for the sun was indeed a circle, as illustrated in the discussion of the Shen- Ring.

Since the king was the representative of the sun on earth, this symbolic connection of pi with the sun would also have applied to him.  Therefore, if the ancient geometers were aware of those constants and the mace designer had used them knowingly, then the pi- component in each of the above mace quantities would most likely have referred to the king as well as to his celestial counterpart the sun.

If pi was the sun, then the most logical role for phi would have been to represent the moon because the mathematical behavior of this constant resembles the observable behavior of that other “eye in the sky”, as shown on the page about some properties of the golden ratio.

The moon appears in three major forms which are waxing, full, and waning.  Many moon goddesses of later times were split into these separate aspects to be venerated as the lovely young maiden, the mighty and mature queen of heaven, and the wise old sibyl who could also act as nasty witch.  The three forms of phi joined with pi on the mace, that is, its square root, phi itself, and its square, may therefore correspond to these three aspects of the moon goddess.

If so, the three combinations of pi with different forms of phi could indicate that the sun king united with the moon goddess in her three manifestations, all presumably embodied by the woman seated before the sun king.

This does not contradict said lady’s simultaneous representation of the sky goddess because the latter was often also identified with the moon which many early religions held to be her interchangeable image as the face of the night sky and thus as bride or also mother of the sun.

The self- regenerating properties of phi would have made this constant also an excellent symbol for the renewal the king expected from his triple union with that “king-bearer” goddess in her shrine formed like the numeral "ten".  Indeed, the ratio between the total of all items and the number of humans included in that total comes close to phi times ten [1], as shown below:

 1,942,000  total / 120,000 prisoners = 16.1833 Difference Ten phi = 16.1803 0.0183%

Narmer's combination of these constants and their implied symbolic values fits the context of his nuptials and renewal so well that it looks as if the designer of his mace had carefully planned its numerological account of the occasion, or else as if Thoth the god of mathematics and magic had directly guided the sculptor's unwitting but obedient hand.

Of course, any single example of apparently intentional composition with numbers could be due to random chance, and no one of them proves with certainty that individual scribe’s awareness or intent, nor any intervention by Thoth.

However, in the case at hand, it is intriguing that the total of magical renewal from Narmer's mace pops up again in the ratios of tomb dimensions from two slightly later kings.  These are the burial pit of the Horus Den (3,007 to 2,975 BCE) [2] and the pyramid enclosure of the Horus Sanakhte (2649 to 2630 BCE):

 1,942,000 / 1,000,000 = 1.94200 Narmer total 12 phi / 10 = 1.94164 Pi divided by phi = 1.94161 Difference Burial pit of king Den in Saqqara : 9.5 meter / 4.9 meter = 1.93878 0.144% Pyramid enclosure of king Sanakhte at Abu Rowash: 330 meter / 170 meter = 1.94118 0.0222%

The aspect ratios of these tomb structures would not have yielded the same digit sequence as on the mace without the lofty powers of ten from there.  They would have been written in unit fractions, as illustrated below:

 Unit fractions: 1 + 1/2 + 1/3 + 1/10 + 1/116 = 1.94195 Unit fractions: 1 + 1/2 + 1/3 + 1/14 + 1/27   = 1.94180 Unit fractions: 1 + 1/2 + 1/4 + 1/8 + 1/15     = 1.94167

However, the ancient Egyptians were very adept at multiplying their fractions with powers of ten, and they would easily have seen that a series such as one of these would yield the total from the mace when shifted across their decimal system.

In any case, it seems that this number node offered an auspicious guarantee of renewal not only for the Heb-Sed celebrations of kings but also for their burials since these were the prelude to another and even more important renewal.  It may therefore be no accident that these very similar numbers are found in the dimensions of those royal tombs.

Comparable compositions of constants and their intersections with integers occur in many other ceremonial number groups and architectural proportions from all periods of pharaonic rule.

These sources of information are usually overlooked by mainstream scholars committed to their received wisdom of Greek priority.  However, the study of such number groups reveals a considerable body of long lost and at least partly recoverable pre- Greek mathematical knowledge.  It shows a consistent pattern and a coherent fit with the contexts that make it hard to dismiss all the findings of constants as merely accidental.

You will see in this book that an inquiry into ancient uses of number magic offers eye- opening insights into how the properties of certain numbers influenced several ancient religions so deeply that some of these influences still survive in their modern successors.

You will also discover that the approximations the ancients knew for these constants were much more accurate than today's historians of science led us to believe.