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]  James Legge, translator: “The I Ching -- the Book of Changes”, 1882, edition consulted Dover, New York, 1963, pages 3 to 9.

[2]  Encyclopaedia Britannica, 1980, Macropaedia Volume 4, page 943: “Combinatorics and Combinatorial Geometry”.

[3]  Encyclopaedia Britannica, 1980, Micropaedia Volume VI, page 484: “Magic square”.

Numerals and constants

tell the creations of numbers and world

Month and year as Narmer's bonus results

The mace designer’s mathemagical artistry does not end with the above combinations of mathematical constants.  The number of goats divided by the sum of the two smaller quantities comes close to one tenth the days in the sidereal month, that is, the time it takes the moon to return to the same background of stars.  And balancing this direct reference to the moon and the night sky, the ratio of the differences between two pairings of the three items yields exactly one hundredth the number of days in the Egyptian civil year which was based on the cycle of the sun.

 Calendar ratios from Narmer’s mace
 1,422,000 goats / 520,000 oxen + prisoners = 2.73462 Difference Star month / 10 = 2.73217 0.0897% 1,022,000 goats less oxen / 280,000 oxen less prisoners = 3.65000 Days in calendar year / 100 = 3.65000 0%

These two complementary periods embedded in the “booty list” extend the number magic of the mace into time and so unite again the moon and the sun in forms even more directly related to these than phi and pi.   They also echo the connection of the Heb-Sed festival with the “meeting of sun and moon” in the pharaonic time reckoning, as proposed in the chapters about the re- enactment of the Heb-Sed festival cycle on the Senet gameboard in my e-book "The Board Game on the Phaistos Disk".

The knowledge of these astronomical periods is not attested in writing until later times, but nothing says they could not have been measured much earlier, as I propose on the pages about the star month and the solar year.

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The Narmer mace designer’s juggling of numbers and their ratios to obtain all these combinations of mathematical and astronomical constants is a masterly work of mathemagical art.  He milked from those three quantities of goats and oxen and prisoners a steady stream of excellent approximations to salient numbers that are as hard to obtain from random entries as striking water from surface rock.  Just try it: pick blindly any three numbers with three or four digits each and see if they produce even droplets of this kind.

Composing a group of numbers to yield with their ratios so many close hits on these constants requires skillful planning and might be compared with the challenge of constructing a so- called “magic square”.

Magic squares are square grids in which the integers from one to the square of the grid side are arranged in such a way that each row and column, and often also each diagonal, adds up to the same sum.  Their name reflects that they were believed to have magic powers, a logical assumption in the symbolic realm where the order imposed on the numbers counteracts disorder and chaos and thus protects from the unpredictable.

The earliest known magic squares appear in the “I Ching”,  the ancient Chinese manual of divination which is said to have originated in China anywhere from “before 1000 BCE”[1] to “around 2200 BCE”[2].  In Greek and Roman antiquity, people used magic squares widely as religious symbols and talismans against disaster, including some Pompeians who may have hoped they would prevent volcanic eruptions.

Despite such blemishes in their performance record, magic squares were still popular in 19th century Europe and America as insurance against fire, sickness, and other catastrophic events[3].  It is therefore easy to imagine that Narmer and his advisers may have attributed similar ordering forces to arrangements of numbers that harnessed not the integers, as in the magic squares, but the even more mysterious and therefore surely even more potent basic constants which hide behind them.

This proposal may be mere speculation, but it seems more plausible than the mace designer hitting such a mother lode of special ratios with random census results from Narmer's booty- counting department, particularly since the mace numbers contain yet another major constant