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by H. Peter Aleff |

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Near the end of the Second Dynasty, the Both of these Heb-Sed statuettes have inscriptions on their base with numerals, next to pictures of slain or drowned people and an apparently swimming bound prisoner. On one of these objects, which is now in the The swimming bound man on this Oxford statuette has five 1000-numeral lotus stalks sprouting from his head, and a slanted line comes from his nose that resembles the “leash” from the nose of a similarly subjugated enemy on the Narmer Palette who has six such stalks growing from his back. Presumably, this reflects Khasekhem’s count of The other statue is in the In both cases, the 1000- lotus- stalks in that number are grouped into two clumps joined at their base, one with four stalks and the other one with three, if we omit the possible scratch on the second statue which seems to add there a fourth stalk to that other clump. The apparent subject of both inscriptions matches labels on stone vases from the reign of this same king that say “Year of Fighting the Northern Enemy”. Yet, the numbers given in them may be unrelated to any actual quantities. Like the entries on Narmer’s mace, king Khasekhem’s Heb-Sed quantities look as if they had been composed to connect the king’s rejuvenation festival with magical numbers that were appropriate for the occasion and expressed its purpose. Here is how: Dividing the 47,209 slain by the 5,000 bound yields
Now multiply the slain with the bound, and you get The fraction starts out as the reciprocal of the cubed Golden ratio phi. Add one unit, and you get two divided by phi. Add two instead, and this fraction transforms the plain square root of four into the highly versatile and again phi- related square root of five, a feat which may have pleased many of the early number mystics. Moreover, when you add three, you get double phi, and if you add four you obtain the cube of that golden ratio as well as said fraction’s own reciprocal. Add one more, for a total of five, and you have twice phi squared. The magic seems to skip at six, but add this quasi- magical fraction to the royal number seven and square the result, and you find twenty times phi squared, a number in which the same fraction returns again, this time multiplied by ten.
This unique fraction, related to the golden ratio and its sibling the square root of five, transforms thus the above integers into ever higher manifestations of the initial entry. To Khasekhem's mathemagician priests, this may have seemed a perfect parallel to the king’s expected transformations into ever newer and higher versions of himself. The size of the number obtained by that multiplication of the slain with the bound further implies that these transformations were to be repeated a thousand million times over, about as close to “forever” as the ancient scribes could convey with their numerals. This was again perfectly suited for a The curious precision of the Those sums of unit fractions which come closest to the correct value are the following, as shown below in red and in the order they get generated from these rules. Their decimal equivalents are shown underneath each unit fraction and added up in the last column:
The fourth one of these solutions comes closest to the actual value. However, the difference between it and the first three arises only in the fifth decimal place. We do not know just how precisely Khasekhem's number crunchers had computed that fabulous fraction, except that they rejected the shorter and much more elegant last solution of 1 / 8 + 1 / 9 which has only two terms and so would have been preferable in their system. This most elegant solution is also almost twice as far from the correct value than those first three. According to the rules Gillings and other scholars deduced from the surviving examples of unit fraction adding, the ancient scribes would probably have picked one of the first three solutions if they deemed it precise enough. They may have liked the third one more than the others for numerological reasons because it has an even divisor in the second position and its second and third divisors are both squares, but any of those first three solutions fits the number of slain in the inscription perfectly: Multiplied by a thousand million and then divided by the 5,000 prisoners, each of those first three solutions produced by the ancient rules yields the number of the slain in king Khasekhem's inscription: they are, in order, The precise coincidence of the king's number with the preferred ancient way for writing that fraction of miraculous transformation appears therefore to confirm that it was derived from this fraction to express the king's renewal. It would not have mattered in that context whether any of those people shown on the statuettes as slain or bound had ever existed or not. Their picture plus the writing of their number made them real enough for the magical purpose at hand.
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