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Ancient Creation Stories told by the Numbers

by H. Peter Aleff

 

 

  

Footnotes :

 

[1]  Aidan Dodson: “The Mysterious Second Dynasty”, KMT, Summer 1996, pages 19 to 31, see page 27 for pictures of statuette and of inscription.

 

[2]  Marshall Clagett: “Ancient Egyptian Science”, Volume 1, Figures 1.11 a and b, page 752.

 

 

[3] Richard J. Gillings: "Mathematics in the Time of the Pharaohs", Boston, 1972, edition consulted Dover, New York, 1982, pages 49 to 52.

 

  

 

 

 

  

 

  

Numerals and constants  

 

 tell the creations of numbers and world

 
 


King Khasekhem's Heb-Sed statuettes
 

Near the end of the Second Dynasty, the Horus Khasekhem (2711 to 2691 BCE) left us the earliest known three- dimensional representations of an Egyptian ruler.  One is a limestone portrait head, and the other two are statuettes of the seated king in his “Heb-Sed” robe.  On both of the latter he wears the White Crown of Southern Egypt but has also the supposedly northern emblem, the Horus falcon, over his name box.

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 Ashmolean Museum at Oxford, the numerals below the three slain bodies and the two hieroglyphs on their level read as47,209 enemies”[1].  The two hieroglyphs are, in Gardiner’s nomenclature, S39 which is a scepter and may express the king's power in overcoming his enemies, and Aa26, which Gardiner calls a “doubtful” determinative for “rebel” and which seems here to designate said "enemies".  

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 5,000 prisoners.  

The other statue is in the Cairo Museum and is less well preserved.  It illustrates the same three slain victims but only the legs of the “swimmer”.  At first sight, its numerals seem to read as 48,205.  However, one of the “1,000” lotus stems in Quibell’s drawing has no head and looks more like a scratch on the statue; also, the place for the missing unit strokes seems to be worn or chipped off, so the number next to the slain victims was probably the same as on the Oxford example[2]. 

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  9.4418 which is a fair approximation to three times pi.  In symbolic terms, where three meant “many” and where we postulate that the circle ratio represented the sun, this would have been equivalent to “many times the sun”.  Since the king was the embodiment of the sun, that ratio of slain to bound may then have promised him many years and/or many lives, both suitable for his magical renewal.

The 47,209 slain enemies and 5,000 prisoners of king Khasekhem:

47,209 / 5,000

= 9.4418

Difference

3 pi  = many suns 

= 9.4248

0.1806%

 47,209 x 5,000

= 236,045,000

 

1,000,000,000 / phi3

= 236,067,978

0.009735%

Now multiply the slain with the bound, and you get 236,045,000.  This is a thousand million times 0.236045 and so matches very closely the remarkable fraction  0.236068 on a different level of the decimal system.  This mathematical parallel to renewal by transformation miraculously mimics the successive stages of growth when you combine it with the first integers:   

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.

Transformations based on the reciprocal of phi cubed

1 / f3

= 0.236068

= 1  / f3

1  +  1 / f3

= 1.236068

= 2 / f

2  +  1 / f3

= 2.236068

= Ö

3  +  1 / f3

= 3.236068

= 2 f

4  +  1 / f3

= 4.236068

= f3

5  +  1 / f3

= 5.236068

= 2 f2

7  +  1 / f3

= 7.236068

= f Ö 20

7.2360682

= 52.36068

= 20 f2

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 Heb-Sed celebration which was meant to be repeated forever. 

The curious precision of the 47,209 entry for the slain appears to further confirm its derivation from a fifth of that miraculous fraction 0.236068.  There are 52 ways to express this fraction in the ancient Egyptian system of unit fractions where the denominator is always one, or 51 ways if we follow the apparent rules for their decomposition which Richard J. Gillings describes in his classic "Mathematics in the Time of the Pharaohs"3 and which limit the size of the largest divisor to less than 1,000.  These rules also favor the least number of terms and the smallest initial divisors, unless a slightly larger one greatly simplifies the expression.  

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:

Approximations to .236068 in the ancient Egyptian system of unit fractions

1 / 5
.200000

+ 1 / 29
.0
344827

+ 1 / 640
.
0015625


= .2360452

1 / 5
.200000

+ 1 / 31
.0
322580

+ 1 / 264
.
0037878


= .2360459

1 / 5
.200000

+ 1 / 36
.0277777

+ 1 / 121
.0082644


= .2360422

1 / 5
.200000

+ 1 / 44
.
02
2727

+ 1 / 75
.0
13333


= .2360
61

1 / 5
.200000

+ 1 / 46
.0
21739

+ 1 / 70
.01
4285


= .2360
25

1 / 5
.200000

+ 1 / 55
.
018181

+ 1 / 56
.0
17857


= .2360
39

1 / 6
.
166666

+ 1 / 24
.
041666

+ 1 / 36
.0
27777


= .236
111

1/6
.
166666

+ 1 / 25
.
0
4

+ 1 / 34
.0
2
94117


= .236
078

1/6
.
166666

+ 1 / 21
.
0
47619

+ 1 / 46
.0
2
1739


= .236
025

1/7
.
1
42857

+ 1 / 14
.
0
71428

+ 1 / 46
.0
2
1739


= .236
025

1/7
.
1
42857

+ 1 / 17
.
0
58823

+ 1 / 29
.0
34482


= .236
163

1/7
.
1
42857

+ 1 / 21
.
0
47619

+ 1 / 22
.0
45454


= .23
5931

1/8
.
1
25

+ 1 / 9
.
111111

 


= .236
111

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, 47,209, then 47,209.2, and 47,208.4. Each one matches the number of victims exactly when you omit the 0.2 killed in the second example or round up the last 0.4 in the third case to count that fractional dead as a whole person.  All the other and less desirable combinations of unit fractions would have produced different quantities.  

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.

Actually, the numbers did not need to be written to exert their divine power, as you can see from the next examples which bring us to king Djoser's step pyramid complex at Saqqara.
 

 

 

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