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 Clifford A. Pickover: “Keys to Infinity”, John Wiley & Sons, New York, 1995, pages 147-151. Pickover calls C.J. Bouwkamp’s 1965 shortcut to this diameter “not for the meek”.

2 Tamara Curnow: “Falling down a polygonal well”, Mathematical Spectrum 26:4 (1993/1994), pages 110 to 118.

3 See, for instance, the review in The College Mathematics Journal, January 1996, of Tamara Curnow’s “Falling Down a Polygonal Well” cited on Figure 3.

4 Hugh Thurston: “Early Astronomy”, Springer Verlag, New York, 1994, page 75.

The mathematics of Genesis 1

in the layout of the Jerusalem Temple

 The outermost regular shape that contains all others of its kind click on  picture  = 14 Kb

The constant of the all- embracing circumcircle is the limiting radius you obtain when you surround a circle of radius one with the successive regular polygons,starting with a triangle and adding one more side each time, so that each of these polygons touches the circumcircle of the preceding one with all its sides.

The radius of the outermost circumcircle grows quickly in the first few steps, but as the polygons come closer and closer to the shape of the circles they surround, that growth slows to an ever more sluggish crawl and then virtually stops. Instead of continuing to expand, that radius approaches a maximum value, as illustrated in the drawing above.

The mathematician and computer researcher Clifford A. Pickover discussed this construction in his book “Keys to Infinity”, in a chapter called “Infinitely Exploding Circles”1 . He explained that its radius is given by the reciprocal of the product

(cos pi / 3) x (cos pi / 4) x (cos pi / 5) x ..... x (cos pi / n)

where n is the number of sides in the outer polygon.

He programmed a computer to begin this tedious calculation and found that the first thousand polygons bring that radius only to 8.657231; even 10,000 iterations come no closer than 8.695745 whereas the actual ultimate value is 8.7000366252081945.....

As Pickover tells the story, modern mathematicians first tried to compute this limit in the 1940s. At first, they came up with a wrong value of about 12 which was mentioned in the literature as late as 1964. In 1965, C. J. Bouwkamp developed a shortcut to the correct constant, producing a formula that Pickover called “monstrous- looking” and “not for the meek”.

Almost three decades later, Tamara Curnow searched for a good approximation2 and came up with 8.7000366545. She, too, used mathematics that a reviewer described as “not for wimps”3.

I am not suggesting that any ancients knew such advanced and intimidating methods. However, if they thought the number was symbolically significant, then the drudgery of computing it the long way would not have stopped the folks who routinely piled up millions of baked bricks or hewn stones for symbolic purposes.

 A typical view of pyramid blocks piled up on the Giza plateau in northern Egypt.  Pyramids were symbolical structures and had no practical use.  Photo from Corel Photo CD #30000 on Egypt

And the symbolic importance of this outermost limit would have been fairly obvious. If any ancient geometers investigated this greatest regular shape that contains all others of its kind, they would probably have identified it with the all- encompassing Heaven and with the Cosmos that contains everything in existence.

To evoke that postulated symbolic significance of this constant to early mathematicians, I call it here C for Containing Circle, and also for its symbolic identity with the all- surrounding Cosmos.  That designation also fits modern nomenclature because C the mathematical limit is as hard to reach as c the speed of light limit in physics where every narrowing of the gap to the final value also makes further progress ever harder.

The colors and hieroglyphs I added to the geometric construction in the picture above are not just decorations. They are meant to suggest that if any ancient number researchers thought about this limit, then they would not have perceived it as a mere nesting of abstract lines and circles, but as a metaphor animated with cosmological and religious meanings.

These meanings, and the association of C with the sky, would have been strongly reinforced by the discovery that it also produces the major cycles of the heavenly bodies.  We do not know which of these cycles the ancients knew with what precision, but if their math is any guide, then chances are they also knew more about astronomy than what they left us in writing.

A mathemagician with good astronomical data would soon have found that the circumference of that ultimate all- containing circle, C x pi = 27.3320, is close to the 27.3217 days in the sidereal month, that is, the time it takes the moon to return to the same background of stars. This combination of pi the sun number with the cycle of both moon and stars alludes again to all that is contained within the circle of heaven.

Moreover, ten times the square root of C approximates the 29.5306 days in the synodic month, that is, until the moon returns to its same phase, and ten times its cube matches the 6,585.32 days4 in the Saros cycle with which the Babylonians measured the time until the moon returns to a similar eclipse. Similarly, the 18.5995- year cycle of the nodes in the lunar orbit relative to the ecliptic is close to 100 phi / C.

So much for the moon; as to the sun, adding the 30 days in an Egyptian standard month to the twelve months in a year, and then multiplying this calendrical total of 42 with C, yields close to the 365.2424 days in the solar year. Here is a summary of these celestial connections:

 C = 8.700036625... and astronomical periods pi x C = 27.3319 d = 0.03733% Star month (days)  = 27.3217 10 x square root of  C = 29.4958 d = 0.1179% Phase month (days) = 29.5306 10 C3 = 6,585.11 d = 0.003189% Saros eclipses (days) = 6,585.32 100 phi / C = 18.5980 d = 0.008065% Moon node cycle (years) = 18.5995 42 C = 365.4015 d = 0.04357% Solar year (days) = 365.2424

These numerical connections between C and so many major cycles in the sky were likely to impress any ancient number mystics who were aware of them.  If any of them went to the trouble to compute this elusive number and to accumulate the relevant astronomical observations, then they would most probably also have explored the permutations of the results and so noticed these matches.

The close relationship of C with the numbers obtained from the sky, in turn, would have further confirmed their mathemagical interpretation for this constant of the all- surrounding circle as symbol of the sky.