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Charles William Johnson

In ancient reckoning, the method of duplatio/mediatio (doubling/halving) numbers appears to have been quite prevalent in the computations. As we have discussed frequently in the Earth/matriX essays, a number series based on doubling shall yield the powers of numbers as well. In this regard, the ancient procedure of doubling/halving numbers may have represented a simple method for computations, which are viewed as being more complex today. Furthermore, we have often emphasized the possibility that the ancient reckoning system may have also employed the method of tripling numbers and hence dividing them by three as well. With that, any number becomes available within the system.

The square root of ten (3.16227766) from the perspective of ancient reckoning, along with its based numbers, 2 and 5 (2 x 5 = 10), become quite significant.

10 = 3.16227766

1 / 3.16227766 = .316227766

In a system of doubling and halving numbers, the numbers two and five are significant regarding multiplication and division:

And, given the fact that the ancients disliked fractions, we may then comprehend the idea of a floating decimal place, which would nullify the decimal point (fractional expression) in favor of a fractal one: 1512 : 151.2. One cannot help but ponder the possibility of the 52c calendar round of Meso-America as representing some kind of symbolic significance for this mathematical expression (5 or 2 as multiplication or division).

From the perspective of math and geometry, the square roots of numbers within the series also becomes relevant. In fact, in such examples as that of the Great Pyramid, as we shall observe below, the square root of the number two (2) becomes all-important. One may even wonder whether the number 707c, which has been cited as an historically significant number may have come from the square root of two. Further, we shall consider the squares roots of the numbers five (5) and ten (10).

2 = 1.414213562		5 = 2.236067978
	10 = 3.16227766

One must remember that the square root of five has also been employed to develop the Golden Section:

	=	1.618033989
		1 / 1.618033989 = .618033989

Square Roots and Reciprocals

The square root of the number two appears in many different places within the ancient reckoning system. We shall discuss only a couple of these aspects significant to our own research.

Within the doubling/halving series of numbers, such as in the Maya Long-Count numbers, the square root of two plays a significant role.

The Maya long-count numbers and categories are cited as:

From one level to the other, the multiplication by two is obvious on the previous long-count series (plus, the addition of one zero). For now, let us simply consider the fractal expressions of these numbers.

The square root of two comes into play regarding the level of roots among these long-count fractals.

576 = 24		24 x 2 = 33.9411255
33.94112552 = 1152 (kinchiltun fractal)

Let us offer another example for clarity of method:

1152 = 33.9411255		33.9411255 x 2 = 48
	482 = 2304 (alautun)

One may wonder whether the ancients comprehended such an elementary relationship among the series of numbers based on the doubling/halving method. Consider the following relationship discerned from other ancient reckoning numbers. The Great Pyramid’s baseline measurement is often cited as 756 feet (189, 378, 756, 1512)....

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