TABLE OF CONTENTS

Introduction

Contemporary societies pride themselves on having attained certain kinds of knowledge that the ancient societies were unable to develope. The exact measurement of the concept of pi () is one of those achievements. Pi represents the number reflecting how many times a circle's diameter may be divided into its circumference: 3.141592654 is generally given on contemporary electronic calculators. The so-called exact expression of pi has been worked out to hundreds of thousands of decimal places; when the first zero appears at the 32nd decimal place (3. 14159 26535 89793 23846 26433 83279 50). In practical matters of mechanics, like measuring clearances in engines, four decimal places are sufficient. Anything beyond that becomes incommensurable in a sense. For that reason, 3.1416 was employed for many years without affecting how things were actually engineered.

The relation of the circle's diameter to its circumference becomes undoubtedly significant in computing distances, areas and volumes. The more exact the figure, the more precise the measurements. However, given the fact that it is essentially impossible to physically measure this relationship, since the very starting point and ending measurement of the lines employed as the diameter and the circumference influence the act of measuring, the numerical representation of pi is achieved through theory; not practice. Hence, the theoretical possibility of finding pi to thousands of decimal places, while in practice the unending decimal places become meaningless.

Because of this theoretical development, then, one may understand why contemporary societies enjoy believing that we have achieved something theoretically possible, while ancient societies may not have perceived such a theoretically abstracted answer. We view the ancient societies as having been excellent physical artisans in making exact measurements in their pyramidal structures, but we deny them any exactness in their theoretical conceptions. When, in fact, we should realize that any exactness in practice means an exact theoretical conception. However, we generally are not prone to recognizing exact theoretical thinking on the part of the ancients.

The numbers registered in the ancient reckoning systems may, however, reflect a precision and exactness in their theoretical posits that we have been unwilling to consider. It does not appear to bother most scholars that ancient stone-age people were able to work stone in a very precise manner; that is an accepted characteristic of stone-age people. But, we are unwilling to even consider that those stone-age peoples may have developed precise mental abilities that led them to theoretical posits much like the ones we believe to represent higher forms of societies (i.e., our contemporary societies).

The numbers of the ancient reckoning systems, however, appear to reflect the theoretical possibility that the ancient peoples, who worked with stone (and other media), knew much more about mathematics, geometry and trigonometry than we are willing to recognize. One particular number that has always intrigued us is that generally cited as the Nineveh number, 2268. This particular number may be viewed simply as 3 x 756 = 2268; or 36 x 63 = 2268, etc. Furthermore, it may be viewed in relation to another number from Nineveh, 1959552/fractal: 864 x 2268 = 1959552. We should remember that 864 is double the Consecration number 432.

In the previous Earth/matriX essay (No.101), we saw that when viewing the Trigono/metriX table of ratios, the 2268 number may be conceptualized as that of

pertaining to the corresponding degrees in a right triangle: 22:68:90 degrees (sum total 180 degrees). The obvious suggestion arises whereby the apparent day-counts of the ancient reckoning systems may also have symbolized the number of degrees in a triangle (or circle).

However, since we have already come to realize that a single number in the ancient reckoning systems may have enjoyed various meanings (depending upon how that number may be related to other numbers), then, the case may be that the 2268 (22:68) number(s) may, in fact, represent still other meanings and options in the computations. It is in this sense that we have come to realize that the number 2268 may also reflect a pi-like number for the 260c (day-count; circle), in much the same way that pi (3.1416) represents such a relationship for the 360c (day-count; circle).

Now, let us examine the different aspects of the concept of pi in relation to the different day-counts (and circles) of the ancient reckoning systems.

Symbolism: Day-counts and Degrees

If the numbers relating to specific combinations of degrees within right triangles may have been used to represent the day-counts of the ancient reckoning systems, then specific computations would have been available. Let us consider only a few for the sake of example.

The first possible combination on a right triangle could be considered to be the 1:89:90 degrees triangle. Interestingly enough, this number resembles that employed for the ancient Mesoamerican calendar round of 18980c days (or 365 days times 52 years). Now, if we bring into account the ancient maya companion number, 1366560 (day-count), we see how this number acts as a transition device for computing between 18990 and 18900 (an ancient kemi number), when we employ the mediatio/duplatio method (of halving/doubling numbers).

The 2268 number viewed as degrees and day-counts becomes also symbolic of other numbers through this method. If we view the right triangle, 22:68:90 degrees, the following obtains:

Or, consider reversing the order:

Now, let us take the 13:77:90 degree triangle and visualize these numbers as representing a day-count number:

Whereby it becomes all too coincidental to imagine that adding half of the Nineveh fractal number 2268 (or 1134) to the maya companion fractal number 136656 would produce the corresponding number related to the 13:77:90 triangle (136656 + 1134 = 137790). However, this is precisely what obtains.

Such symbolic logic may have been designed into the angles of the Great Pyramid. For example, if the angles of the Great Pyramid of Khufu are 51.5:77:51.5 degrees, then other angles are being suggested. The 77-degree angle is suggesting the other angles of a corresponding right triangle: 13:77:90 degrees. The 51.5 x 2 (angles) implies a total of 103-degrees. Obviously, the coincidence of number 13-degrees and 103-degrees both reflect the integers one and three (13, 103, fractal-like numbers). We should also bring to mind the fact that the number chosen for the day-count on the Aztec Calendar is 13-acatl as the main symbolic glyph. Symbolically, at least, the 13-acatl glyph at the top of the Aztec calendar, and the 77-degree angle at the top of the Great Pyramid reflect complementary angles within a right triangle. Such coincidences of numbers in the ancient reckoning systems imply a correspondence that may have implications for the computations that we shall illustrate here.

Were we to visualize the 18990 number as including the 189c (189, 378, 756) kemi series with a sacred nine, then we would see how the maya long count numbers may evolve in relation to the 432 consecration number:

Or, consider, the calendar round number:

An even number (160000) differentiates the 1959552/fractal of Nineveh from the Mesoamerican calendar round number (19435520 / 18980 = 1024).

Conceptualizing the day-counts as degrees on right triangles (or viceversa) takes on interesting computations. Consider the Venus synodic cycle number (585c) for the right triangle 5:85:90, in the following figures:

Here, we see how the maya long count number (1872000) enjoys a difference of another maya fractal number (2880000 pictun). The previous illustrations reveal how the day-counts of the ancient reckoning system relate directly to the degrees within right triangles.

It is impossible to review the infinite computations that become available by viewing the numbers of the ancient reckoning systems in relation to the degrees of right triangles as found in trigonometric relationships. However, we shall examine more options as we review the ratios and the concept of pi in the following analysis.

The Trigono/metriX Table and
Ancient Reckoning: the Concept of Pi

In order to discuss a possible relationship between the ancient reckoning systems and the numbers of degrees in right triangles, we have first considered computations for some of the historically significant numbers of the ancient reckoning systems and the degrees of right triangles. As we have illustrated above, a coincidence exists for achieving computations between the ancient day-counts and the degrees of right triangles.

Now, given the fact that the numbers of right triangles hold a significance for the ancient day-counts, we may also consider these sets of number combinations in relation to the number system (of degrees) in a circle. In previous essays, we have discussed the possibility that the ancient reckoning systems may have employed distinct number systems for circles, such as the accepted 360c (360-degrees). But, we have also analyzed the possibility of circles having been divided into 378 segments (degrees) and 3744/fractal segments (degrees) [consult, Earth/matriX Nos. 87 & 89].

Obviously, a circle may be divided into an infinite number of segments (or degrees). Historically, we have come to accept the 360c-degree circle. Another historically significant day-count in ancient Mesoamerican reckoning systems was that of 260c (days/years/cycles). With the change of the number of segments (degrees) in a circle, then one could understand that the concept of pi itself would vary. Although, the numerical representation of pi for a 360c circle would be ca.3.1416 (for the sake of example), for a 260c circle, then, the number would vary. It is understood that the diameter of any circle, no matter how many degrees it may possess, would divide into its circumference ca.3.1416 times, still, for computations of the number of degrees, the pi-like number would vary proportionately.

One could then ask, if 3.1416 is to 360c, what would the number be to a 260c circle. Essentially, one only has to do the following computation to obtain an answer:

3.1416 / 360 = .008726667 then,

.008726667 x 260 = 2.268933333

Interestingly enough, the 2268/fractal Nineveh number makes it appearance. But, many more considerations must be made in order not to simply view this as a matter of coincidence. The point being, that we may observe in a single computation the two historically significant day-counts (260c, 360c) of ancient reckoning systems, alongwith the historically significant 2268 Nineveh number; in relation to the concept of pi.

In other words, if the ancients employed 2268 as an historically significant number for computations in their ancient reckoning system, and this harbors a relation of direct proportion to the 360c and the concept of pi, then, possibly the very existence of the 2268/fractal means that the ancients knew the concept of pi to its exactness as known today. Having singled out 2268 for significance may mean that they knew pi. We must remember, they also employed the 360c circle. Therefore, 2268c may represent a pi-like proportion for the 260c circle, taking into consideration the fact that the ancients appear to have preferred employing round numbers.

In reviewing the following analysis, we should remember that some of the differences found in historically significant numbers may be due to adjustments having been computed. This is a significant concept. For example, the 360c, 364c, 365c day-counts distinguish different computations. From this perspective and understanding of the historically significant numbers and counts, one may realize that there is no single answer to some of the questions being raised in our analyses. For, in fact, many different answers may exist whereby the distinctions represent adjusted computations.

Nonetheless, when series of numbers appear to be relational throughout various computations, we may be observing the manner in which the computations were obtained. Hence, the differences and distinctions may not mean that some of the numbers cited are erroneous, but simply represent different procedures for obtaining those computations. Given that the perspective may change regarding how a particular number/fractal is viewed, then the results also change. Furthermore, specific numbers may be attempting to symbolize different perspectives assumed in the analyses. For example, the different measurements of the apparently exact length measurements of the sides of the Great Pyramid of Khufu, as we shall observe below, may have been designed to symbolize different adjustments in the computations. It could be that the four side measurements of the Great Pyramid (755.43, 755.77, 755.88, and 756.08 feet) may have been exactly portrayed, or represent approximations to a possible common 756c measurement. We may consider all of the possibilities and their related implications as of the numbers coming out of the ancient reckoning systems.

Even though we may never know for certain what was the originally conceived measurement(s) for the side lengths of the Great Pyramid, we may review how the numbers/fractals perform within the ancient reckoning system's apparent computational design. Many statements, then, must be conditioned by the word if such and such number were considered, then possibly such and such number may have been the result. We must always state, however, that we are working within the realm of theoretical speculation. However, the numbers/fractals themselves suggest to us that some of the possibilities discerned in the computations reflect a possible procedure that may have been represented and symbolized by the ancient reckoning system's design.

With these reservations stated once again, let us view the numbers relating to the ancient reckoning systems and the Trigono/metriX table, and the implied procedures for obtaining certain computations.

As we review the following computations, let us remember that the 260c implies the number series: 13, 26, 52, 104, 208, 416, 832, etc.; and, the 360c implies the number series 9, 18, 36, 72, 144, 288, 576, 1152, 2304, etc. (the latter being common to the numbers/fractals of the maya long count). The 260c is often cited as representing 13 months of 20 days; and, the 360c as representing 18 months of 20 days. This is significant for us to remember that the corresponding numbers/fractals of the degrees of a triangle or circle occur in these multiples.

Now, let us observe the distinct numbers relating to the sine and tangent of one degree and fractals thereof:

From the above data, we may observe that as we approach the minimum degree that may be possibly expressed on a circle, we see how the sine and the tangent expressions become apparently equal; and, ultimately incommensurable in practice. At one-degree the distinction is observable; at lesser amounts the distinction disappears as far as numbers are concerned. Although, we know that if it were possible to actually draw or make a circle and lines based on these measurements (for example, one-hundreth-of-a-millionth of a degree), then a physical distinction would actually exist.

For all practical purposes, then, the sine and tangent of a .00000001^o are the same, whereby the radius of the circle approaches the measurement of what the hypotenuse would be.

Hence, in computing the .000000001-degree figure, we obtain a number corresponding to b and c on the right triangle (circle), which represents the number known as the radian (the length of the radius on the circumference of the circle). In other words, since a of the triangle (the tangent) has approached the point touching the radius, the difference is negligible.

In this manner, we may now employ the 360c numbers/fractals corresponding to the triangle and the circle.

.017452406 x 180	=	3.141433159
.017455065 x 180	=	3.141911687

But, when the angle of degrees is almost indistinguishable, then,

One must note that 3.141592654 is the number offered for pi on any electronic calculator today. In this case, the tangential line of the triangle represents .00000001 degrees of the circumference of the circle; hence, the expression of pi equivalency. In fact, the radius (b) and the hypotenuse (c) become indistinguishable at 100th of a degree, but the numerical relationship to pi is 1/one-hundred millionth of a degree.

Now, when we take the .017452406 number and multiple it by 13°, as effected by most trigonometric tables today, we find the result .226881284 obtains. Then, 18° times .017452406 equals .314143308, close to the pi figure. Such adjustments mean that the corresponding 26-degree figure reminds us of another historically significant number 453762556 (ca.4536); and, the 36-degree figure represents nearly 2: .628286616/fractal. Once again, we see how the 260c corresponds to a 2268-like number, and the 360c corresponds to a pi-like number.

If we visualize the 18:72:90 degrees of a right triangle, as reflective of the maya long count number/fractal, 1872000, then we see how the ratio 18:72 degrees reflects the -like number (3.14143308). If we were to make an adjustment, whereby the 18:72 degree level were to reflect pi exactly, as 3.141592654, then the sine number would be .017453294, and the 13:77-degree number would be .226892822. From this perspective, the numbers/fractals related to the Great Pyramid of Khufu may be representative of these two major systems with corresponding adjustments.

The base length of the Great Pyramid has been computed and measured in differing amounts over the years. The height of the Great Pyramid has been calculated to be around 481 feet. The calculated base length of the four sides, at 756 feet, divided by 962 (twice the height measurement) would yield:

756 / 962 = .785862786

.785862786 x 4 = 3.143451143 as a symbolic representation of

The 756c is suggestive of the 2268c Nineveh number/fractal.

2268 / 756 = 3

962 / 26 = 37

481 / 13 = 37

Had the ancient kemi known the corresponding ( number, 3.141592654, then, they would have obtained the proportion of .785398163, thereby possibly producing the following measurements:

	962	x	.785398163	=	755.5530332
or	756	/	.785398163	=	962.5690963
					481.2845482

Considering that the measurements given by professor Edwards for the four base lengths are as follows, then the corresponding numbers would be:

755.43 / .785398163 = 961.8433498 (or, 480.9216749)

755.77 / .785398163 = 962.2762512 (or, 481.1381256)

755.88 / .785398163 = 962.4163076 (or, 481.2081538)

756.08 / .785398163 = 961.3977159 (or, 480.678858)

One may wonder which one is correct; yet, they may all be correct, that is, correctly computed adjustments. Such computations evidently depend upon the measurements provided by professor Edwards. The baseline of the Great Pyramid, we should remember, has been measured distinctly over the centuries, anywhere from 693 feet to 765 feet in length. The 756c figure has been more widely accepted. Aside from the speculation dealing with specific numbers/measurements, we are analyzing the computational methods behind the possible numbers.

For the previously cited reasons, one may understand why many scholars have posed the idea that the Great Pyramid represents the concept of pi. However, as we have seen, the measurements not only allow for considering the possibility that the ancient kemi knew the concept of pi related to the contemporary 3.141592654 figure, but that they also represented the pi-like number of 2268 from Nineveh.

The 2268 pi-like Number

Let us observe how the 2.268 number/fractal performs regarding some basic computations.

For example, if we take the formula for the area of a sphere, 4², with these historically relevant numbers, the following obtains:

4², where pi is represented as 2.268

4 times 2.268 times 756²

4 x 2.268 x 571536	=	5184974.592
		2592487.296

whereby we see appear a number related to the Platonic cycle (25920c).

Another interesting example, involves the 260c itself in relation to the 2.268 pi-like number.

260 x 2268	=	589680
		1179360	-	1366560	=	187200 (maya long count-like)

Then consider the 2268 fractal/pi-like in relation to the Venus synodic count 585c:

585 x 2268	=	1326780 - 1366560	=	39780
				79560
585 x 2268	=	1326780 - 1385540	=	58760

and, further, 58760 - 39780 = 18980 (calendar round). In our view, the 756c and 576c like numbers are interesting.

Then, the 2268 fractal in relation to the constant number series (3, 6, 12, 24, 48, 96, etc.) is also significant for the 756c.

2268 x 48	=	108864 while,		108864 / 360	=	3024
3024 / 756	=	4

Even the maya companion number relates well to the 2268 pi-like number.

1366560 / 9 = 151840

151840 x 2.268	=	344373.12
		688746.24
		1377492.48	-	1366560	=	10932.48
						5466.24
						2733.12
						1366.56 fractal

At another level, the 2268c reveals a direct relationship to the 36c and the 63c:

36 x 63 = 2268

whereby the 2268 reflects these mirror-like numbers, remembering that the historically significant numbers 693, 756 and 819 (k'awil) all lie on a 63c.

In this manner, the Mesoamerican count 819c, relates to the 1296c of ancient kemi:

1872, 936,	468
	468 x 2268 =	1061424
		1061424 / 819 = 1296 fractal/kemi

In previous essays, we have discussed a Sothic cycle number (693) encoded into that ancient calendar (Cfr., Earth/matriX No. 73), as representing 1649.457812, based upon the chosen days of the calendar. If we consider the formula for the surface area for a cylinder as 2(rh, and employ the numbers of the Great Pyramid with the 2.268 pi-like number, the following obtains:

2 x 2.268 x 756 x 481 = 1649452.896 fractal.

This number reveals only a slight difference from the 1649457812 number cited in our work. Again, we may be speaking about adjustments in the computations.

And, if we consider the 2268c pi-like number in relation to the sacred seven often cited in the literature, then a most interesting computation obtains:

7 / 2.268	=	3.086419753
		6.172839506
		12.34567901	(reflecting a relation of 1 / 81 = .012345679)

This particular computation reminds us of the significance of the 25920 Platonic cycle number; whereby we could consider an adjustment of the 2.268 pi-like number to be changed to a 2.5920 number:

1 / 2.5920	=	.385802469
		.771604938 1.543209877 3.086419753 6.172839506 12.34567901

We know that the radian is 57.29577951, the square root of which is:

5729577951 = 75693.97566 (a 756c fractal!)

Now, what if we were to consider the maya long count fractal 576c (calbatun), as a radian-like number for computations: 57.6 instead of 57.29577951.

But, consider:

Taking into consideration the aspect of remainder mathematics, one could effect computations with the differences listed above from one system to another. (Remembering that the square root of 1366560 is 1168.999572, and could thereby be substituted for the adjustments.)

In other words, the 57.6c could have been employed as a radian-like number for the 360c system. Consider the following computational adjustments:

180	/		=	57.29577951	(114.591559)
180	/	3.125	=	57.6	(115.2)
180	/	3.2	=	56.25	(112.5) whereby	112.5 225 450 900 1800 3600 7200 14400 28800 57600

Yet, when we multiply instead of divide these same numbers the following obtains:

They become switched in order, given the fact that 5625 lies on the 576 series as shown above. Hence, if the numbers 3.125 (constant number series 3125, 625, 12500, 25000, 50000, 100000, 200000, 400000, 800000, 1600000, 3200000...) or, 3.2 (constant number series 2, 4, 8, 16, 32...) are employed as pi-like numbers, the results are the same.

If one employs the 3024 (756 x 4) kemi pi-like number, then the computations are relational.

180 x 3.024	=	544.32
		54432 -57600	=	3168
		54432 - 56250	=	1818 (3636)

Consider, then, 3636 - 3168 = 468 (936, 1872). The computational adjustments are unending:

54432 x 5625

=

306180000 612360000 1224720000 2449440000 4898880000 9797760000 19595520000

(Nineveh fractal)

Observations

The numbers/fractals of the ancient reckoning systems appear to reflect the relationships observed on the numbers corresponding to the ratios of the trigonometric table of right triangles. This correspondence also lends itself to the logic of the different circles that may have been conceived and devised as of the different day-counts: 260c, 360c, 3744c, 378c, etc.

In this essay, we have seen that the numbers/fractals corresponding to distinct day-counts of the ancient reckoning systems may also reflect the numbers/fractals regarding the combinations of complementary angles within right triangles.

Furthermore, although the relationship of the diameter of the circle to its circumference may produce a specific number/ratio (3.141592654), as we have come to know it, it is possible to conceive of relational numbers/fractals in distinct ways as the number of segments or degrees of a circle vary. Therefore, although we have come to associate with the 360c circle, the ancients may have employed the 2.268 number/fractal as a pi-like number related to a 260c circle, given that their relationship is proportionately similar.

Furthermore, as the numbers relating to the divisions/segments/degrees within a circle may vary, meaning we can divide a circle into as many degrees as is infinitely possible, then different numerical alternatives exist for achieving the computations. Within the ancient reckoning system, then, there may appear to exist historically chosen numbers, selected on individual merit, while other number series appear to reflect constants. When, in fact, any number, depending upon the computational value and position within the reckoning, may represent either a selected variable or a constant. In this sense, even the number representing ( could have been conceived to be a selected variable or a constant. And, therefore, even the numbers representing pi could have been interchanged according to the needs of the computations.

In our contemporary system of computations, where the concept of is fixed at a specific value (i.e., 3.141592654), we may have difficulty in understanding that the concept of pi in the ancient reckoning system could have been represented by almost any number, depending upon the computational problem under analysis. For us there is only one possible concept of pi; for the ancients, pi could have been any number.