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Secrets of old Egypt

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julia
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« on: June 19, 2007, 02:56:23 pm »

Deep Secrets

The Great Pyramid, The Golden Ratio and The Royal Cubit



 

This site provides a new, and perhaps for some a controversial, explanation for the rationale behind the exterior and interior design parameters of the Great Pyramid of Giza. Learn here: 1) the historical significance of the "golden ratio" and of the equal-sided pentagon (and pentagram); 2) a new theory for the derivation of the ancient Egyptian Royal Cubit; 3) a diagrammatic method by which the square root of any number can be derived; 4) how to diagrammatically derive a trigonometric table; 5) a relatively easy to follow presentation of Euclid's derivation of the 36 angle; 6) and now a newly added section detailing the derivation diagrams for the interior design parameters of the Red, Bent, Khafre, and Great Pyramids.

Introduction

As one delves into the exterior design details of the Great Pyramid, two striking numerical correlations emerge from the data, and these compel the serious student to either explain them as being nothing more than odd coincidence or to deal with their obvious implications. These findings are: First, that the pyramid's cross-section, as defined by its slant height (= 611.5 feet) divided by one half the length of a side (= 377.9 feet), embodies a numerical finding equal to the enigmatic ratio popularly known as the "golden ratio" (for a schematic of this situation refer to triangle ABC in Diagram 18 in the Pyramid section); and Second, that twice the perimeter of the Great Pyramid (that is, 377.9 x 16 = 6,046 feet) is a length that is precisely equal to the length of a minute of latitude as measured at the Earth's equator.

There appears to be no dispute regarding the reality of the above two pyramid lengths, as they have been determined by modern day survey. Rather, disagreement arises over whether or not the ancient Egyptians had a means by which they could produce the constant known today as the golden ratio, and whether or not they had developed the capability to measure the size of the Earth (and present their findings in degree measurement, no less). In other words, did they intentionally build the above knowledge into this pyramid, or do these numbers merely appear as a result of chance?

For the most part, archaeologists and historians assert that the existence of these relationships must be due to coincidence. The reasoning is that there is no incontrovertible proof to support intentionalism behind either numerical correlation. Still, there are some who consider this dismissal to be too abrupt, and who feel that the known surviving evidence is both too suggestive and too limited to preclude the possibility that the ancient Egyptians had gained certain awarenesses which were otherwise apparently ahead of their time. Although there has not been found a "smoking gun" to provide a universally accepted level of proof to support intentionality for the correlations mentioned, I believe it fair to say that there is likewise at present no conclusive evidence that the ancient Egyptians could not have found a way (such as is outlined in this essay) to achieve these results.

This issue of intentionalism is a question that continues to hold fascination, in no small part due to the specificity of the two numerical correlations mentioned here. In the process of becoming familiar with the design parameters of this pyramid, therefore, one can be excused for probing about for a cohesive and rational explanation, other than chance, that would account for these findings. As I began my own explorations into this situation it became evident that much could be gleaned by approaching the material from a position of devil's advocate - that is, starting from the assumption that the Egyptians of Old Kingdom Egypt had, in fact, intentionally built into the Great Pyramid the knowledge in question. With this as a point of beginning, entryways into the problem can then be established by focusing on two fronts: 1) How would it have been possible for the ancient Egyptians to have been able to arrive at these understandings using only the tools and advancements they are known, or can reasonably be inferred, to have had? and 2) With the architects having achieved these awarenesses, why would they then have designed the pyramid to incorporate this information? 

What follows in these pages is first an outline, and then a development, of the conclusions I have reached, with footnoted references to material providing greater depth and documentation. What I have derived, although admittedly not without some intracacy, is, I believe, logically consistent within the context of Old Kingdom capabilities, and within the realm of credible possibility. I have tried to present the ensuing discussions and diagrams as clearly and as directly as my abilities would allow, and I trust that those with an interest in the topic will persevere if and where necessary. The theories herein presented are speculative, and should be seen as such by the reader. My only claim is that these pages present a provocative and contextually consistent explanation for the stated Great Pyramid design realities, and perhaps for other related Old Kingdom design choices as well.

 

The core of this presentation proposes that the most compelling explanation for the intentional appearance of both the golden ratio, and of the length of a minute of latitude, in the design of the Great Pyramid is that Old Kingdom researchers had gained the ability to derive a fairly complete knowledge of trigonometric relationships some 2,000 years prior to the currently recognized date for said development by the ancient Greeks. Such an Old Kingdom accomplishment would likely have been achieved via the empirical exploration of geometric diagrammatic relationships in general, and the insights made available by the inscribed pentagon, in particular. Furthermore, with an aptitude for careful work and a trigonometric capability, shadow lengths could then have been used to determine the size of the Earth (as indeed they were by the later Greeks). Having attained the knowledge I describe, it is credible to reason that it was then held restrictively secret at the highest priestly levels in ancient times for reasons concerned with the centralization of power.

Since the relationship termed the 'golden ratio' is central to the diagrammatic derivation of the inscribed pentagon (and thereby to a continuum of trigonometric relationships), it is perhaps for this reason that it would have figured prominently in a building that was being designed to incorporate a primary unit of Earth measurement.

As is quite clear, the above summarization is meant only as a thumbnail sketch. The various aspects of the problem, in depth explanations, and much relevant documentation, form the remainder of this report. Tying into the integral context of the theory is also a discussion of a possible rationale behind the ancient Egyptians' choice of a length equal to 20.63 inches (= .524m) to be the length of their standard unit of measurement, the Royal Cubit.

 The Circle & The Square
"But now the sight of day and night, and the months and the revolutions of the years have created number and have given us a conception of time, and the power of inquiring about the nature of the universe. And from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man."
Plato. Timaeus, 47 a .

The Circle

 Conventional wisdom holds that geometry, as a true discipline, was the creation of the Greek mind. A fair amount of scholarship has been devoted to showing in what ways the surviving written record upholds this belief. Oddly enough, this is not what the Greeks themselves had to say about things. Greek writers such as Isocrates, Plato and Diodorus all credit Egypt as the source of Greek geometrical studies.1 Sir Thomas Heath, in his work A History of Greek Mathematics, adds that "the Egyptian claim to the discoveries (of geometry) was never disputed by the Greeks".2

The existence of the pyramids themselves, with all of their complexities and precision, also argue in favor of there having been some form of sophisticated geometrical capability on the part of their architects. I will therefore be accepting as a given that by the time of the building of the Great Pyramid, the art of diagrammatic geometry had become fairly well developed in ancient Egypt.

 

The circle is a perfect, though mysterious, shape. It appears daily in the heavens as the sun; and monthly, as the moon. It can appear as an artifact of nature on the surface of the Earth as can be seen, for instance, in a perimeter made in grass by a tethered grazing animal. The ancient Egyptians had apparently gained an awareness of the fact that there is always a constant relationship between the circumference (perimeter) of a circle and the diameter of that same circle.3 This is the relationship we now call Pi. It is a value equal to 3.14159...., the dots signifying that the fractional part of this number has no known finite end.

There is a surviving document which shows that the Egyptians were - knowingly or not - placing into service a value for Pi that was equal to the square of (8/9 x 2), a value which is almost exactly 3.16 in decimal notation. (See footnote 3 for a further explanation). The scribe writing this document states that he is making a copy of an older work, a work that is judged to have been written in about 1850 B.C. In his treatise on this papyrus, Professor T.E. Peet states, "Surely the complicated fabric of Egyptian mathematics can hardly have been built up in a century or even two, and it is tempting to suppose that the main discoveries of mathematics should be dated to the Old Kingdom.....the Golden Age of Egyptian knowledge".4 The Great Pyramid was constructed sometime near the date of 2550 B.C., at the height of the Egyptian Golden Age that Peet refers to.

The implications then appear to be that the Egyptians: 1) had developed an accurate means of approaching the Pi relationship; 2) had chosen to employ this relationship in a way that was easy to remember; and 3) had chosen to employ this relationship in a way that could also be easily mathematically manipulated by the everyday scribe. The level of pragmatism apparent in these findings strongly implies that a surprisingly sophisticated understanding of the relaionships inherent in the circle had already been mastered by this early time.

Returning to the above mentioned tethered grazing animal, there is no diffculty in observing that a cord which is allowed to pivot on a post can be employed as a means of marking out a circle. If one makes the circle's diameter a royal cubit in length, then because the royal cubit has 28 "finger" subdivisions, the circumference of this circle can be found to apparently measure exactly 88 of these "fingers" in length.5 This finding immediately leads to the understanding that the relationship of a circle's circumference to its diameter is 88 units to 28 units, which when divided by 4 becomes 22/7ths - the familiar ratio often used to approximate Pi. There is no known surviving written proof that the ancient Egyptians had reached this specific or general understanding, but given the nature of their royal cubit measurement system, it would be surprising if they had not.

In any event, the accompanying insight that the circumference to diameter relationship remains constant, regardless of how big or small the circle, is for the moment the more crucial point. It is not a result that one might initially have expected prior to its proof by a chance occurrence, or by the exploration of a curious mind. This is a constant numerical relationship which occurs freely in the natural world, and its discovery would surely have aroused further curiosity. Pondering over the reality of the Pi phenomenon, it would only have been a matter of time before some insightful priest or scribe began to wonder whether there might be any other such findable constants. From this moment forward, the game was afoot.

 

The Square

A next logical shape to consider would have been the square. In fact, the evolution of the square from a circle is almost inevitable. The circle is halved, and then halved again, and the points at which the diameters intersect the circle are joined to form an interior square. To perform this task with accuracy, a tool functioning as a
drafting compass, or dividers, would be needed.
A straightedge, that need not be calibrated, is also
necessary.6 Using such equipment, one can empirically
assure that all sides of the square are of equal length.

One can also empirically show that each of the four interior triangles are identical, and that each central angle is 1/4th of a full rotation. (Such an angle is referred to as a "right angle". Since we use a system in which a full circle is considered to contain 360, a right angle is then an angle having 90).

 
 


The issue, then, is to find whether there is a numerically constant relationship between a side of a square and the diagonal of that square.

The following diagram shows how this relationship may have first been discovered.7

 
The diagram begins with the drawing of the smallest square to the left. Each side has a length of one unit, and so the area contained within that square is said to be 'one square unit'. (Area within a square - or any rectangle - is determined by multiplying the length of a side by the length of an adjacent side. In a square, the lengths of these two sides are equal, and so the formula becomes: Area = S x S = S2. Since S equals 1 in this first square, the area equals 1 x 1 = 1 square unit.) The next step is to draw in the diagonal to this first square, and when done, it is seen that the diagonal creates two equivalent right angled triangles within this first square, each of which contain an area of one half of one square unit.

What happens now if a square is drawn using this diagonal as a side? As the diagram shows, this next square (in green) will contain exactly four of these same triangles. As a result, it is visually apparent that the area of this second square must contain two square units. From the understanding that Area equals the length of the square's side multiplied by itself, for this second square we get : Area = S2 = 2 sq. units. As a result, for this second square "S" must then equal that number which when multiplied by itself yields the number 2. This number is then the "square root" of 2, and is written as 2. It can be closely approximated in decimal notation by the value of 1.4142.

The diagram above continues to show that the length of the diagonal for any square can be found by simply multiplying the length of that square's side by the 2. Notice that each side of the green square has a length of 2, and its diagonal equals 2. Each side of the orange square has a length of 2, and the diagonal of this square is 2 x 2, and so on. We have here, then, another freely occurring numerical constant in nature. The first, Pi, was found in the context of a circle, and now we find 2 in the context of a square.

Two very important questions now arise and must be addressed: 1) Did the ancient Egyptians have an awareness of the 2 relationship between a square's side and its diagonal? And 2) Did they have the mathematical capability to determine square roots? (For instance, did they have a way to find that the 2 has a value equivalent to 1.4142?) The answer to both of these questions is "yes".

That the Egyptians had an essential awareness of the 2 relationship is evidenced by the fact that they had a unit of measurement, the "double-remen", which they defined as being the length of the diagonal of a square whose sides were a royal cubit in length. (In other words, the double-remen equaled the length of a royal cubit multiplied by the 2).8 There is also ample evidence that, not withstanding their reliance upon a computational system much removed from our own, they were indeed quite capable of handling the sometimes complex chore of finding square roots.9 (I include in the Appendix a diagrammatic square root method which fits in remarkably well with the ancient Egyptian unit fraction system, and which could quite conceivably have been devised at the time in question. In addition, utilizing this square root diagrammatic method, I present an interesting coincidence of number between the ratio of the perimeter of a square to its diagonal and the Egyptian representational choice for the relationship we call Pi. See the Square Root of 2 derivation in the Appendix.)

 

Having become aware of the two constant relationships of Pi and the 2, would not the thought have occurred that perhaps there were still other similar constants to be found? Would not those involved with line drawing and building design have next wondered whether there might be some way to discern a relationship between the sides and diagonal in non-square rectangles? I believe it entirely possible that not only would this question have been pursued, but that it would have been pursued in a manner similar to the method used in the previous diagram.

 
Instead of starting with a square, Diagram 3 begins with a rectangle (AB). The diagonal (here denoted "C" ) is drawn within this rectangle, and a square is drawn using this diagonal as the length of its side.

If one could find the area contained by the square on C in terms of the values A and B, the length of the diagonal C could be determined by taking the square root of this area amount.

Diagram 3 shows how the right triangle ABC may distributed to best advantage inside the square on C. It is clear that the area of this square can be represented as being equal to the area of four times the area of triangle ABC plus the area of the small square on (B-A). Since each triangle ABC contains half the area of the rectangle AB, and since the area of the rectangle AB equals A multiplied by B (i.e., AxB), the area of four times the area of triangle ABC is the same as the area of twice (AxB), or 2AB. It remains now to find the area of the square on (B-A).10

To do this, it is likely that the empirical advantages inherent in the drawing of a gridded square would have been utilized.11


In Diagram 4, line B and line A are shown at the top of a square whose side is the length B + A. Lightly drawn in are the squares on each of these line segments.

In the following diagram, Diagram 5, the square on the line segment B - A has been added.

This diagram may perhaps appear at first glance to be a little more forbidding than it actually is. An AB rectangle has been placed horizontally inside the green-sided square on B, and the magenta-sided square on (B-A) has also been delineated inside the square on B. Left unaccounted for inside the square on B, then, is the small rectangle containing six small-square units to the right of the square on (B-A). Investigation shows that this amount, when added to the area of the square on A (in the diagram, the area of the square on A has been sketched in above these six units as a dotted orange-sided square), proves to be exactly enough to constitute a second AB rectangle.

 
As a result, it can be empirically seen that if one adds the area of the square on A to the area of the square on B and then subtracts the area of two AB rectangles, what remains is the equivalent of the area of the square on (B-A).

In formula form, this reads: (B - A)2 = A2 + B2 - 2AB.

As we saw earlier, the square on a rectangle's diagonal equaled two times the area of AB plus the area of the square on (B - A). Expressed as a formula, this previous statement reads:

C2 = (B - A)2 + 2AB
If we now substitute in the just derived value for (B - A)2, we get :

C2 = A2 + B2 - 2AB + 2AB
The 2AB's cancel out, and we are left with a formula that most will recognize:

C2 = A2 + B2
This is, of course, what is today known as the Pythagorean Theorem. The truism that is represented in this finding is simply that there is a definable relationship between the lengths of the sides in every right angled triangle (i.e., in a triangle which is, in effect, one half of a rectangle, be the rectangle a square or non-square). If the lengths of two of a right triangle's sides are known, then there is one and only one length that the third side can have, and that length is determinable via this equation.

The discovery of this theorem would not have immediately satisfied the goal of finding other naturally occurring constants such as are found in Pi and the 2, but as we will see, it would have provided the means by which this pursuit could have been vigorously continued. First things first, however. If, supposedly, Pythagoras (in 500 B.C., approximately) discovered the formula that bears his name, then how is it possible the ancient Egyptians are not generally credited with having known of it some 2,000 years earlier?

To start with, it must be noted that there simply is not a large body of ancient Egyptian mathematical material that has survived down to our era. Additionally, the few papyri that have survived are largely student training exercises, and they present us with an incomplete picture of the full capabilities of the master scribes. Some scholars do feel, however, that reasonable inferences can be drawn from existing data to support the possibility of Egyptian knowledge of the workings of the Pythagorean Theorem.12

It is also of interest to note that a Babylonian clay tablet, dating from roughly the same time period as the afore-referenced Rhind Papyrus, reveals that an extremely sophisticated understanding of the Pythagorean Theorem and associated number theory existed during the Old Babylonian period.13 This tablet leaves no doubt that the Pythagorean Theorem was well understood by this contemporaneous civilization at least 1,200 years before the birth of Pythagoras. I will later present evidence of substantial contact between this culture and that of ancient Egypt prior to the Rhind Papyrus date.

Lastly, I want to stress the inherent simplicity and power of an empirical diagrammatic approach. Movement from one understanding to another proceeds organically, with each new result open to easy corroboration either by direct measurement or by a comparison of shapes. Such a diagrammatic technique is well within the context of ancient Egyptian capabilities and understandings. The Egyptians may not have used exactly the same diagrams as derived in this essay, and they may not have written their results in an identical symbolic formula format, but the same basic understandings as we have reached thus far, and will reach in the following sections, would have been well within their grasp.

Following on the heels of an analysis of the proportions inherent in a square, the search would certainly have continued to seek out what other shapes can be created within the context of a circle, and to see what (if anything) could be learned from them.


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