The geometry of the Giza's plateau

@Pharaoh

Published in 

Egypt

 · 29 Nov 2023

The survey of the Giza site

After the work of Colonel Howard-Vyse and Perring in 1837, it is generally agreed that the first serious archaeological work on the Egyptian pyramids was "The Pyramids and Temples of Giza" by Sir W. M. Flinders Petrie, published in London, 1883. The archaeologist, equipped with great determination and the most advanced equipment of the time, conducted the first detailed topographical survey between 1880 and 1882, obtaining data that was confirmed by the surveys and studies conducted subsequently, and therefore to a large extent still accepted today.

The following tables, taken from Petrie's work, show the most relevant data concerning the positioning and dimensions of the pyramids of Giza. The measurements are expressed both in the unit originally adopted by the scholar - English inches - and in meters. The fact that Petrie's findings have not yet been substantially questioned, more than a century after his masterly work, allows us to tackle the analysis of the "geometry of Giza" with the reasonable certainty of resting the foundations on a sufficiently solid ground.

AXIAL DISTANCES BETWEEN THE PYRAMIDS OF GIZA

DIMENSIONS OF THE PYRAMIDS OF GIZA

Is there a Giza project?

According to traditional theory, the three pyramids of Giza were built in succession by the pharaohs Khufu, Khafre and Menkaure over the course of about a century; it is also a common belief that Khafre and Menkaure were unable to compete with their predecessor, in the grandeur of the work, due to a lack of sufficient resources: hence the need to settle for smaller, simpler and even less accurate structures. This would seem particularly evident in the third pyramid.

This theory is in clear conflict with the hypothesis of a general plan for Giza: the very idea that each pharaoh would have aspired to surpass his predecessor, if only he had had sufficient resources at his disposal, would have been at odds with the the objective of pursuing a general scheme which, once completed, would go beyond the individuality of individual sovereigns; and it remains to be demonstrated, furthermore, that Khufu's successors lacked sufficient resources. According to traditional theories Khafre, during his reign, would have built not only a pyramid almost as large as that of his predecessor, but also the Sphinx. All things considered, Khafre could very well have surpassed Khufu, but he didn't: why? Perhaps because he had to respect a general plan, the same plan that Menkaure would later respect in the construction of his "small" pyramid.

The possible existence of a master plan for Giza and of a socio-political structure capable of pursuing an objective of this magnitude over the span of a century, completing it in such a spectacular manner, is something that leaves us incredulous to say the least and that constitutes an enigma in itself. Yet, as soon as we begin to analyze the configuration of the Giza complex in more depth, the idea of a general project emerges with ever greater force; but, one wonders, were it really the pharaohs of the 4^th dynasty who gave shape to this incredible project? It must be said that it is at least legitimate to ask the question, even if here, however, I will carefully avoid answering it: the terrain is too insidious and I would end up exposing myself to easy criticism, with the risk of compromising the credibility of the research and the results I intend present. Moreover, venturing now into the endless torment of "how, when, by whom" the pyramids of Giza were built is of no importance: as we will be able to see, it will become clear that the pyramids speak for themselves and in such a precise - mathematical language and geometric - that cannot be misunderstood.

My analyses, therefore, will let the pyramids speak, limiting themselves to highlighting a series of absolutely objective correlations, verifiable by anyone, where too much space is not given to individual interpretation. Only at the end, if ever, will each of us be able to venture into speculation regarding "how, when, by whom" this astonishing architectural project was created.

The diagonal alignment of the three pyramids

At a first superficial glance, the three pyramids of Giza appear to be arranged along a diagonal alignment starting from the Great Pyramid (fig. 1); the second aspect that appears evident is the precise orientation of all three pyramids with respect to the cardinal points. A scheme of this kind does not necessarily imply the existence of a precise general design: it is sufficient to advance the reasonable hypothesis that each of the pharaohs who succeeded Khufu had considered his predecessor's pyramid and the cardinal points as fundamental references in the construction of his own.

There would therefore be nothing shocking for the conceptions of traditional Egyptology.

Fig. 1 - The diagonal axis of the Great Pyramid

However, there is something unconvincing about this hypothesis. The alignment of the three pyramids with respect to the diagonal axis is not very precise, much less precise than one might expect even with only rudimentary control and detection instruments, and not even the morphology of the Giza plateau can be called into question to justify this inaccuracy, since there are no rock formations or hills in the plain that could have forced the builders to deviate from the desired alignment.

From now on, for convenience, we will refer to the pyramids of Khufu, Khafre and Menkaure as the "first", "second" and "third" pyramids respectively; we will also use the expressions "P1", "P2" and "P3" to refer to the respective centers of the bases of the pyramids.

Compared to the simple diagonal alignment scheme, the discordance is quite small as regards the position of the Khafre pyramid (whose center P2 is 13.75 m off-axis), but decidedly more sensitive in the case of the Menkaure pyramid (the whose center P3 is off-axis by 116.49 m). If we consider that the distances P1-P2 and P1-P3 are 486.88 m and 936.16 m respectively, it follows that for Khafre's pyramid the alignment error would be equal to 2.8%, while for the pyramid of Menkaure would be equal to 12.5%.

Is it possible that these are simply errors due to the lack of sufficiently precise topographic instruments? If you reflect for a moment, it is difficult to believe that this could be the reason: just consider the extraordinary accuracy demonstrated by the builders of the Great Pyramid, whose base sides are exactly identical (except for an error of less than 0.1%) and exactly aligned with the cardinal points (unless an error of less than 3' of arc). In short, the ability to achieve such high standards of precision in the construction of the first pyramid suggests the idea that the positioning of the second and third pyramid could be affected by such large errors completely unacceptable, if the intention had really been to align the three pyramids along a diagonal axis.

It seems inevitable to draw the conclusion that if the three pyramids of Giza are not perfectly - or almost - aligned, then they should not have been so even in the intentions of the ancient builders who, if they had wanted, would certainly have been able to do better.

So is there a project for Giza? We don't know it yet, but, if it exists, it is certainly something very different from a simple diagonal alignment scheme: for example, the Giza complex could constitute a sort of map capable of representing something else, and this is precisely the central point of the theory of stellar correlation (correlation between the pyramids of Giza and the Orion's Belt), a theory supported by Bauval and Hancock. This is a fascinating theory, which I myself have developed and explored elsewhere (see the article The First Time of Sirius); but here I intend to take a step back and address the Giza site "from the inside", that is, within the limits of an exclusively geometric analysis. As we will see, this will be sufficient to demonstrate the absolutely intentional nature of the planimetric scheme created at Giza.

Hidden geometry

Any pretense of considering the configuration of the Giza site as the casual result of the successive interventions of three pharaohs, without the guidance of a general overall project, is immediately swept away by an irrefutable fact. In a zenithal planimetric view (fig. 2) the three pyramids can be contained within a rectangle that goes from the north-east corner of the first pyramid (that of Khufu) to the south-west corner of the third (that of Menkaure); based on Petrie's analysis, the lengths of the sides of this rectangle are 742.37 m (east/west) and 907.12 m (north/south), corresponding respectively to 1,416.6 and 1,731.0 Egyptian royal cubits (we assume for the real cubit the value of 0.524 m established by the measurements carried out by Petrie during his study campaign in Egypt).

Fig. 2 - The rectangles of Giza

Now, it so happens that 1,416.6 cubits equals \sqrt{2} \cdot 1000 cubits (the error is +2.4 cubits equal to approximately +0.2%) and 1731.0 cubits equals \sqrt {3} \cdot 1,000 cubits (the error is -1.1 cubits equal to approximately -0.1%).

These are such small deviations as to exclude simple randomness, and we can therefore affirm that, in all probability, the ancient builders intended precisely inscribe the three pyramids within a rectangle whose sides measured \sqrt {2} \cdot 1,000 cubits and \sqrt {3} \cdot 1,000 cubits; this also means that the length of the diagonal of this rectangle (equal to 1,172.17 m, i.e. 2,236.8 cubits) can be expressed with good approximation as \sqrt {5} \cdot 1,000 cubits (the error is +0, 7 cubits, much less than +0.1%).

This circumstance, highlighted by John Legon more than twenty years ago now, is already very significant in itself, since it demonstrates the knowledge of irrational numbers on the part of the builders, but that's not all. First of all, the value \sqrt {5} is closely linked to another fundamental number of geometry, the FI number (the "golden" number), which is exactly \frac {( \sqrt{5}+1)}{2} \approx 1.618 . Secondly (a fact that not even John Legon seems to have noticed) it so happens that the sum of the numbers \sqrt{2} and \sqrt{3} constitutes an excellent approximation of PI ( \pi ) (the "pi" number): in fact \sqrt{2} + \sqrt{3} is worth 3.146, a value very close to 3.142 (the real value of PI); from this it follows, therefore, that the perimeter of the rectangle (equal to 6,295.2 cubits) is equivalent, with good approximation, to the circumference of a circle with a radius of 1,000 cubits (the error is +12.0 cubits equal to approximately +0 ,2%).

Ultimately we have that the three pyramids of Giza are inscribed within a rectangle whose dimensions refer to the irrational numbers \sqrt{2} , \sqrt{3} , \sqrt{5} , the golden number FI and the transcendent number PI: a condensation of meanings geometric shapes that cannot be neglected.

Yet we are only at the beginning: Giza truly seems to be an inexhaustible mine of mathematical-geometric relationships. Let us now take into consideration the rectangle identified by two opposite corners coinciding respectively with the centers P1 and P3 of the bases of the pyramids of Khufu and Menkaure (fig. 2): we will call this the "minor rectangle of Giza", to distinguish it from the "rectangle greater than Giza" already examined.

The sides of the smaller rectangle are 574.45 m long (east/west) and 739.19 m (north/south) corresponding respectively to 1,096.2 cubits and 1,410.5 cubits, while the diagonal is 936.16 m long corresponding at 1,786.4 cubits.

The first thing that catches the eye is the length of the north/south side, very close to the length of the east/west side of the larger rectangle and even closer to the value \sqrt{2} \cdot 1000 cubits. In practice, it seems that the number \sqrt{2} \cdot 1000 was taken as a reference both for the length of the east/west side of the larger rectangle and for the length of the north/south side of the smaller rectangle and the circumstance is completely evident if we calculate the value that represents the average between the two lengths. This value is (1,416.6+1,410.5)/2 = 1,413.6 cubits and differs by just 0.6 cubits from \sqrt{2} \cdot 1000 cubits (the error is much less than -0.1%). The following table shows the dimensions of the Giza rectangles with the corresponding reference measurements, as well as the related approximation errors.

DIMENSIONS OF THE GIZA RECTANGLES (in royal cubits)

For many traditional Egyptologists all this would be a coincidence; but certainly Chance, for these gentlemen, must have worked a lot at Giza...

Let's also bring the Sphinx into play and see what happens: well, if we point the compass at the center of the diagonal of the larger rectangle and draw a circle with a radius equal to half of the diagonal (i.e. we trace the circle that circumscribes the rectangle) we see that this circle completely crosses the Sphinx (fig. 3). Coincidence or design?

Fig. 3 - The circle that circumscribes the pyramids of Giza and crosses the Sphinx

Hidden arithmetic

This picture should already be surprising to say the least for those who still do not want to believe in the existence of a general project in Giza, but further surprises are reserved for us if the same are converted into Egyptian palms (a royal cubit is divided into seven Egyptian palms). Measurements reported in the previous paragraph are depicted in the following table.

DIMENSIONS OF THE GIZA RECTANGLES (in palms)

From this table it is clear that the two diagonals are multiples of the length PI \cdot 1,000 palms, while the sides are multiples of the length 1,100 palms; but this length, reconverted into real cubits, provides the value 157.1 cubits which once again refers us to PI through the numerical ratio 22/7 which is an excellent approximation, indeed, the best that can be obtained with operators no more than two digits.

Let's see how: 1100 palms, reconverted to real cubits, is equivalent to 1,100/7 = (22/7)*50 ~ 157.1 ~ PI*50 cubits (the error is less than +0.1%). The knowledge of the 22/7 ratio, and therefore of the almost exact value of PI, is traditionally attributed to the Greek mathematician and philosopher Pythagoras (who lived in the 5^th century BC), but evidently the builders of the Giza site must have been aware and made extensive use of it: we find it in the case just described with the existence of a module of 1,100 palms corresponding to PI*50 cubits; we find it in the fact that the lengths of the sides of the two larger and smaller rectangles can be expressed as 11M, 9M, 9M and 7M (where M = 1,100 palms ~ PI*50 cubits) and consequently the ratio between the largest and the most small of these sides is approximately equivalent to 11/7 ~ PI/2; we still find the aforementioned relationship in the dimensions of the Great Pyramid themselves, as has long been known, and precisely in the relationship between the semiperimeter of the base and the height.

It is equally known that the Great Pyramid also encodes the number FI in its dimensions through the ratio between the apothem and half of the base side. Coincidence or conscious intention?

More hidden arithmetic...

The analysis of the Giza site from a mathematical-geometric perspective risks being an adventure comparable to the exploration of a bottomless well: at each level of reading new relationships emerge which in a game of cross-references, as in a labyrinth of mirrors, create an astonishing symphony of numbers where PI and FI constitute a sort of obsessive leitmotif, as if they were signals intentionally placed to capture our attention...

Fig. 4 - The pyramid of Khafre in relation to the diagonal of the greater rectangle of Giza

We must still be surprised if, tracing the diagonal of the larger rectangle and projecting perpendicularly onto it the position of the center P2 of the second pyramid (fig. 4), we discover that the diagonal itself is divided into two segments with a length of 1,000.8 cubits and 1,236.0 cubits? Now, with minimal approximation the first of the two segments can be expressed as 1,000 cubits, while the second as 2*618 cubits; but 618 cubits is nothing other than the golden ratio of 1,000 cubits, or 1,000 cubits divided by the golden number FI ...

And again, we must be surprised (fig. 5) if half the base side of the Khufu pyramid (equal to 115.17 m) is almost exactly one fifth of the east/west side of the smaller rectangle (the error is +0.2%), while half the base side of Menkaure's pyramid (equal to 52.75 m) is almost exactly one fourteenth of the north/south side of the same smaller rectangle (the error is -0.1%)?

Fig. 5 - The dimensions of the first and third pyramids in relation to the sides of the smaller rectangle of Giza

We could continue, but by now what we wanted to demonstrate is well demonstrated: the configuration of the Giza site, far from being the result of subsequent interventions unrelated to each other, is generated by an extremely sophisticated project that denotes profound knowledge of mathematics and geometry.

However, I would like to highlight a further geometric relationship, which on the one hand confirms the existence of the module M = 1,100 palmi, and on the other highlights the particular importance attributed to the diagonal axis of Giza, even if not in the sense hypothesized by some, namely that this axis could represent the direction of alignment of the three pyramids (which has proven, as we have seen, simplistic and incorrect).

If we trace the axis inclined at 45° passing through the center P1 of the Great Pyramid, then we draw the square with a vertex on the center P3 of the third pyramid, the opposite vertex on the same diagonal axis and sides parallel to the cardinal directions, we observe that the length of the side of this square is 82.38 m, equivalent to 157.2 cubits or 1,100.4 palms: that is, practically identical to the module M discussed in the previous paragraph (fig. 6). This fact on the one hand is a clear confirmation of the intentionality of the M module, on the other hand it seems to particularly highlight the diagonal axis from which the third pyramid stands out in such a non-random manner; and in fact this axis plays a decisive role in the stellar correlation scheme developed and explored in depth in the article The First Time of Sirius.

Fig. 6 - The position of the Menkaure pyramid related, through the M module, to the diagonal axis of the Great Pyramid

The circle of Giza

It should not have escaped the attention of the attentive reader that the set of geometric relations just described leaves Khafre's pyramid decidedly aside. But it is certainly not possible that the Giza project - assuming it exists - ignores a pyramid that rivals the one built by Khufu in size and which, moreover, is connected, via one of the raised ceremonial streets, with the temple downstream near the Sphinx, another of the key elements of the Giza monumental complex.

One way out of the impasse may be to verify whether geometric relationships of a non-linear type exist, unlike those just described, which are precisely of a linear type. We can, thus, start from the very natural observation that three points on the plane, when they are not perfectly aligned, uniquely determine an arc and the circle to which the arc belongs.

Therefore, the centers P1, P2 and P3 uniquely identify an arc and a circle (fig. 7), which from now on we will call respectively "arc of Giza" and "circle of Giza". The diameter of the circle is 4,711.43 m, its length is 942.43 m, while the central angle P1-C-P3 (which from now on we will call "Giza angle") is 22.9218°.

Do these values have meaning or are they simply random?

Fig. 7 - The arch of Giza

It seems sensible, to begin with, to examine what the diameter of Giza is when expressed in Egyptian units of measurement. The Egyptians had a rather complex system of units of measurement: usually the royal cubit is identified as the main unit within the system, and Petrie's studies assign to it, as we have said, a value equal to 0.524 m, generally accepted. Other units of measurement arise from subdivisions of the real cubit: a cubit contains seven palms, while each palm contains four fingers; again, other units of measurement are multiples of the finger, while others are multiples of the cubit; finally, a further unit is represented by the side of a square whose diagonal is equivalent to a cubit.

In the first of the following tables we find a scheme that summarizes the system of units of measurement of ancient Egypt, with the equivalences compared to today's metric system, while in the second table the parameters of the Giza circle are shown (diameter, arc and circumference) expressed in each of these units, including the main modern units of measurement, namely the meter and the Anglo-Saxon inch (which is equivalent to 0.0254 m).

ANCIENT EGYPT UNIT OF MEASUREMENT

THE PARAMETERS OF THE GIZA CIRCLE EXPRESSED IN DIFFERENT UNITS OF MEASUREMENT

A careful analysis of the data reported in the second table shows that, indeed, there are several significant correlations: in the following table, these correlations are shown explicitly and clearly.

CORRELATIONS BETWEEN THE PARAMETERS OF THE GIZA CIRCLE AND SPECIAL NUMBERS

Well, the astonishing conclusion is that the parameters of the Giza circle show clear correlations with round numbers and with the numbers PI and FI; the reduced margins of approximation (not exceeding 0.2% in absolute value) and the relevance of these correlations completely exclude pure and simple randomness, and rather lead one to think of a precise design intention. Since the different units of measurement of the Egyptian system are linked to each other by very specific relationships, it is evident that the ancient builders were able to establish only one of the correlations listed above, while the others derive from necessity. In my opinion, the choice of the ancient builders was to trace the circle of Giza with the radius of length equal to PI multiplied by 10,000, expressed in palms: this is undoubtedly the simplest, most direct and convincing correlation of all those seen.

It must be underlined that it would be completely unlikely to attribute this correlation to chance. There can be no doubt about this: just consider that the determination of a circle starting from three points given at random on the plane is a highly unstable process, in the sense that even small variations in the position of the points themselves can generate large variations in the result final, especially when the three points are close to alignment (as in the specific case). To clarify the concept, let's assume that the pyramids of Khufu and Menkaure are fixed, while that of Khafre can move freely, giving rise to infinite possible circles of Giza. If the pyramid of Khafre moved, from the position in which it actually is, about 47 meters towards the south-east, it would end up being exactly on the line joining the centers of the other two pyramids, and therefore the circle of Giza would have an infinite radius; if instead it still moved 47 m, but in the opposite direction, the Giza circle would have a radius of approximately 16,215 palms, i.e. just over half of the real one; but even if it moved just one meter to the east, this would already be enough to destroy the fragile balance on which the correlation is based, since the radius of the Giza circle would be worth approximately 30,953 palms, a completely anonymous value.

The corner of Giza

There is one additional fact that emerges: the Giza arch is approximately one fifth of the diameter, with an extraordinarily small margin of error, less than +0.02%. This last observation leads us to wonder whether the Giza angle could also hide some mathematical correlation.

The answer is affirmative: this correlation lies in the relationship between the entire circumference and the Giza arc, which is naturally equivalent to the ratio between the round angle (360°) and the Giza angle (22.9218°). This ratio is valid:

To be precise, the exact value of 5*PI is 15.7080, from which a theoretical center angle of 360°/15.7080 ~ 22.9183° would follow, while the exact value of 6*(FI+1) is 15.7082, from which a theoretical center angle of 360°/15.7082 ~ 22.9180° would follow. In summary, for greater clarity, we have the following relationships:

• (1) Giza angle ~ 22.9218° = gamma
• (2) 360°/5*PI ~ 360°/15.7070 = 22.9183° = alpha
• (3) 360°/6*(FI+1) = 360°/15.7082 = 22.9180° = beta

As you can see, it turns out that alpha and beta are identical to less than a real trifle (three ten thousandths of a degree).

This means that the relationship

• (4) 5*PI ~ 6*(FI+1) ~ 15.7081±0.0001

represents a very effective approximation of PI as a function of FI (and vice versa), the best approximation achievable with linear functions. The intermediate angle between alpha and beta (let's call it mi) is 22.91815°: well, gamma (i.e. the Giza angle) differs from mi by less than four thousandths of a degree...

Even in this circumstance it must be underlined that it would be completely unlikely to attribute the correlation to chance. This can be verified with the example illustrated above: it would be sufficient for Khafre's pyramid to have been built just one meter further east (the other two remaining the same) for the circumference thus identified to be 15,051.11 m , that the arc was 942.22 m and that the relative ratio was 15.974; this number divided by 5 gives the value 3.195, which no one would ever dream of passing off as an approximation of PI.

Let's take a moment to reflect. The Giza angle, therefore, refers simultaneously to PI and FI, and in particular to a relationship that links the two numbers to each other. What are the chances that such a circumstance could have arisen by chance? It is necessary to compare the range of possible values of the Giza angle with the margin of error just calculated. If we assume that the three pyramids were positioned randomly, the range of possible values of the Giza angle goes from almost zero (three pyramids aligned) to 240° (three pyramids arranged at the vertices of an equilateral triangle). Consequently, taking into account that the margin of error must be doubled as it can be both positive and negative, the probabilities that such a circumstance could have occurred by chance are:

Conclusions

The set of correlations that we have shown, and which constitute only a part of what I have been able to discover (the rest will be shown elsewhere), cannot be attributed to pure coincidences. No researcher with a minimum of intellectual honesty could be satisfied with such an explanation, which also explains nothing at all: it is better, then, to arm oneself with the necessary courage and fully draw out the consequences of what is clearly evident.

First of all, whoever built the Giza complex knew the value of PI with an even better approximation than that provided by the ratio of 22/7, traditionally attributed to Pythagoras, as mentioned; he also knew the meaning and value of the golden number FI, and he knew the best linear expression of PI as a function of FI. All this knowledge, as we have seen, is as if engraved on stone, represented through the particular configuration of the three pyramids.

Secondly, whoever built the Giza complex possessed a mastery of construction techniques and topography operations, such as to be able to place enormous objects on the ground - such as were not built until our century - creating with them configurations of territorial scale, with precision standards that are difficult to achieve even with the tools available today.

Do we want to believe that it was the Egyptian people - the one known to us from historical documents - who created an architectural device of such perfection? Whoever wants can continue to believe it. But this is not the fundamental point. The question is: why did the planners of Giza - whoever they were - arrange such a redundancy of mathematical and geometric signals? Just to perform an astonishing virtuosic exercise, and thus show one's skill to posterity? Certainly not. As we said, if they did a lot to attract our attention, the purpose was to make us understand that it was worth questioning the stones of Giza, because they had something important to say. Something extremely important, judging by the effort the builders had to make.

In other words, everything we have shown so far falls, no more, no less, into the first phase of communication. We could give the example of the search for intelligent signals coming from the cosmos: in this phase we have discovered that an intelligent signal has reached us from space; but we still don't know what the message conveyed by the signal is. All we hear is a crackling sound, certainly artificial, but which will remain incomprehensible until the code is discovered.

To return to us, what we need is the Giza code, and fortunately we don't even have to go very far, since it has already been discovered: it is the correlation between the three pyramids and the three stars of Orion's Belt. The stellar correlation, discovered by Robert Bauval and then developed together with Graham Hancock in the theory of the "First Time of Orion", was reformulated by me in an original way, therefore correct and in-depth to the extreme consequences, as I showed in the article First Time of Sirius, to which I refer.