The results of the two most reputable surveys of the Giza pyramids, those of W.M.F.Petrie (1881) and J.H.Cole (1925, Horizon Pyramid only), are shown in Appendix C. Since these surveys were conducted, certain peculiarities in the form of the Horizon Pyramid have come to light, not the least being a very slight indentation of the sides which results in the distance across the middle of the base (BC in Figure 3-3), between opposite apothems (AB and AC in Figure 3-3), being shorter than the lengths of the sides parallel to it, by about 1.2 metres. The indentations went unnoticed until discovered by aerial photography and are visible only when the Sun is shining across a face, throwing half of that face into shadow. Their presence testifies to the high degree of precision with which the Horizon Pyramid was built.

Figure 3-4. Stellated Pyramid

The first person to comment on the apparent p (pi) relationship between the height of the Pyramid and the perimeter of its base was John Taylor, working from measurements made by John Perring during Howard Vyse's expedition in the 1830s. Petrie later opined that it formed the basis of one of the best theories regarding the intended dimensions. Of the angle and elevation of the Horizon Pyramid, Petrie says[12]

“On the whole, we probably cannot do better than take 51° 52' ± 2' as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side.
“The mean base being 9,068.8 ± 0.5 inches, this yields a height of 5,776.0 ± 7.0 inches.”

This gives a range of ratios of base-width to height, varying between (9068.8 - 0.5)/(5776.0 + 7.0) and (9068.8 + 0.5)/(5776.0 - 7.0), or 1.5681 and 1.5721, with a difference of 0.004, or roughly 0.25%. It can be seen that the values p/2 (~1.5708) and 11/7 (~1.5714) both fall within this range.

This means that, whatever the true or intended dimensions, the height is equal to the radius of a circle whose circumference very closely approximates the perimeter of the base.

Figure 3-5. Squaring The Circle ­­­­­– Geometry of The Horizon Pyramid

The attribution of the building of the Horizon Pyramid to Khufu was documented first by Herodotus, who took it as fact on the basis of assertions made by the priests of Sais.

The modern evidence is equally suspect; some red ochre paint daubed on a block of stone in one of the cavities above the metrological cavity, the so-called King's Chamber, allegedly showing the cartouche of Khufu. There were, however, rumours of paintbrushes and cans of red ochre paint, still used by Egyptian quarry workers nowadays to mark stones, being carried into the Horizon Pyramid during Howard Vyse's expedition coincident with Vyse's need to raise new funding. This he succeeded in doing following the announcement of his convenient ‘discovery’ that the Horizon Pyramid could indeed be attributed to Khufu and therefore confidently dated ­­­­­– happily for everyone ­­­­­– in accordance with both Egyptian kinglists and the Torah's dating of ‘Creation’.

Moreover, the ancient cemeteries and boat pits adjacent to the Horizon Pyramid seem unlikely to have belonged to the builders or original keepers of any part of the Giza complex since they encroach upon the Horizon Pyramid's precinct, as can be seen quite clearly in satellite images. Such travesties would simply never have been permitted by the builders since all they have ever done is to despoil and desecrate the complex, as can also be seen quite clearly in satellite images. The later pyramid cults to which the cemeteries bear witness appear to have had as much to do with the building of the Giza pyramids as latterday ersatz ‘druids’ have to do with that of Stonehenge. The remains of the precinct walls of the two other main Giza pyramids can also clearly be seen in satellite images.

* * *

The first person known to have looked for a linear modulus of the Horizon Pyramid in the dimensions of the metrological cavity was Sir Isaac Newton (1642-1727), working from measurements of it made separately by John Greaves (1638) and Tito Livio Burattini (1642). Through a particularly abstruse line of argument, Newton concluded that this modulus was equal to a tenth part of the width of the cavity. It was the dimensions of the Earth that Newton was looking for in order to confirm his theory of gravitation and he had acquired from an unknown source the belief that the builders had encoded these dimensions in those of the Horizon Pyramid. As Tompkins records[3]

“...it was from Greaves's data that Sir Isaac Newton deduced that the Great [Horizon] Pyramid had been built on the basis of two different cubits, one of which he called ‘profane’ and the other which called ‘sacred’. From Greaves's and Burattini's measurements of the King's Chamber, Newton computed that a cubit of 20.63 British inches produced a room with an even length of cubits: 20 x 10. This cubit Newton called the ‘profane’, or ‘Memphis’, cubit; whereas a longer, more arcane cubit appeared to measure about 25 British inches.

“This longer, or ‘sacred’, cubit Newton derived from the Jewish historian Josephus's description of the circumference of the pillars of the Temple at Jerusalem. Newton estimated this cubit to be between 24.80 and 25.02 English inches, but believed the figure could be refined through further measurements of the Great [Horizon] Pyramid and other ancient buildings.

“All of this Newton wrote up in a small and now hard-to-find paper called A Dissertation upon the Sacred Cubit of the Jews and the Cubits of several Nations; in which, as taken by Mr. John Greaves, the ancient Cubit of Memphis is determined.

“Newton's preoccupation with establishing the cubit of the ancient Egyptians was no idle curiosity, nor just a desire to find a universal standard of measure; his general theory of gravitation, which he had not yet announced, was dependent on an accurate knowledge of the circumference of the earth. All he had to go on were the old figures of Eratosthenes and his followers, and on their figures his theory did not work out accurately.

“By establishing the cubit of the ancient Egyptians, Newton hoped to find the exact length of their stadium, reputed by classical authors to bear a relation to a geographical degree, and this he believed to be somehow enshrined in the proportions of the Great [Horizon] Pyramid.”

Newton was working backwards from a purely hypothetical Israelite cubit allegedly described by a now largely discredited historian and of which no physical example existed, via the ratio 6:7 (that of a hypothetical ‘profane’, or ‘common’, cubit to a hypothetical ‘sacred’, or ‘royal’, cubit and a Diophantine approximation to √3:2, that being the exact ratio of the dimensions of 30th Parallel and Equator on a sphere), to a division by 10 of the width of the floor of the metrological cavity of the Horizon Pyramid thence to postulate a ‘Memphis’ cubit that had existed since the foundation of Egypt. Clearly this line of argument is far-fetched, to say the very least. Nonetheless, subsequent authorities bowed to that of Newton and, for example, Dr. Charles Piazzi Smyth lauded Newton thus:[4]

“How thankful should we be that it pleased God to raise up the spirit of Newton amongst us; and enable him to make one of the most important discoveries of his riper years ­­­­­– though the opposition of the Church of England has caused it to remain unread almost to the present day ­­­­­– that while there undoubtedly was in ancient times a cubit of 20.7 inches nearly... ...and which Newton calls ‘the profane cubit’ there was another which he equally unhesitatingly speaks of as the sacred cubit, decidedly longer.”

Petrie also accepted without question the use of Newton's Israelite cubit in the construction of the Horizon Pyramid [9]

“Thus the total length of the plug-blocks would be about 203 inches, or very probably 206 inches, or 10 [Newton's] cubits, like so many lengths marked out in that passage.”

And then Prof. Livio Stecchini adds his stamp of approval;[6]

“It was Newton who, on the basis of the survey conducted by Greaves, realized that the King's Chamber measures 10 by 20 [Newton's] cubits. Having established this fact,...”

Others, however, have noticed the insubstantial nature of Newton's cubit, Derek Gjertsen observing in respect of it that[14]

“As with many such works, both before and after, however tight the reasoning, the reliability of the conclusions will still depend upon the accuracy of the initial assumptions. As many of these were, in fact, no more than guesses, Newton's conclusions lose much of their initial plausibility.”

Despite its dubious pedigree many contemporary authors continue to treat Newton's cubit as if it was genuine.

The only clear facts are that the proportions of the floor of the metrological cavity are 1:2 and that the length is a whole-number division of both the width and the height of the pyramid, making these 22 and 14 cavity-lengths respectively. Hence, for the floor, the longest common linear extent is the width of the cavity, 206.3 ±0.2, English inches (5.24m). This is indeed close to Petrie's reported length of the plug-blocks, as noted above. The width of the cavity is to its height as 9:10 and taking this into account the longest common length that can be derived from all three of its dimensions is a unit equal to a tenth part of the height of the cavity, again unlike Newton's cubit, at ~22.9 English inches (~582mm). In terms of this linear modulus, or Horizon cubit, the width of the base and the height of the Horizon Pyramid come out at 396 and 252 respectively. The figure for the width of the base is remarkably close to that suggested by French scientist Edmé-François Jomard (1777-1862), 400 pyk belady, that being the number of cubits to a stade.[5]

“In vain Jomard argued that he had found an even more surprising coincidence in that the four-hundredth part of his base of the [Horizon] Pyramid gave a figure of .5773 meter, which was exactly the length of a longer modern Egyptian cubit called the pyk belady.”

Jomard's measure of the width of the base was, however, slightly excessive since 1/400th of the actual width is 0.5759 metre (22.67 inches), taking J.H.Cole's mean value of 230.364 metres, where 1/396th gives 0.5817 metre (22.9 inches). Moreover, the square stade is the area of the sacred precinct, not of the temple which stands within it. Hence the Horizon stade would be expected to exceed the width of the Horizon Pyramid. Four hundred Horizon cubits gives a stade of some 9,160 inches (232.66 metre). Curiously, 1/400th of Perring's erroneous figure for the width of the base, 764 English feet, is the pyk belady of 22.9 English inches.

Although the pyk belady was in use in Egypt in Jomard's time nothing is known of its origin and history therefore no firm conclusion can be reached about the age of the Horizon Pyramid from the fact that it is the modulus. All that can be said of the modulus is that it must of necessity predate any structure which incorporates it. The name ‘pyk belady’ combines the Egyptian word for a cubit, etymologically related to the Greek ΠΗΧΥΣ, pekhus and ΠΥΓΩΝ, pygon, both also meaning a cubit, with that for ‘our land’, as in the modern Egyptian national anthem ‘Belady, Belady, Belady’. Hence ‘pyk belady’ can be taken to mean ‘national cubit [of Egypt]’.

Figure 3-6. Theoretical Cross-Section of The Horizon Pyramid

It might be thought that metrological standards in ancient times followed the same principle as those of today and that, therefore, a given unit would have measured exactly the same absolute quantity no matter what the context. However, as was seen in the case of the Parthenon stade, the unit used for the temple, although approximating it, was not the same as that of the general tradesman. There seems to be no intentional conversion ratio between the two standards, which merely approximate one another, varying only by a few inches; varying nonetheless. It would appear that in the field of monumental architecture units were consistent only within the context of a particular undertaking and that an architectural guild was at liberty to contrive its own linear standards project by project within a fairly narrow range of lengths as long as they conformed to a uniform fixed system. The particular set of standards for each pyramid was created and ritually consecrated specifically for that pyramid then ritually desecrated and made useless once the building was complete. A remnant of this ritual destruction of the project standards can be seen in the modern ceremony of cutting the ribbon at the official opening of a new public building by some or other dignitary. The ribbon symbolises the primary standard, the stade rope, used in the construction.

The averaged, corrected measures of the metrological cavity are shown in Table 3-1. Petrie's words concerning the condition of the cavity as he found it must be borne in mind.[10]

“The King's Chamber was more completely measured than any other part of the Pyramid; the distances of the walls apart, their verticality in each corner, the course heights, and the levels, were completely observed. On every side the joints of the stones have separated, and the whole chamber is shaken larger. By examining the joints all round the 2nd course, the sum of the estimated openings is, 3 joints opened on N. side, total = .19 [English inches]; 1 joint on E. = .14; 5 joints on S. = .41; 2 joints on W. = .38. And these quantities must be deducted from the measures, in order to get the true original lengths of the chamber. I also observed, in measuring the top near the W., that the width from N. to S. is lengthened .3 by a crack at the S. side.

“These openings or cracks are but the milder signs of the great injury that that the whole chamber has sustained, probably by an earthquake, when every roof beam was broken across near the South side; and since which the whole of the granite ceiling (weighing some 400 tons), is upheld solely by sticking and thrusting. Not only has this wreck overtaken the chamber itself, but in every one of the spaces above it are the massive roof-beams either cracked or torn out of the wall, more or less, at the South side; and the great Eastern and Western walls of limestone, between, and independent of which, the whole of these construction chambers are built, have sunk bodily. All these motions are yet but small ­­­­­– only a matter of an inch or two ­­­­­– but enough to wreck the theoretical strength and stability of these chambers, and to make their downfall a mere question of time and earthquakes.”

“The floor of the chamber, as is well known, is quite disconnected from the walls, and stands somewhat above the base of the lowest course. It is very irregular in its level, not only absolutely, but even in relation to the courses; its depth below the first course joint varying 2.29 [English inches], from 42.94 to 40.65.”

This is an example of the marriage of certain irrational and mutually incommensurable numerical constants, such as π (pi) and √2 (square root of 2), using a system of whole numbers and their ratios. In order to do this, some rough-and-ready approximations must be incorporated. The primary approximation expressed by the gross form of the Horizon Pyramid is that of 6 ²/₇, or the perimeter of the base divided by the height, as a representation of the value 2π. Hence there can be no exact expression of any of the constants referenced in any of the physical approximations embodied in any of the structures. In fact, to find a single exact value anywhere would be to exclude all possibility of finding comparable references. The architect has created a compendium of equally variant Diophantine approximations.

Figure 3-7. Theoretical Proportions of The Metrological Cavity of The Horizon Pyramid

In terms of the proportion 9:10:18, employing the derived Horizon cubit of ~22.9 English inches:- The theoretical surface area of the metrological cavity is 864 square Horizon cubits. This is equal to the surface area of a cube of edge 12 Horizon cubits. The volumes of the cavity and its cube of equal surface area are in the ratio of a naturally tempered musical semitone, 15:16. This is also the ratio of both the widths of the bases (variance 0.34%) and the capacities of the coffers of the Grand and Horizon Pyramids. The sum of the lengths of all the edges of a 12-cubit cube is to that of the cavity as 36:37. The theoretical volume of the Horizon Pyramid is 13,172,544 cubic Horizon cubits.

13,172,544 = 2⁶ × 3⁵ × 7¹ × 11² = 7 × 33² × 12³

That is exactly equal to 7,623 12-cubit cubes; cubes, that is, of surface area equal to that of the theoretical metrological cavity. Now, these 7,623 cubes can be arranged as a stack of 7 equal square tiers of width 33. With the width of the tiers equal to the width of the base of the pyramid, the height of the resulting rectangular block is exactly 1/3rd of the height of the pyramid. Altogether, the figure clearly embodies in whole numbers the arithmetical method for calculating the volume of any pyramid; one third of the height times the area of the base.

Hence it may be concluded that the Horizon Pyramid was laid out and constructed using the pyk belady as its modulus. However, because nothing is known about the history of this unit of measure beyond its use in the construction of the Horizon Pyramid and its general use in Egypt at the time of Jomard's visit, its age and origin cannot be established and, therefore, no firm conclusion can be drawn about the age of the Horizon Pyramid from its use. Jomard's observation is seemingly the only connection to any memory of its existence, since all other mentions of this unit themselves refer back to him as their source. Nonetheless it is clear that the linear modulus of the Horizon Pyramid possesses a unique name and is no mere hypothesis. Not only that but the Horizon Pyramid linear modulus can confidently be labelled the Egyptian cubit, for that indeed is what it is.

Figure 3-8. The Egyptian Stade

Taking the disk of the Earth as a circle and the value of pi as 22/7, in Egyptian stades of 400 cubits the circumference so obtained is exactly 171,072 Egyptian stades and the radius 27,216. A Greek stade on the same circle would be exactly equal to 99/125 Egyptian stade (terrestrial circumference = 216,000 Greek stades).

The Horizon Pyramid width, at 99/100 Egyptian stades, is equally interesting since the terrestrial circumference comes out at 172,800 or 100 times the cube of 12. Moreover, again on the same circle, it is exactly equal to 1¼ Greek stades, the Greek metric standards having been supplied to Solon by the Temple of Neith at Sais around 600 BC, suggesting an unbroken tradition stretching from the building of the Giza complex to that of the Parthenon of Athens. The Greek system of linear measures was derived from the Egyptian and both include a digit (finger) as the smallest unit, a foot of 16 digits, a cubit of 24 digits and a stade of 600 feet.

* Peter Tompkins' website was removed from the World-Wide Web in February 2008, a year after his death in January 2007. This author has a copy saved in his archives.

3-3. The Grand Pyramid

The mean width of the base of the Grand Pyramid is almost exactly 370 Horizon cubits (pyk belady), the variance being roughly 1 part in 6,000 or 0.016%, and here the rounded figure will be assumed to be the intended width. This pyramid was apparently intended to conform to the proportions of a Pythagorean {3-4-5} triangle, giving a theoretical height of 246⅔ cubits or 370 feet.

Figure 3-9. Theoretical Cross-Section of The Grand Pyramid

The theoretical volume of the Grand Pyramid is exactly 37,989,750 cubic Horizon feet.

37,989,750 = 2 × 3 × 5³ × 37³ = 6 × 185³

The length of each edge of the 6 cubes so defined is equal to half the height of, and to one third the width of, the pyramid.

3-4. The Divine Pyramid

Petrie's mean for the width of the Divine Pyramid is 4,153.6 English inches, or roughly 181.34 Horizon cubits with a probable intended figure of 181⅓ cubits, or 272 Horizon feet exactly, differing from the mean by ­­­­­–0.004%. For the angle of the apothem Petrie's “best conclusion” is 51°0' ± 10'. This results in a height somewhere between 2549 and 2580 English inches, or roughly 167 and 169 Horizon feet. Taking the mean gives a height of 168 feet and this will be found to share a common factor of 8 with the theoretical width. Hence the proportions are 21 rise on 17 base, revealing a clearly intended allusion in the division of the width by the height to Φ (Phi), the Golden Mean or Divine Proportion, differing from it by a mere +0.06%. In fact the ratio of height into width, 21:34, displayed by the Divine Pyramid is the 8th term of the Fibonacci Sequence, which converges to Phi.[18]

Figure 3-10. Theoretical Cross-Section of The Divine Pyramid

The theoretical volume of the Divine Pyramid is exactly 4,143,104 cubic Horizon feet.

4,143,104 = 7 × 34² × 8³

* * *

Figure 3-11. The Ratio between The Heights of Horizon And Divine Pyramids

The ratio between the heights of Horizon and Divine pyramids is the same as that between the length and breadth respectively of the platform of the Parthenon of Athens - 4:9. This ratio belongs to the Lambda, itself the root and source of the dodekaphonic musical scale, as described in Chapter 1, suggesting that the Giza complex comprises an integrated design and that its overall planning was completed before a single stone was quarried for a single pyramid.

Figure 3-12. The Pyramids of Giza (2)

3-5. The Southern Shining Pyramid

Figure 3-13. The Southern Shining Pyramid

The reader can forget the Egyptological dogma that seeks to explain the compound slope of the Southern Shining Pyramid as a decision by the builders to switch slopes in mid-project, the rationalization of this usually being that they discovered that the internal stresses of the structure prevented completion of a pyramid with the continued slope of the lower section. How exactly they discovered this is never made clear and no evidence supporting the stress-factor hypothesis is ever presented. It is of course a flight of fancy, for from the outset the Southern Shining Pyramid was designed to be a flattened obelisk, as the following analysis of its geometry will make very clear.

7,286,976 = 12³ × 4217

The outer form contains an implied inner, or hidden, pyramid.

Figure 3-15. Theoretical Slope of The Hidden Pyramid

The theoretical volume of the Hidden Pyramid is exactly 6,298,560 cubic Southern Shining cubits.

6,298,560 = 12³ × 9³ × 5 = 5 × 108³

The length of each edge of the 5 cubes so defined is equal to three fifths (0.6) the height of, and to one third the width of, the Hidden Pyramid.

3-6. The Rising Sun Pyramid

Figure 3-16. The Rising Sun Pyramid

The Rising Sun Pyramid, also known as the Red Pyramid because of the colour of its exposed rocky core, is located in the Dahshur pyramid field at 29° 48' 30.9" N × 31° 12' 22" E. This is a large pyramid with a rock core and, like every pyramid and mastaba, has been stripped of its limestone casing by stone robbers, leading directly to the complete and utter collapses of those many pyramids which had been built with mud brick cores. However, with the cores exposed aerial photography and now satellite imaging reveal that those of both the Horizon Pyramid and the Rising Sun Pyramid are stellated, something that would have remained undetectable and completely unknown had they retained their unstellated casings. This means only that both pyramids are unquestionably products of the same guild of architects, assuming that no single architect working in isolation could have raised all the stone-cored pyramids given both the magnitude of each project individually and the number of other pyramids with similarly stellated stone cores.