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Last of the ancient wonders IV: The pyramid face angle
Thursday, May 29 2008 @ 12:00 AM CDT
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Assem Deif investigates the face angle of Egypt's pyramids. Although it relies on the culmination of Sirius, the brightest star in the sky, it is rounded to match the nearest Egyptian unit, seqed
When the Egyptians witnessed the rising of Sirius just before dawn (known as the heliacal rising), they knew that the Nile would soon flood; for they depended upon the flooding for the agriculture and fertility of their land. The heliacal rising, which falls close to the summer solstice, marked the beginning of the New Year coinciding with the month of Thoth.
Since Sirius brought prosperity to Egypt, tracing of the star was crucial. At the Isis-Hathor Temple of Denderah there is a statue of Isis which was oriented to the rising of Sirius. When the priests saw the rays from Sirius penetrating the temple to fall upon the gemstone she wore, they announced that a New Year had begun. In the temple appears the inscription, "Her majesty Isis shines into the temple on New Year's Day, and she mingles her light with that of her father on the horizon."
Sirius did not only bless the living by bringing wealth to the Egyptians, but it also blessed their dead. It was believed that Sirius was the doorway to the afterlife, so the ancient Egyptians abstained from burying their dead at the time of the year when the star was hidden from view, which lasted about 70 days. This led Herodotus to assume that the mummification process could take up to 70 days, yet it is commonly known that it used to last 40 days. Presumably, what he meant was that it would not take more than 70 days, as burying the deceased involved other rituals as well.
The Egyptians, being blessed by Sirius, buried their dead such that the head lay in the direction of the star's culmination point. In other words, the corpse lay on the meridian with the head pointing due south. Visitors to the King's Chamber in the Great Pyramid will notice that the sarcophagus is fixed in this direction. They will also notice that it is broken on the southern corner, meaning that the robbers who plundered the tomb knew in advance that it was this side that contained the treasure. All the evidence suggests that the upper chamber of the pyramid was the one that contained the king's tomb, and that no other room existed anywhere else in the building, therefore dissipating the dream of many pyramid enthusiasts.
Sirius was thus revered as the Nile Star or the Star of Isis by the Ancient Egyptians and they would build temples in its tribute. Lockyer found seven temples oriented to the rising of Sirius. It was also represented on monuments with a triad (the hieroglyphic symbol for Sirius): the star, a ben-ben symbolising the emergence of existence from non-existence, and a pyramid; meaning that the latter was consecrated to this god, that is a place in which the king passed by a phase of ascension and transformation to the afterlife. Many nations paid equal tribute to the star, like the ethnic Dogon tribe of Mali and the pre- Islamic tribe Khuza'ah in the Arab peninsula, who referred to it as the idol known as "Manat". This is why God in the Qur'an in Surat "Al-Nagm" addressed those atheists by calling Himself "Lord of Sirius"; i.e. the Lord of the star they worshipped; meaning it was more appropriate that they worshipped Allah who created Sirius rather than the star itself. He said in Ayas 49- 51: "And that He, Ever He is the Lord of Sirius; And that He caused to perish the earliest A'ad, And Thamud, so He did not spare any of them."
Concerning the shafts in both the King's and Queen's chambers, a little reasoning persuades us that they were part of the original plan, that the Great Pyramid was meant to be designed this way. They are not ventilation shafts -- for whom? Once the burial was completed, the pyramid was supposed to be shut for ever. Besides, they cannot serve the purpose of ventilation because those of the Queen's Chamber fall short by a few meters from the outside of the pyramid.
To assume that the air shafts were tunnels for communication, an "intercom" between the inside and outside workmen, is also untenable; for why the need to be exactly inclined at the specific angles of 45 or 32 to the horizontal to end up exactly and precisely at the same horizontal level. And why were four channels needed from both the north and the south sides? Since they were constructed during the erection of the structure, they could not be so well aligned except perhaps by the tracking of the stars. Badawi described the stellar destiny of the pharaoh, in reference to alpha-Orionis "Betelgeuse", one of the brightest stars in the southern sky, as a kind of prince among the stars, with which the spirit of the king is to be united. This stellar theory has gained wide acceptance among today's Egyptologists and seems plausible; especially in that an indication to it appears regularly in the Pyramid Texts. We shall therefore assume that the theory put forward for the king's southern shaft is correct.
What about the other air passages? The northern ones are thought to point to some circumpolar stars (alpha-Draconis "Thuban" was the pole star in this period, along with stars like Kochab). It is possible that these were the stars that helped align the edifice.
Of all the shafts, it is the Queen's southern shaft which is in my view the most controversial. Falling short by six metres from the outside of the pyramid makes it even more controversial. Egyptologists presume that it is meant to point to Sirius. Most likely it does; especially in that many astronomers traced this issue back to the epoch and could tell the configuration of the sky with the shaft pointing to the star. But what they reckon is only partially true. For it seems that the builders, while working, made sure that the shaft was continuously pointing to the star's culmination point. This is most unlikely, because the northern shaft has an equal inclination and has nothing to do with Sirius. Further, both shafts do not run outside the pyramid, so how did they track the stars? Again, they are not straight, but angle up after traveling horizontally for a few metres. It looks as if both were meant to fall perpendicular to the pyramid's face; i.e. connecting virtually the so-called Queen's Chamber to the outside with the shortest path possible. Again, that the shafts' purpose is to align the edifice is not possible either, because alignment was supposed to be done before and not during erection. That they are an outlet for the king's ba is also far-fetched, because the latter was known to pass through doorways. In my view they are meant to be of a symbolic nature only, and therefore it is not at all necessary that they are totally straight. Had they served a crucial purpose we would have seen them in other pyramids.
In my view, it is the other way around. The builders designed their structure so that at culmination Sirius's rays fell perpendicular to the pyramid's south face; thus giving its maximum blessing to the deceased king and his ka. In other words, Sirius transmitted its benediction from the climax of its throne to the deceased. This view, which is credited wrongly to the German engineer Neuberger in 1919, is in fact a property of Mahmoud Hamdy Pacha, k Al-Falaki, the famous Egyptian engineer and astronomer who lived in the mid 19th century, and founded the first Egyptian observatory. Among his several books is one written 60 years prior to Neuberger and devoted to the Pyramids of Giza in which he calculated the age of the great edifice. The results he published in 1862 in the Belgian Royal Academy put him on the lead in the science of archeoastronomy. For after carrying out measurements for the Great Pyramid during the vernal equinox, he discovered that its four faces are inclined equally towards the horizon; meaning that they possess the same slope. Al-Falaki concluded that this could not be coincidental. The planes of the pyramid were also inclined in such a way that the rays of Sirius, while at culmination, fell at a right angle with the southern face; a necessity that must be ascribed to the religion. So upon measuring the face angle and calculating its complement, he proceeded backward to calculate when Sirius attained its maximum altitude. He found that the Great Pyramid must have been erected c. 3303 BC, deviating greatly from the agreed date. For although he already knew that the face angle is 51:45; being slightly incorrect (the true face angle was still unknown), Al-Falaki preferred to use the value 52:30 as the average face angle of all the pyramids on the plateau, and that is his main mistake. Besides, his calculations could not be accurate; being carried out by hand. Perhaps, if we could run a sky program and provide the expected altitude angle of Sirius at culmination which is 38:9:26 (complement of the pyramid face angle), then we would be able to reach a better estimate.
Before pursuing our investigations, we should point out that it is immaterial whether Petrie or Gantenbrink provided a slightly different measure for the Queen's Chamber southern shaft inclination, because one must expect the practical implementation of the design to incorporate some admitted tolerances on the part of the builders. True, that the shaft could have been pointing to Sirius at best all the way through the construction phase, and equally true that it must give in principle the age of the edifice, but who said that the Sirius culmination point was meant to be so accurately determined; especially as the ancient Egyptians must have known that Sirius passes through phases of highest and lowest culmination and the edifice must be designed to account for this change and not just at the time of burial. Besides, the builders were not interested as such in Sirius's altitude, rather in its complement in order to set their design of the faces' angles. Can one possibly believe that they were so keen to measure exact complements of angles or exact altitudes of stars; notwithstanding the fact that they measure only in Egyptian fractions ?
What seems more plausible is that they already knew an approximate altitude of Sirius at culmination in this epoch to be around 98/11 in seqed units (palms/cubit) and set consequently the seqed of the faces to 11/2. If one studies the pyramids of the Fourth and Fifth dynasties in the vicinity of Memphis, one notices that most replicate the same angle with a tiny variation. But while at work a multitude of factors intervened, or more likely there was a hidden reason for this slight variation. True, the Bent Pyramid has a much less angle of elevation, but this was because the builders had already an unfortunate experience in Meidum.
A problem of interest therefore is to be able to calculate approximately the epoch when the Great Pyramid was erected, given the culmination of Sirius at that time. Since the maximum altitude of the star is given from the relation: Amax = 90 -- latitude angle of the pyramid + declination of Sirius, where Amax is taken as the complement of the pyramid face angle, and the latitude angle of the Great Pyramid is 29.97921389 (from Google Earth and matching with M. Lehner results on the Giza Plateau), it follows that the declination of the star at the time of erection equals approximately -21:52. From the table of Sirius declination, the Great Pyramid must have been erected in c. 2825 -- 2800BC; offsetting Al-Falaki's results.
Most Egyptologists however accept the date c. 2600 -- 2500BC for the age of the edifice. What can then be wrong? We shall try to forward a plausible explanation for this difference. We shall show that the face angle of 51:50:34 provides a rounding figure for the seqed of the pyramid with the least error from the ideal fractional seqed which guaranteed that Sirius rays had fallen perpendicular to the pyramid's face.
In my view, the builders must have resorted to rounding values for the seqed of their structure. Suppose the Sirius at culmination fell perpendicular to the pyramid's face with an angle of inclination of 38:28 which is the exact shaft inclination, giving a seqed per cubit for the face inclination of 5.56p being equal to 5p 2.24d. The builders must design the seqed of the pyramid ranging from 5p 2d to 5p 3d and more likely rounded to the first. The first matches with c. 2825 -- 2800BC, whereas for the second choice one obtains c. 2450 -- 2425 BC for the age of the edifice. It is therefore unlikely that one can estimate precisely the age of the monument, but since the builders rounded their seqed value to its lower bound, then one is inclined to suppose that it dates c. 2650BC.
The assumption that Sirius's rays were to fall perpendicular to the pyramid face does not entail its validity for the rest of the pyramids. It was crucial here due to the upper burial of the corpse and it proved beneficial to be able to bless both the mummy and the ka. So while Sirius blesses the ka though the southern Queen's shaft, the King's points towards the Orion belt or, as Petrie suggested, Ra's rays enter the King's Chamber twice a year. He noted that the Sun shone down the southern shaft of Khufu's burial chamber at midday on two particular days of the year, 2 November, and 8 February 1881 (time of his excavations). As V. Brown noted: the bringing of the sun into the tomb contained the significantly potent symbolism of rebirth and rejuvenation; for this to occur inside Khufu's Akhet chamber purely by accident would be extremely fortuitous. It must have been intentionally factored into its design.
Given that Sirius's rays were expected to fall perpendicular to the Great Pyramid's southern face, one is curious about the face angle of the rest of the pyramids. It looks as though there were no standard or unique rule set for this purpose. Yet one expects -- if my hypothesis is viable -- that it must depend on the highest and lowest culmination of the star. Assuming Sirius's maximum altitude is 40 and minimum 38 at culmination between the years 2850 -- 2250BC, it follows that pyramid face angles should be confined to the following bound: 50 is less than the pyramids' face angle which is in turn less than 52. Scanning all of Egypt's pyramids we notice that their face angle is confined more or less to these bounds, or that no Egyptian pyramid throughout history exceeds the above two limits. In fact, most possess a face angle varying between 51 and 53. Taking the Great Pyramid as a datum point, considering the theory that its face angle is the complement of Sirius's altitude during erection, and knowing that the star does not seem to be altering much of its altitude over a long period of time, only two degrees, then it is expected that the face angles of Egypt's pyramids alter within plus or minus this value. However, this does not prevent any king from altering this tradition for some hidden considerations; yet if he did, it would be according to a certain canon which we must discuss.
First, we have to come to terms with the Egyptologists' measurements. Most worked them out either in feet or metres. Since the Egyptians measure in cubits, colossal edifices of stupendous size like the pyramids must adhere, or at least be designed, such that their dimensions are integer values of a cubit with no fractions in them. So either we transform the data collected for the pyramids -- presuming their correctness -- into the nearest integers of a cubit, as for the Great Pyramid, when Egyptologists settled for 280c for the height and 440c for the base side-length, or else that the slope of the pyramids' faces be set to the nearest seqed units; i.e. palms/cubit. Both choices are equally valid.
Now the seqed of Khufu's Pyramid is equal to 220/280, the equivalent of 5.5 palms; in other words for each cubit rise, the run is five palms and two digits. During this period of erecting the three edifices of Giza, which lasted approximately 75 years, Sirius presumably did not change much in altitude. Since its maximum fluctuation did not exceed two degrees, and probably much less, then the next pharaoh wanting an adjustment would automatically choose the next seqed in value. Therefore one would assume that Khafra would either have to choose for the seqed of his pyramid five palms and one digit or five palms and three digits for each cubit rise. It seems he settled for the first choice, giving his pyramid a higher slope angle (with no explanation on my part that it should be less due to the increase in Sirius declination). By calculating the face's slope which is the reciprocal of one seventh of its seqed, it will be equal to 1/[(1/7)(5.25)] = 4/3, giving a face angle of 53:7:48 being the exact figure for the second pyramid. It was found that that such an angle, which is inscribed in a 3:4:5 triangle, was adopted by at least eight known kings as their pyramids' inner triangle. This led Pythagoras, when visiting Egypt, to call it the "Holy Triangle". Only two pharaohs used Khufu's face angle of 51:50:34: his father Sneferu at Meidum and Niuserre of the Fifth Dynasty. The same angle was also adopted in two of the smaller satellite queens' pyramids to the east of Khufu: Hetepheres and Meritit.
For Menkaura, it is not easy to fit the data, because Egyptologists did not agree on its dimensions. The base itself possesses two different side-lengths; that is having a rectangular shape of 102.2 x 104.6 m and the height is 66.45 m from which they worked out a face angle of 51:20:25 which fits into a Seqed of 5 palms and 2.4 digits; being highly improbable. Most likely he adopted his grandfather's dimensions. On the other hand, Egyptologists report a face angle of 54:27:44 for Seneferu's Pyramid at the South of Dahshur which coincides exactly with a Seqed of 5 palms. Due the unfortunate incidence -- so it is said - that the ground had sunk causing subsequently a destruction of the chambers' system, the pyramid's angle had to be reduced at the height of 49.1 m from 55 to around 43:22, a little less than 7 palms and 2 digits per cubit, resulting in its characteristic bent shape. After another 15 years of his reigning period and getting convinced of this new angle, Sneferu duplicated the same angle for his Red Pyramid at north of Dahshur. Khufu on the contrary had more courage in decreasing his Seqed by 2 palms reverting back to the canon of Meidum.
The reader is entitled to enquire about the three odd cases of the Red Pyramid, Sahure and Unas Pyramids, and whether the masons had been able to handle such fractions of Seqeds? In fact their face angles have slopes more amenable to work with than the other integer values. For example in the case of Sahure, the pyramid has a exact tangent of 1.2 for its face angle (Khufu say has a tangent of 1.2727...), whereas for Unas the tangent is exactly 1.5. So it all depends on the size of the stone block and its height, controlling by how much the next strata should dip. Besides, the Egyptian cubit rod was partitioned into 28 digits, each subdivided into an ascending no of sections; the lengths of 1/3d and 2/3d are equal respectively to one and two partitions of its second digit.
* The writer is a professor of mathematics at Cairo University and Misr University for Science and Technology
Temple of goddess Hathor; Isis
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