## The Geometrical Pyramids

If the widely accepted chronology is to be trusted, the first geometric stone pyramids were constructed at Dashour, immediately after the Meidum pyramid, by Sneferu. Abruptly, thus, around 2600 BC, Egyptian architects set about building with successive, horizontal courses of heavy blocks of stone. They erected five—two at Dashour and three at Giza—and began two others, one at Abu Roash and another at Zawyet el Arian, that were never completed. And then they simply stopped. All subsequent pyramids are greatly inferior, not just in size but, more importantly, in the quality of workmanship (for this reason they are in very poor condition today).

Describing one of the five geometrical pyramids is relatively simple. The basic structure is simply a series of layers of large limestone blocks (normally the size of three washing machines put side by side) weighing 2 to 3 tons, laid one on top of the other and fixed with mortar. The layers were tiered so as to create a fixed gradient, and once this nucleus had been completed, the pyramid was covered with a casing of slim limestone blocks, sculpted in such a way as to look smooth. Unfortunately, many of the blocks forming the casing were removed, probably during the Middle Ages, and today the unique nearly intact casing is that of the south pyramid at Dashour; at Giza, we have left a few blocks around the top of the second pyramid, and a huge block on the north side of the Khufu pyramid (the casing of the Menkaure pyramid is discussed below). The pyramids had no underground foundations, and the only expedient used when beginning the layering work was the digging of a ditch in the rock to accommodate the first course. In some cases, however, preexisting natural features were incorporated into the lower layers to make them more stable (examples of this can be found in the core of the first Giza pyramid and in one corner of the second).

Ramps must have been used to move the blocks when working up high, although it is unclear what kind of ramp or combination of ramps was used. But it would not have been possible to use a single, straight, external ramp

for carrying materials to the top, which would then have needed be lengthened and raised as the work progressed, because the ramp would have been an engineering feat comparable to the building of the pyramid itself. A more likely possibility seems to be a spiraling ramp or a zigzagging sloping surface, partially covering the pyramid itself during construction. The French architect Jean Pierre Houdin even suggested that the spiral ramp of the Great Pyramid may have been internal, a kind of tunnel made inside the structure, which should therefore be still in place, over a kilometer long and probably still practicable. However, the whole length of the corridor suggested by Houdin would have had to be relieved from the weight of the huge mass above by the building of an inverted V-vault, also a kilometer long. I thus believe that we are still far from understanding exactly how the pyramids were built.

The unit of measurement used in Egypt, since the Old Kingdom, was the cubit, equal to 52.5 centimeters (a splendid example of a cubit, covered in gold, can be seen in the funerary work of the architect Kha, today at the Egyptian Museum in Turin). We prefer to work and think in integer multiples of a meter, and similarly the Egyptian architects thought in terms of integer multiples of a cubit. As a result, almost all the measures of the pyramids can be written as integer multiples of a cubit. However, I shall continue to use the meter to keep things simple.

Pyramid construction necessitated the solution of numerous geometrical problems, although the Egyptians only had to make implicit use of trigonometry, since they regularly used ratios between integers. In particular, to define the tangent of an angle (i.e., the ratio between sine and cosine), and therefore the slope of a pyramid, the architect only had to give the legs of a triangle as integers. For example, the tangent is 14/11 for the pyramid of Khufu and 4/3 for the pyramid of Khafre. In this way, to cut the casing blocks with the correct angle, the quarrymen did not have to use the cubit. For Khufu all they had to do was to count 14 (arbitrary) units vertically for every 11 of the same units counted horizontally. It was even easier for the pyramid of Khafre because the triangular section of the casing blocks formed a Pythagorean triangle with all integer sides 3-4-5 ("sacred triangle''), thus the hypotenuse could be checked immediately by counting five units on it (in many publications about the Great Pyramid it is noted that, due to the 14/11 slope, the ratio between the base perimeter and the height is 44/7, a number which coincides with 2n up to the third decimal; this fact has given rise to a series of theories, including a famous one claiming that the pyramid was conceived of as a model of a sphere and, by analogy, as a model of the earth, which are nothing but sheer nonsense).

All casing block cutting operations, probably effected with copper saws and abrasive sands (the limestone used is not particularly hard, so this method is effective), were in any case carried out with the utmost care since any unevenness of corners, however minimal, would accumulate in such a way that they could not be remedied retrospectively. It is likely indeed that the pyramid was only partially visible during the building work, since it was enveloped in ramps. It would thus emerge, like a jello from a mold, only when all the support apparatus was dismantled. The trickiest problem of all was undoubtedly the cutting of the corner blocks, that is, the casing blocks placed on the corners of the pyramid, joining two faces. To see how tricky this is, try drawing one of these blocks and calculating its measures; it is quite an instructive exercise.

Each pyramid had it own name; for example, the Khufu pyramid was called the Horizon of Khufu. We know these names from inscriptions that do not date from the time of building, but from a somewhat later date; in particular, relevant inscriptions are to be found in the mastaba of Qar, an important royal functionary at the time of Pepi II, about 200 years after the construction of the third pyramid of Giza.

Pyramids were fitted with architectural "annexes" that began to appear at Dashour and became standard at Giza. There were other much smaller pyramids, including queens' pyramids—though not all of them have inner rooms suitable for housing a sarcophagus, and no mummies have ever turned up in any of them—and the tombs of dignitaries and kinsfolk of the pharaoh who were hoping to participate in his rebirth. On the east side of each pyramid, a structure known as the "funerary temple'' was erected. It was connected by a causeway, that is, an elevated monumental path several hundred meters long, with another structure, the "valley temple,'' normally located at the outer edge of the Nile flood plain or on the shore of an artificial lake linked by canals to the river, allowing convenient arrival by boat. This also facilitated the transport of building materials to the site and perhaps also the approach of the royal funeral. These temples and causeways were architectural masterpieces in their own right; the complex in best condition today and also the most celebrated and studied is the one surrounding the second pyramid of Giza.

To summarize, building a pyramid entails excavating limestone blocks, transporting to the site, and placing them one on top of the other. But doing this for 2,200,000 blocks, as with the Khufu pyramid, requires teams working in shifts for 24 hours a day for 365 days a year, excavating, transporting, and laying one block every 5 minutes, if it was to take the 20 years traditionally attributed to the construction of the monument; simultaneously the workers also had to slave away on the construction of two megalithic temples, made of blocks weighing hundreds of tons, as well as a causeway stretching several hundred meters.

Listing all the problems they had to face in order to bring off such a feat would be too much of a digression here. It will suffice to mention that to do similar demanding work today using electric machinery, we require a similar length of time and equally enormous organizational and technical efforts. As for the pyramid builders, we are still very far from knowing how they solved their problems; we only have some hypotheses, which would have to be tested seriously, not in the haphazard way of some recent debatable satellite television ventures (about which I will say no more).

## Telescopes Mastery

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