Pyramids

This page is not in final form yet, but should still be interesting

"Pyramid" immediately brings to mind the three great Egyptian pyramids at Giza. There are others, of course, in the western hemisphere and China and other places, but none as impressive as these. They have survived because of their great bulk, but with a sadly defaced appearance, though still an impressive one. Much of what we know about them comes from Herodotus, who visited Egypt when the pyramids were still pristine, though even then about two millennia old. That is, in the life of the pyramids up to the present day, Herodotus visited at the mid-point. He asked the Egyptian priests what they had to say about the purpose and construction of the pyramids, and reported that in his history, with frequent warnings that it was only what they said, and was not necessarily so. He did not speak Egyptian, relying on intepreters, and was not allowed to visit the interiors of the pyramids. For a description of the pyramids and the present state of knowledge concerning them, please refer to the British Museum link in the References.

The first thing we must consider is the origin of the name. It does not come from the Greek he pyr, "fire," but rather from ho pyros, "wheat." It seems that a sweet cake made from honey and wheat was often a prize in athletic or literary contests, and this cake was called ho pyramous, and apparently had a pointed top. A similar geometric figure was named after these cakes, and called he pyramís, pyramídos, where the second form is the genitive singular and shows the stem for the other cases. This is, idos ending is a diminutive, so the the word apparently means simply "little wheat cake." The name of this geometric figure was then carried over, in all probability, to the immense Egyptian constructions, when they were first seen by Greeks. The Egyptian word for a pyramid, mn (vowel unknown), is something completely different, so "pyramid" is probably not derived from a non-Greek word, as suggested by Lidell and Scott. It was also probably not coined on the basis of any similarity between an Egyptian pyramid and a wheat cake.

Geometrically, a pyramid is formed from any plane polygon by connecting some point not in the plane, called the vertex, with straight lines to each of the vertices of the polygon. The distance between the vertex and the plane of the base is the altitude. A regular pyramid is formed from a regular polygon and a vertex above the center of the polygon. The Egyptian pyramids are regular quadrilateral polygons. The volume of any pyramid is one-third the product of the area of the base and the altitude. The lateral area of a regular polygon is one-half the product of the perimeter of the base and the altitude of each triangular face, considered as a triangle.

King Djoser of the third dynasty had a stack of mastabas, or elevated platforms, made such that each was smaller than the one beneath, a structure now called a step pyramid, around 2780 BCE, intended as a burial monument. The fourth dynasty extended this idea to the smooth-faced pyramid. Cheops (or Khufu), second king of the fourth dynasty, built the first, and largest, pyramid around 2700-2500 BCE. This date is about the same as, or even earlier than, the construction of Stonehenge in Britain. His (grand)son Chephren (Khafra) had a slightly smaller pyramid made, guarded by the Great Sphinx, and Chephren's son Mycerinus (Menkaure) completed the trio with a considerably smaller pyramid. These three great pyramids, with smaller pyramids and other buildings surrounding them, were built on a high ridge west of the Nile, opposite present-day Cairo.

Note that two names are often given in these accounts. The first is the Greek name, written down by Herodotus after hearing them pronounced in Egyptian by the priests. In parentheses is the modern academic name, derived from study of the hieroglyphic representations. Knowledge of ancient Egyptian and hieroglyphs was thorougly stamped out by Christians, who considered it impious, and only recovered recently after much tedious work. The vowels are still not known with any surety, since hieroglyphs do not record vowels. Furthermore, Arabic names also may be used, and the Greek names may be transliterated the traditional way, or according to more revolutionary modern conceits. In general, the reader should beware, since Egyptology is a mixture of science and creative writing, and its practitioners seem to be somewhat arrogant. Much, actually, cannot ever be known for sure. In modern work, a plausible story is often paraded as fact, and the stories change with time.

The three great pyramids are all approximately halves of regular octahedra, whose faces are equilateral triangles. The slope of the faces are ideally 54°, which the great pyramids nearly approach at 52°. For such an equilateral pyramid, the length of a side is √2 times the altitude. The altitudes of the great pyramids are 146, 144 and 66 metres, approximately. A little height has been lost due to deterioration, but these are the quoted heights when constructed. The base of Chephren's pyramid is at a higher level than Cheops', so it appears taller, but it is not. All are quite accurately oriented with the sides of the bases in the cardinal directions. They are approximately equally spaced along a NE-SW line. They were accompanied by temples on the east sides, now vanished, and with stone causeways leading down to the Nile, where again there were temple buildings. The ancient entrances to the interior passages of the pyramids were on the north side. The burial chambers were about in the centre of the pyramid, reached by inclined passages. There were also one or two other chambers. Cheops' pyramid had a subterreanean chamber, but it is apparently unfinished. There seem to be no extensive subterranean excavations, as the priests informed Herodotus there were (but some have apparently recently been discovered). Boat pits, for ceremonial "solar boats" made of wood have recently been discovered near Cheops' pyramid. The southern boat has been excavated and reconstructed, while the western boat has been left in place. I do not know why the burial chambers were not more completely walled off; all have been robbed, of course, apparently in early times.

The mass of the pyramids is composed of sandstone or sandy limestone blocks quarried a short ways to the southeast, and fitted dry with great accuracy. Each block weighs about two tons, and there are 2.3 x 10⁶ such blocks in Cheops' pyramid (or so they tell me). Such blocks are quarried by wedging them free, taking advantage of natural fractures, and then smoothed laboriously with chisels. Only bronze (impure copper) tools were available, but they were evidently satisfactory. The blocks were moved by dragging them generally downhill on a wet clay surface. A coefficient of friction of, say, 0.1 would mean that a tractive force of 400 lbs each would be sufficient, easily obtainable with a crew of 100 to 200 men. At the construction site, the blocks could have been raised either by pushing them up a ramp (requiring a good deal more force), or by machines made of timber. The piling up of the pyramid was a big job, but not an impossible one.

Some attempts at reconstructing the means of handling these blocks have supposed that there were timber rollers beneath them, or sledges. People who have actually tried this expedient on TV shows have found out just how unsatisfactory it is. The rollers sink into any soft road (one not paved) and disintegrate rapidly. This means of moving heavy objects is a myth. It is much easier simply to drag them on a slick clay surface. Also, the lack of steel tools is merely an inconvenience. The Egyptians were experts in stone, about the only permanent construction material available to them. They could even use the hard syenite of Syene; sandstone would have been easy to handle.

Once the sandstone core was finished, the pyramid was sheathed in white limestone from Tura across the Nile in the Arabian Mountains. These blocks, smaller than those used for the core, were dragged to the Nile, put on boats to cross the river, and then dragged up the stone causeway to the construction site. The causeway provided a hard, smooth surface of constant slope that made the job possible. The casing blocks were then raised by timber machines (so say the priests) from step to step to their final positions. This work was done from the top down. When it was done, the pyramids were smooth and gleaming white. Only Chephren's pyramid still shows some casing at the top.

Thin metal was used to sheathe the pyramidion, probably gold or electrum. This kind of top was also used on obelisks, a later structure that is really a very slender pyramid made from a single stone. It does not take much metal to make a very pretty top. The greedy Christian or Arab who first went up to strip it off was probably deeply disappointed by the small amount of booty and the amount of work required to seize it.

The limestone sheathing was nearly all removed to be burnt for lime, for constructional purposes, or used directly on acid soils, in later years. Limestone temples and other structures met the same fate. The Sphinx, because it was sandstone, survived, as did the cores of the pyramids. Most of the impressive stone structures of ancient Egypt survived intact until the end of the Roman period, after which they rapidly degenerated.

The main constructional material in ancient Egypt was not stone, but dried mud brick or adobe. This was a mixture of clay, straw and dung properly selected and mixed, and rammed in a mold. When thoroughly dry, it formed a very serviceable constructional material. Nearly all secular buildings, from the Pharoah's palace down to the serf's hut, were made of adobe. This includes even what Herodotus thought the most magnificent building in Egypt, the Labyrinth, with 3000 rooms. The Labyrinth, and all such buidings, are now only shapeless mounds of earth, as indeed are all the famous cities of ancient Egypt. Only the somewhat unusual sacred stone tombs, temples and monuments survive, strongly biasing our archaeological knowledge of ancient Egypt.

Egypt had very few natural mineral resources. There was a little copper in the northeast, and a valuable deposit of trona in the northwest. Trona is, in Greek, to nitron, which should not be confused with nitrates. It is a sodium hydrogen carbonate, a weak alkali. The ancient world was severely restricted by a lack of strong alkalies and acids, so that many easy chemical processes were not available to them. There was no alunite in Egypt, and so no alum. Gold was available from Nubia, and there was more copper in Sinai, for which Eqypt sent expeditions to acquire. Amethyst and turquoise were also sought. Metals, gems and wood all had to be obtained mainly by trade.

Sandstone was a widely available building stone, relatively easy to quarry and use. Sandstone varies widely in hardness, from almost unworkable quartzite to loose, friable rock useless for building, so good quarries were not found everywhere. Limy sandstone or marl occurs in varieties intermediate between sandstone and limestone. A variety of coarse-grained sandstone called graywacke was also quarried near the Red Sea. White limestone was available from the quarries at Tura, on the Arabian side of the Nile opposite Giza, to be used for finish work. Alabaster was also available from Hatnub, east of the Nile about halfway between the Fayyum and Abydos. This is a fine-grained gypsum, beautiful and easily worked but too soft for external use. Near Elephantine, at the first cataract, syenite was quarried. This is a dark, very hard igneous rock of great durability. Near the second cataract, at Buhen, diorite, a similar rock, was available. Such rock is often called just granite, but the geologist would notice the almost complete lack of quartz and call it syenite or diorite instead.

Mycerinus sheathed the lower half of his pyramid in what Herodotus calls "Ethiopian rock," which means some kind of this syenite or diorite, floated down the Nile to his pyramid. This pink rock made a nice contrast with the upper limestone casing. Some of the pink rock still survives around the modern tourist entrance.

Many of the 80 pyramids in Egypt were made with adobe cores, which was much easier than a sandstone core, since the blocks could be made in place and did not have to be dragged to the site. These cores, stripped of their limestone facing, have fared worse than sandstone cores. At least one pyramid (at Meidum, probably Snofru's) was made from limestone alone, and so was avidly robbed in later days, leaving a rectangular limestone pillar. This was earlier than the pyramids at Giza, possibly the original design.

The quarrying, working, transport and raising of the large blocks was a difficult job, but one well within the capabilities of ancient Egyptians. The most remarkable part of building the pyramids was not these essential activities, but the planning and control of the works, something about which we have very little knowledge. If, as the priests told Herodotus, it took twenty years to build Cheops' pyramid of over two million blocks, this was only 7305 days if every day was a working day, so about 274 blocks had to be placed on a typical day. Modern scholars scoff at the priest's claim that 100,000 men were at work, but this would only be 365 men per block placed. If the work was done only when the Nile was high, the large number of workers seems about reasonable, in fact.

These workers would certainly not be slaves in the modern sense, or even in the Greek sense, but simply serfs put to profitable occupation instead of idling about getting into trouble. Egyptian society was, in a sense, the large extended family of the king. The workers were apparently supplied with all they needed, including beer, food, shelter, clothing and medical care, in addition to tools and rope. There was, no doubt, considerable grumbling, but probably not much worse than this.

It's easy to make a pyramid model from construction paper or light cardboard. Besides this material, you'll need a compass, triangles, scissors, pencil, paste and the sharp point of dividers or equivalent for scoring. These instructions yield a regular quadrilateral pyramid. Set the compasses to draw a circle of radius equal to the length of a side of the base, which is √2 times the altitude. The figure at the right shows the construction, which can be made without resetting the compass. Although the right angle can be constructed with the compasses, it is easier just to use the triangles. To simulate the original appearance of the Pyramids, use white or cream construction paper. Also draw the sides of the pyramidion if you are going to paint one on, aided by a small circle drawn around the centre with a radius of about 1/20 the side length.

Cut out the pattern with scissors, allowing tabs for pasting along the three free sides of the base and the available side of a lateral surface. Score on the outside of each folding line with a divider point or similar. Be careful to do the scoring accurately; the whole appearance of the model will depend on it. First fold the lateral surfaces, and paste along the tab, both sides of which are accessible to apply pressure. This establishes the pyramidal shape, so that the base needs only to be tacked on so it does not fall off. Using gold paint, carefully paint on the pyramidion.

The very pleasant British Museum website on Ancient Egypt will be found at Ancient Egypt. Click on the pyramid for information on the pyramids. The external view contains links to explanations of the several features; just click on the feature. This feature is not (apparently) pointed out explicitly. There are, however, no numbers here.

Among the great number of websites with pyramid information, Egypt Sites seems benign and helpful. Many US .com sites, purporting to contain educational information, are predatory and should be avoided, even that of PBS, which pushes multiple cookies for some nefarious purpose known only to them. Other sites deposit webdroppings of various kinds. Many sites are devoted to the mystical rubbish surrounding the pyramids, such as extraterrestrial or Atlantean builders, or resemblances to the earth or sky. The pyramids were oriented by means of simple astronomical observations, apparently accurate enough that precessional effects can be detected.

Composed by J. B. Calvert
Created 2 December 2003
Last revised 4 December 2003