The Slope of the Sides of a Pyramid

The ancient Egyptians knew how to computed the slope of the sides of a pyramid which we know from the Rhind Mathematical Papyrus and other ancient scrolls. The value they used was known as the seked which was the base of a right triangle whose height was a unit length. It is similar to an Egyptian fraction which were the sums of unit fractions. The fractions were just a series of divisors.

The procedure for computing the seked was to take half the base of the pyramid and divide by the height. This makes computing the slope of the Great Pyramid of Giza especially simple since the ratio of the base to height was 11/7. The seked is just 11/2 or 5½ (palms) since the unit length, the cubit, is 7 palms.

The pyramid designers needed to keep track of the slope of the pyramids since they were working near the limits of the construction material. The Meidum Pyramid may have collapsed due to internal stresses which can result in cracks within the structure. A sand dune similarly has a critical slope. It is believed that the Great Pyramid was reentered for inspection after it had been sealed because there is a tunnel from beneath the pyramid to the Grand Gallery that was concealed afterwards.

It is doubtful that 11/7 had anything to with π but may have just been a convenient slope. Still the scribes were devoted to Thoth and may have shared his attitude on secrecy. Their education consisted of figuring things out for themselves. Problems in the scrolls were worked out in detail for the reader but he had to determine the general method. The problems just illustrated the procedure involved.

The Royal Cubit & Foot of the Fourth Dynasty

A significant event in the history of Ancient Egypt was the unification of Upper and Lower Egypt. This was accomplished by focussing on similarities rather than differences. The subsequent events show what happens if two people work with each other instead of against each other. The central administration of the Old Kingdom resulted in a new class of scribes and a general advance in society over previous times.

But our main focus is the royal foot (remen) whose use can be seen in the design of the pyramids of the Old Kingdom. Let's take a look at the Great Pyramid of Giza whose original dimensions are estimated to have a base of 230.5 m and a height of 146.6 m. What does this tell us? Well, the ratio of the base to the height is 1.57231 which doesn't say much by itself but we can look at its continued fraction.

Ignoring the sequence after 1,1,1,2 tells us that the ratio is nearly equal to 8/5 while ignoring that after 1,1,1,2,1 suggests a ratio of 11/7. Comparing these multiples of the royal foot and cubit with the dimensions of the pyramid gives the following information,

which shows that one gets a best fit for 11/7 with a scale factor of 40 cubits or 70 feet. This implies that in Khufu's time,

1 royal foot = 299.35 mm

1 royal cubit = 523.86 mm

which is very close to the values for the cubit stick from the 18th Dynasty.

An interesting coincidence is that,

11/7 = 1.5714

π/2 = 1.5708

π/2 is the arc of a circle corresponding to 90°.

Empowerment

There is an ancient Egyptian fable about the Book of Thoth which illustrates the sense of empowerment. It is from the Ptolemaic period when the Greeks ruled Egypt and so is several periods removed from the time of Pyramid building. It is not known how old the story is but was written on papyrus in Demotic and found in the grave of a Coptic monk at Thebes. Thoth is a god of wisdom and is attributed as the source of writing and the arts and sciences. What is interesting about the story is that Thoth goes to Ra to be empowered to seek vengeance for the theft of his book. The moral seems to be that even kings are subject to the will of the gods.

Another aspect of the story is the secrecy associated with the Book of Thoth. It was hidden away and protected by a series of defenses. But no matter how well something is protected the defenses can always be defeated.

Thoth appears in the story to have a dark side. While Ra, the Sun god, is the source of truth and light Thoth, the Moon, is more obscure. To obtain vengeance on the thief he first gets power over him by taking him and his family to the land of the dead. Thoth is jealous of his power and it is not easily taken.

Pharaoh

The title of Pharaoh was not solely a personal honorific. Literally it means "great house" so the power of the Pharaoh extended to those who served him. It is somewhat akin to the "Crown" in England and the "Presidency" in the US. He was the Egyptian Chief of State. Servants were referred to as "arms". Their power came from the Pharaoh whose power ultimately came from the gods.

An Emblem of Authority

The cubit, foot and other units of length can be viewed as a tool to extend the Pharaoh's power throughout Egypt. The sacred foot was an emblem of power as the ideographs below indicate. There is an ideograph similar to that of the tcheser (foot) in which the hand is holding what appears to be a scepter. The royal cubit may therefore be a symbol of excellence.

One of the duties of the scribes was to distribute goods. When precision was required the royal measures were beyond question.

Tcheseru

The royal cubit was considered sacred and used to determine the dimensions of temples. I found the hieroglyphics for a tcheser length in Wallis Budge's Egyptian Hieroglyphic Dictionary which confirms its special status. The name also suggests a connection with the Eye of Horus fractions mentioned earlier. It could be that tcheser conveys the meanings hallowed, exalted and royal. There is a sense of truth and justice associated with it also. Copies were made of the royal cubit for the use of workers and compared with the original monthly. The penalty for tampering with the royal cubit was death. For more information see Thoth: Architect of the Universe by Ralph Ellis.

The Royal Cubit in the 18th Dynasty

On 18th Dynasty Egyptian cubits the 20th finger is labeled a remen. The 16th finger is indicated as being a "t'eser" and the hieroglyphic symbol on the 15th and 16th fingers is similar to the remen but overlaps the border between them. The glyph appears to be that of a forearm with the hand grasping a rod which can be seen more clearly on the cubit at the Museo Egizio in Turin, Italy. There is a picture of this cubit stick in Mathematics and Measurement by O. A. W. Dilke on p. 23.

t'eser (foot) indicated as being 16 fingers in length

Mathematics in the Time of the Pharoahs by R. Gillings p.220

If one does an actual measurement one does get a value very close to a foot. If you want to check this measurement I would recommend an assistant since balancing the tape measure while taking a picture is rather difficult.

measuring the foot