Pi, Phi and the Great Pyramid - UPDATED

Al Ahram Weekly (Assem Deif)

UPDATE:

Author Assem Deif emailed me yesterday to ask me to post a correction that was not made in the original article

The editor of Al-Ahram wrote a sentence in line 4 of the 2nd paragraph, he said:
They worked out that 3 1/8 is less than Pi....

The original sentence in my original article was:
They can work out that....

Thanks very much to Mohamed Amin for the above link. It is an older Al Ahram article.

Assem Deif investigates the values -- not the symbols -- of the last of the Wonders of the Ancient World

We can forget all the ideas crediting Atlanteans or space aliens with building the Great Pyramid of Giza, and instead imagine ourselves travelling back in time in H G Wells's time machine to try and work out not how the ancient Egyptians built this enormous edifice, because this lies beyond our present understanding, but rather what we can best judge to be its most appropriate proportions. Then, however, there were no electronic calculators, only ropes and rods.

Constructing right angles at the four corners of a pyramid is easy. To do it, history tells us that the Egyptians were aware of the ratios 3:4:5 as the side-lengths of a right-angle triangle. Many old kingdom pyramids adhere to these ratios. The Egyptians also knew a rough value of Pi (the value, not the symbol) as the ratio between the circumference of any circle and its diameter. They worked out that 3 _ is less than Pi, and Pi is less than 3 1/7, i.e. Pi lies between the rational number 22/7 and the Babylonian value. This can be done by constructing a circle of diameter AB and laying the latter on its circumference, starting from A, once until C then D then E, to conclude that Pi is greater than 3. The remaining part EA from the circumference is laid down again on the diameter AB, so seven times EA is less than AB which in turn is less than eight times EA, or EA/AB is greater than 1/8 and less than 1/7.

To work out an Egyptian value for Pi from the dimensions of the Great Pyramid, they had a unit of length called the Royal Cubit (about 0.524m). By transforming the pyramid's height and base from western units (feet or metres) into cubits, it becomes evident that the designed height measures 280 cubits and the base 440. These figures have been worked out by Egyptologists, and in my view are the only plausible dimensions. Neither -- from the logical point of view -- were any fractions added to these values, considering the large dimensions of the structure. By dividing half the base by the height (cotangent of the slope angle) one reaches the ratio Pi/4. It then follows that the Egyptian value for Pi is 3.142857143 being equal exactly to 22/7, and this is the value, in our view, that was reported later by Archimedes, who studied in Alexandria.

What about the Egyptian value for the Golden Ratio Phi?