As shown in some YouTube videos, the AE used a binary form of mathematics for multiplication and division. Let's try 311 divided by 99. Comes out to 3 and 14/99ths. This was the Egyptian form of pi used by the priests (in my opinion). Commoners might use 22/7 for most purposes, because lower numbers are easier to deal with in the real world if you don't require high accuracy. For something like the Great Pyramid, though, they would have made the extra effort. 22/7 is the 6th in the list of fractional approximations of pi and 311/99 is the 13th, vastly more accurate.

How did they know about these fractions? By simple trial and error. They would have tried using various numbers of units for the diameter of a circle and see how many of those units fit around the perimeter. With a diameter of 7 you would find that 22 units fit around the circle nicely but if you made a big circle, like 100 cubits diameter, you would notice the discrepancy because it would be about 1 palm at that scale. So you would find that at large scales 311/99 works a lot better.

Now you might wonder why they stopped at that one instead of going to 333/106 or 355/113. Hard to say. Maybe because the remainder, 14/99, is remarkably close to sqrt 2 divided by 10, because it is .141414 repeating. So you might say that the circumference of a circle with diameter of 1 unit is 3 units plus 1/10th of the diagonal of a square with sides the same length as the circle's diameter. It's easy to work out 1/10th of the diagonal when using cubits because there are 40 digits in the diagonal of a 1 cubit square, so you just add 4 digits for every cubit of diameter after multiplying that diameter by 3. You have to be aware that the Egyptian digit was not 1/28th of a cubit, as commonly thought, but 1/40th of its diagonal. Petrie explains that in the "Values of the Cubit and Digit" section of his book about Giza [www.ronaldbirdsall.com] .

So it turns out that this 311/99 version of pi is quite convenient when using Egyptian units of measure. That's probably why it was used for the Great Pyramid's shape, instead of the much cruder 22/7. Its base is divisible by 311 but not 22. 622 remens is exactly 311 x 2 while 439.82 cubits is significantly less than 22 x 20, which would be 440. Therefore, it is obvious that they used 311/99 instead of 22/7 for the pi pyramid of Khufu.



Edited 1 time(s). Last edit at 08/10/2010 07:28PM by northstar2595.