Tuesday, August 25, 2009

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.

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