domingo, 11 de maio de 2008

Pi, Phi and the Great Pyramid

may 11, 2008

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.

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