## Ancient PI (π)

While a number of slightly different approximations of π have been used for practical purposes since ancient times, French mathematician François Viète is typically credited with perfect mathematical approximation of the constant. Like the Greek mathematician Archimedes, Viète understood that a circle’s circumference could be approximated by an infinite number of polygons inscribed inside a circle(1), as illustrated below:

Understanding that a circle’s circumference could be approximated by a polygon perimeter with an infinite number of sides, Viète consequentially deduced the following formula for calculating π in 1593:

This formula allowed Viète to accurately approximate the constant π to nine digits, as 3.141592654.

Long before the Greek mathematician Archimedes used polygons to approximate π, it is apparent that Egyptian and Babylonian builders recorded more practical approximations of π. For example, a Babylonian clay tablet (~1900-1600 BC) implies a value of 25 / 8, which is equal to 3.125. Also, the Rhind Papyrus of Egypt treats π as 16 / 9, which is equal to 3.1605. Nearly a thousand years later in India, Sanskrit texts recorded π as 9785 / 5568, which is equal to 3.088. Indian mathematicians also approximated π as √10, or 3.1622. After 100 AD, Ptolemy expressed π to four digits as 3.1416 - close enough to build almost anything in the ancient world.

The Great Pyramid of Giza in Cairo, Egypt

Perhaps of greater relevance to the history of π is Egypt’s Great Pyramid. Giza's wonder of the ancient world was built with a perimeter of about 1760 cubits and a height of about 280 cubits. Some have noted that the ratio was equal to 44 / 7, which is two times the familiar 22 / 7 ratio, which is still used today to conveniently calculate π to two decimal places, or 3.14. This Giza Pyramid geometry is summarized in the illustration below:

The PI Constant as Conveyed in the Great Pyramid

While some Egyptologists try to suggest this recorded “ratio in stone” is but a matter of mere coincidence, it would be absurd to propose that people building on a scale such as Giza would not be able to arrive at 3.14 by means of empirical measurement – provided that their engineers and mathematicians were remedial. After all, the Pyramid builders aligned sides of the square pyramid base to true north within 4 minutes of arc (equal to a total of 0.067 degree), which implies that the accuracy with which they measured or surveyed was twice that of their convenient π approximation! If they had the technology to place over 2 million blocks while controlling the base dimensions of the structure to greater than 0.1% accuracy, surely they could have measured the ratio between a circle and its diameter with equal or greater precision.

Keeping Things Down to Earth

Extra! Extra! Read all about it! News flash: The earth is round. Actually, this isn’t a news flash, it's old news. In fact, the Egyptians knew this idea and they knew it well.Demonstrating their mastery of earth measurement, the Egyptians built the four sides of the Great Giza Pyramid very deliberately, making the sides of the structure concave using the very same dimensions. Obviously, incorporating the spherical arc dimensions into what otherwise is perceived to be a perfectly flat surface (i.e., one of the faces of the pyramid) took deliberate foresight and knowledge. After all, the features remain practically invisible to the naked eye, and only become visible during optimal lighting conditions. Although the Pyramid's concave features became apparent via 19th century survey results, the discovery is often attributed to a World War II airplane pilot, who captured the image below during a flyover of Cario during optimal sunlight/shadow conditions:

Photo by P. Groves, British Air Force pilot (1940)

This concave geometry becomes more apparent with the aid of satellite imagery, as shown below:

Ikonos satellite image of the Great Pyramid

While skeptics can easily dismiss the relationships between the structure's and the 22/7 ratio for π, skeptics would be hard pressed to explain how the Egyptians determined the curvature required to mimic the earth dimensions. It's not as if the builders could simply sweep and arc with a string sized equal to the earth's radius. To the contrary, they had to calculate it, and they would have needed to understand the π constant in order to carry out their construction plans.

Although several scholars have debated Israel's involvement in creating the Giza Pyramid, few would suggest that the Pyramids were not standing at the time of Israel's Exodus. Moses was said to have been "Learned in the wisdom of the Egyptians". Clearly, it's hard to believe that Moses would have left Egypt at 40 years old not knowing how to calculate the circumference of a circle.

Next Page: God of PI (π)

References:

1. http://commons.wikimedia.org/wiki/File:Archimedes_pi.png

2. http://www.theglobaleducationproject.org/egypt/studyguide/gpmath.php