A Reconstruction, Assessment, Error Analysis and Simulation of a Method for Measuring {\pi} That Could Have Been Used 3000 Years Ago

TL;DR

This paper explores a plausible ancient method for measuring Pi, based on historical clues, and uses simulation to assess its accuracy and likelihood of success with ancient technology.

Contribution

It proposes a specific iterative measurement method for ancient times and evaluates its feasibility and accuracy through computer simulation.

Findings

The method could have been used 3000 years ago with about 75% success rate.

Simulations show the method's errors are within acceptable bounds.

The coded value 111/106 can be obtained through this ancient measurement approach.

Abstract

There is little known about the methods used by the ancient Babylonians and Egyptians to arrive at their recorded estimates of the value of Pi. A surprisingly accurate estimate of Pi was recently revealed coded within a verse in the book of 1 Kings, the value of which suggests how it might have been measured. The coded value is 111/106, which is a continued fraction representation of Pi/3. This suggests that the value may have been measured using an iterative measurement of remainders when comparing the two lengths C (circumference of the circle) and 6R (6 times the radius). This article describes a method that could have been used 3000 years ago to make such a measurement, the expected measurement errors, and a computer simulation that assesses the chances of such a method succeeding in obtaining the coded value of 111/106. The result indicate that with the technology available at the time the proposed measurement method would have been possible and would have had about a 75% chance of producing the result 111/106 after a few hundred measurements.