How to Measure π in the Sands of the Nile?
by Corina Murg
Why is my first story about π? After all, what mathematical concept is more popular than π? On March 14th we will celebrate Pi Day, and this brings to mind Petr Beckmann’s book, A History of π, and his beautiful rendition of measuring π in the sands of the Nile.
You are sitting on a sandy beach on the banks of the river Nile, in Egypt. The year is 2000 BC. Your homework is to find the ratio of the circle’s circumference to its diameter. You are given the following aids: ropes, sticks, and charcoal. No pencils, no paper, no compass, no calibrated ruler. Oh, I forgot, there is one more tool you can use: the sand.
Start by finding a flat patch of wet sand and drive a stick in. This will be your central stick that will create the center of your circle. Attach the end of a rope to the central stick and tie the other end to another stick. Keep the rope stretched and draw a circle in the sand by moving the second stick.
Choose a point A on your circle. Use a longer piece of rope and stretch it from A across the center O until it touches the circle at B. Use charcoal to mark the length AB on the rope. This is the diameter of your circle and your unit of length.
Use your diameter rope to measure the length of the circle (ie circumference) in terms of AB. Lay the rope into the groove in the sand, starting at A. Make a mark after the first unit of length is measured along the circle, at C. Lay the unit of length a second time, from C to D, and then a third time, from D to E. So far you’ve showed that the circumference is three times the diameter, plus the extra length EA.
To measure the length EA mark it on a piece of rope. Stretch this rope along the length of the diameter AB. EA will go between 7 and 8 times into the length of the diameter. Your approximation of π is between 3 and 3, not insignificant given the rudimentary tools you were restricted to work with.
Congratulations! You just found π in the sands of the Nile.