Egyptian Pi

Hello reader!

I am here to explain another incredibly important aspect of our lives, that has affected our society for centuries:

No no no, not that type of pie (although it is quite pleasing to the stomach).

Ah, yes. One of the most famous irrational numbers in mathematics, pi.

So, what IS pi? What sentence would define pi instead of a number? People say pi is the ratio between the circumference of the circle and the diameter (this can be easily found with knowing that C= π * d, and proceeding to find π = C/d)

This lesson will be mostly a history lesson about mathematics. How have people in history calculated the exact value of pi if it cannot be expressed as a common fraction (that is, an integer numerator AND denominator)?

SIDE NOTE: In your circle geometry homework, the questions may ask you to express pi as 22/7, but if you put that exact fraction in the calculator, you won’t get exactly 3.14159… however, it is a close approximation of the irrational number.

Pi has some interesting properties: first, it cannot be expressed as a sum of any n-root of any rational numbers. Second, if using a compass/straight-edge, the area of a square can never equal the area of a given circle.

So, easily found approximations of pi. Greeks and Egyptians were some of the ancient civilizations that had found a fairly close estimation of pi, and both involved some sort of polygon approximation. We’ll look at the Egyptians’ method right now, because it’s a fairly simple method.

First of all, what DID the Egyptians estimate pi to be? 256/81, or approximately 3.16. This is fairly close to pi, being about 3.14.

How did historians find out what the Egyptians did exactly? Well, they found a manuscript from ancient history and within some hieroglyphs they found this diagram:

The Egyptians approximated pi by first drawing a 9x9 square, and inscribed a circle within it (therefore having a diameter of 9): (please excuse my horrid drawing skills, these diagrams are NOT to scale and are meant for a helpful representation)

Then they put points to divide the side length of the square into 3 parts and connected some of these points to each other to form an OCTAGON:

The Egyptians then hypothesized that this area of the octagon is roughly about the area of the circle (just by observing, it could be quite possibly true). To find the area of the Octagon, you simply took the entire area of the square (9 x 9 = 81) and subtracted the isosceles triangles on each corner of the square (which we know has a side length of 3). Having four of these triangles, the total area is 4.5 x 4 = 18.

81-18= 63 (which is the area of the octagon).

The Egyptians then proceeded to say that by taking one row and column away from the square (9+9=18) then this would also be equal to that of the square:

HOWEVER…

You see how in the bottom right corner of the square, there is one part being overlapped, and is counted twice, which makes the real value only 17 squares being taken away, leaving an 8x8 square behind. This may have caused the Egyptians to over-calculate their approximation of pi (just goes to show how a little mistake can immediately drive you off the right path!). Still, their result was only about 0.02 off, and this was several hundred (thousand?) years ago, which is impressive if you ask me.

Knowing the area of a circle is πr², the Egyptians then wrote:

π(9/2)² = 8²

And therefore:

π = 8² / (9/2)² OR π = 4 x 64/81= 256/81.

VOILA! An Egyptians interpretation of pi.

Of course, now that technology has advanced, we have calculated pi to about 10 TRILLION digits, and people have been memorizing the digits all the way up to 67000 digits. The human brain is certainly accelerating, which gives all the more opportunities to observe and question the world around you!

So long for now!

~The Octopi

Here’s a selection of pies to reward you for actually reading this: