Pi and the Great Pyramid of Giza
Author:
Anvesha Kushwah
BSc.
Mathematics (Hons.)
The oldest of the Seven Wonders of the Ancient World, The Great Pyramid of Giza, also known as the Pyramid of Khufu or the Pyramid of Cheops, situated in Cairo, Egypt, is really a thought provoking enigma.
The value of the mathematical constant \(\pi\) seems to have designed into the Great Pyramid to a value of about 3.1419. But we know that this value of \(\pi\) was not discovered at that time with such accuracy. A very interesting point here is that how did the Egyptians know or use an approximate value of \(\pi\)? The explanation of this question involves a simple scientific logic. Let us find it out.
Measurements of the Great Pyramid of Giza
We can’t get the exact measurement of the Great Pyramid because of the removal of the outer limestone layer and the top capstone for other construction projects. So the measurements vary slightly according to the different sources.
The Egyptians used a different unit of length measurement, royal cubit. A royal cubit is the distance from the elbow to the extended middle finger.
\(1\ Royal Cubit = 52.37cm\)
The original height of the Pyramid was about 146.64m or 280 cubits before the removal of the top capstone blocks. The length of the four sides of the Great Pyramid of Giza vary from 230.25m to 230.45m or about 440 cubit long (or 230.25m, 230.36m, 230.39m, 230.45m).
Now with these details, if we take half of the perimeter of the Pyramid’s base
And if we divide it by its height, the result is very close to \(\pi\). \[\frac{460.275m}{146.64m} = 3.1419 = \pi \]
Or alternatively if we use the cubit unit measurements we get \[\frac{440\ cubit * 2}{280\ cubit} = \frac{22}{7} = 3.1429\]
The key to the mystery of how the Egyptians incorporated \(\pi\) was derived by the another mystery of The Great Pyramid of Giza. How were they able to make the lengths of all the four sides of the Great Pyramid so precise (within 10 cm) to each other over a distance of 230m? The only technology that existed at that time that would give the Egyptians the precision they needed to match all the four side lengths was a trundle wheel. If \(\pi\) was involved, it indicates that a wheel, which has \(\pi\) built-in, was somehow involved. A trundle wheel is just a wheel that is rolled a number of times in order to measure distance and it is a technology that is commonly used even today on sports fields and by surveyors.
It is believed that the Egyptians made a trundle wheel carved out of rock with diameter of exactly 1 cubit. Then they rolled it on flat ground so the trundle wheel rock rolled exactly 140 times around for each of the four sides of the Pyramid. Therefore, it was rolled a total of 280 times around from one corner of the Great Pyramid to the farthest corner on the other side. Through this they obtained precise locations of the corners of the Great Pyramid and then they started to pile the massive blocks to build the pyramid. They piled the blocks of the pyramid until they obtained a height of 280 cubits. The point to note here is that they couldn’t easily roll the trundle wheel straight up, so instead, they counted the same number of cubits to obtain the height. So they counted 280 turns of the trundle wheel from one corner to the opposite corner and measured 280 cubits high from the ground to the peak. Because they used a trundle wheel which has the value of \(\pi\) built into it with the relationship between the circumference and the diameter, the value of \(\pi\) was automatically built into the Great Pyramid without the Egyptians knowing the value of \(\pi\).
But the next point to observe is that even with a base having four exact measured sides, the Egyptians could have made a rhombus not a square. According to one source, the difference in the distances between the opposite corners is within 17cm, which means they indeed obtained corner angles very close to \({90}^{\circ}\). They got this angle by measuring the diagonals to be of matching length or used the Pythagoras theorem.
The Pyramid’s dimensions suggest that the Egyptians may have known about some Pythagoras Triangles. According to one source they called such triangles as “Holy Triangles”. But if we apply the Pythagoras Theorem, we obtain 197.99 turns for the diagonal, or hypotenuse, which is not an integer value but still very close. Perhaps, the Egyptians didn’t know about Pythagoras triplets, but instead knew “Pythagoras Isosceles Triplets” which are close to being integers. And as a matter of fact, there will never be exact Pythagorean isosceles triplet because the hypotenuse will be \(\sqrt{2}\) times the triangle side and \(\sqrt{2}\) is irrational. They used 70-70-99 triplet option to make the pyramid design much more feasible.
One more interesting fact about this pyramid is that 140/99 = 1.4141 . . . which is close to \(\sqrt{2}\), so were the Egyptians aware of this number? But this is the case with all the Pythagorean Isosceles Triplets and of course these patterns will naturally always be built-in with a square base.
But if the Egyptians used the above design of 70-70-99 triplet, they would have used exactly 99 turns of the trundle wheel for the diagonal which would have made the expected right angle inaccurate because of the error in Pythagorean Isosceles Triplets. Using the Cosine Law: \(c^{2} = a^{2}+b^{2}-2ab \cos C\) which gives \(C={90.00585}^{\circ}\), which is slightly more than right angle. This implies that the Pyramid would be slightly indented half way on each side towards the middle of the pyramid because the angle in the middle would be double of \({90.00585}^{\circ}\) which is \({180.0117}^{\circ}\). And this indent actually exists in The Great Pyramid of Giza. It is the only Pyramid to have this characteristic. So the above 70-70-99 design seems to confirm the hypothesis of the construction of The Great Pyramid of Giza.
