Phi, Pi and the
Great Pyramid of Egypt at Giza
There is still some debate as to whether the Great Pyramid of Giza in Egypt, built around 2560 BC, was constructed with dimensions based on phi, the golden ratio. Its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty.
There is compelling evidence, however, that the design of the pyramid embodied these foundations of mathematics and geometry:
- Phi, the Golden Ratio that appears throughout nature.
- Pi, the circumference of a circle in relation to its diameter.
- The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c².
First, phi is the only number which has the mathematical property of its square being one more than itself:
Φ + 1 = Φ²
1.618… + 1 = 2.618…
By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this:
This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272. The ratio of the height to the base is 0.636.
According to Wikipedia, the Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet). This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy. If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.53567, so the difference of only 0.3567 meters appears to be just a measurement or rounding difference.
The Great Pyramid has a surface ratio to base ratio of Phi, the Golden Ratio
A pyramid based on a golden triangle would have other interesting properties. The surface area of the four sides would be a golden ratio of the surface area of the base. The area of each trianglular side is the base x height / 2, or 2 x Φ/2 or Φ. The surface area of the base is 2 x 2, or 4. So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base.
The Great Pyramid also has a relationship to Pi
There is another interesting aspect of this pyramid. Construct a circle with a circumference of 8, the same as the perimeter of this pyramid with its base width of 2. Then fold the arc of the semi-circle at a right angle, as illustrated below in “Revelation of the Pyramids”. The height of the semi-circle will be the radius of the circle, which is 8/pi/2 or 1.273.
This is less than 1/10th of a percent different than the height of 1.272 computed above using the Golden Triangle. Applying this to the 146.5 meter height of the pyramid would result in a difference in height between the two methods of only 0.14 meters (5.5 inches).
Its near perfect alignment to due north shows that little was left to chance
Some say that the relationships of the Great Pyramid’s dimensions to phi and pi either do not exist or happened by chance. Would a civilization with the technological skill and knowledge to align the pyramid to within 1/15th of a degree to true north leave the dimensions of the pyramid to chance? If they didn’t intend the precise 51.83 degree angle of a golden triangle, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees? If the dimensions of the pyramid were not based on both phi and pi, would it not be most reasonable to assume that phi was used since it is based on the visible base of the pyramid and not an invisible circle with the same circumference as that base?
Other possibilities for Phi and Pi relationships
Even if the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter and the golden ratio that appeared in nature, it’s difficult to know if they truly understood the actual decimal representations of pi and phi as we understand them now. Since references to phi don’t appear in the historical record until the time of the Greeks hundreds of years later, some contend that the Egyptians did not have this knowledge and instead used integer approximations that achieved the same relationships and results in the design.
A rather amazing mathematical fact is that pi and the square root of phi can be approximated with a high degree of accuracy using simple integers. Pi can be approximated as 22/7, resulting in a repeating decimal number 3.142857142857… which is different from Pi by only 4/100′s of a percent. The square root of Phi can be approximatey by 14/11, resulting in a repeating decimal number 1.2727…, which is different from Phi by less than 6/100′s of a percent. That means that Phi can be approximated as 256/121.
The Great Pyramid could thus have been based on 22/7 or 14/11 in the geometry shown about. Even if the Egyptians only understood pi and/or phi through their integer approximations, the fact that the pyramid uses them shows that there was likely some understanding and intent of their mathematical importance in their application. It’s possible though that the pyramid dimensions could have been intended to represent only one of these numbers, either pi or phi, and the mathematics would have included the other automatically. We really don’t know with certainty how the pyramid was designed as this knowledge could have existed and then been lost. The builders of such incredible architecture may have had far greater knowledge and sophistication than we may know, and it’s possible that both pi and phi as we understand them today could have been the driving factors in the design of the pyramid.
Construct your own pyramid to the same proportions as the Great Pyramid
Thanks go to Jacques Grimault for these insights, and for other fascinating facts and speculations presented about ancient pyramids in the movie on “The Revelations of the Pyramids.”