Being the oldest site among the original Seven Wonders, it is not surprising that the Great Pyramid of Giza attracts more than 14 million tourists a year. Seeing these huge structures in the vast, sandy desert, many of these 14 million tourists do not realize the mathematical precision used to build the pyramids. The pyramids reveal the advanced knowledge that the ancient Egyptians possessed in the fields of geometry, trigonometry, and many other mathematical principles. Or, are they just coincidences?
It was John Taylor in 1859 who first offered the idea that the pyramids are more mathematical than they seem. After him, many other mathematicians agreed and elaborated on his statements. However, more recently, other mathematicians have cast doubt on these statements, claiming the results are more likely to be a coincidence. But whether they are a coincidence or not, the mathematical constants in the pyramids present an interesting topic.
Some dimensions and ratios of the Great Pyramid, namely, seem related to three fundamental mathematical constants: π (pi), e (Euler’s number), and ϕ (the golden ratio).
The Ratio of Height to Base:
Let’s start with the dimensions of the pyramid. The original base length of the Great Pyramid is estimated to be 230.4 meters (756 feet). Consequently, the perimeter is 926.1 meters (3038 feet). The original height is estimated to be 146.6 meters (481 feet). Taking half the perimeter of the pyramid and dividing it by the height results in 0.3159, a number remarkably close to π. Another way of stating this is by using the slope. Each face has an angle of 51 degrees 52 minutes, or 51.87 degrees in decimals. Using the tangent, this converts to a slope of 1.2738, extremely close to 4/π, which is approximately 1.2732!
Now continuing with these numbers: The sum of the degrees of each angle in a triangle is 180 degrees. Following this, we know the upper angle is 76.28 degrees (180 - 2 * 51.86). Now, 4 x 51.86 / 76.28 = 2.7195. Does this look familiar? It is immensely close to e (approximately 2.7182). Turns out nicely!
Golden Ratio
Another interesting fact about the pyramids relates to the golden ratio. A unique quality of the golden ratio is that its square is exactly one added to itself: ϕ + 1 = ϕ^2. Combining this result with Pythagoras's theorem, one can create a right-angled triangle with base 1, height √ϕ, and hypotenuse (diagonal) ϕ. Note then: (1)^2 + (√ ϕ)^2 = 1 + ϕ = ϕ^2. This is known as the Kepler triangle. The Kepler triangle has a height-to-base ratio of 0.6360. Now looking at the dimensions of the Great Pyramid of Giza, one can find a height-to-base ratio of 146.6 : 230.4 = 0.6360. In fact, the dimensions of the Great Pyramid vary by only 0.025 percent from a perfect golden triangle pyramid!
Some historians take this point a bit further. Placing the moon perpendicular to the earth on the North Pole, the distance between the center of the earth and the center of the moon is 8,116 kilometers. The ratio between the distance from the center of the earth to the center of the moon and the diameter of the earth is 0.6362. Again, very close to the Kepler triangle and, thus, the Great Pyramid of Giza. Some dare say the pyramid is thus modeled off the dimensions of our moon and earth.
Cardinal Points
A seemingly smaller, yet still very interesting fact is the pyramid’s alignment with the cardinal points. Each edge faces either exactly north, south, east, or west, up to a fraction of a degree. The northern edge is only 1/15th of a degree off from the true north. Were the ancient Egyptians astronomers as well?
Speed of Light
A constant that is more widely compared to the Great Pyramid is the speed of light. The speed of light in a vacuum is 299,792,458. The Great Pyramid lies on the latitude coordinate 29.9792458°N. Now, with this constant comes the note that it is only one of two necessary coordinates needed to pinpoint an exact location. Hence, it is true that the Pyramid of Giza lies on latitude with the speed of light coordinate, but so do many, many other places around the world. Besides, there are many other latitudinal coordinates that run through the Great Pyramid. Keeping to the 7 decimal places would allow for more than 20,000 coordinates to run through the pyramid. Of course, however, mentioning the pyramid lies on 29.9246788°N, for instance, wouldn’t be half as exciting.
Although we may never know whether these interesting results were intentionally applied or just a sheer coincidence, they do keep us intrigued. Whether intentional or practical, the mathematical elements within the Great Pyramid of Giza contribute to its alluring and mystique nature. Let’s hope that historians and mathematicians of the future will continue their interest in the pyramids and uncover many more mathematical secrets!