Originally called Khufu's Horizon, the Great Pyramid of Giza is so famously beautiful that it has spawned a thousand theories about its origin, from it being built by the fictional empire of Atlantis to it representing the work of aliens from outer space. But a much more plausible explanation has superseded them all: The Egyptians were extremely advanced in mathematics for antiquity, working in proximity to equations and ratios that wouldn't be written down for centuries. Here's a peek behind the building of Egypt's Great Pyramids and how their relation to phi, pi, and sacred geometry only add to their mystique and enduring legacy.

About two thousand years before Pythagoras would be first credited with his now-famous triangle formula, King Khufu of Egypt's Old Kingdom was just about to kick-off the golden age of pyramid building. Skilled workers from around Egypt gathered in a temporarily constructed city around 2,580 BC to begin work on Khufu's masterpiece, with all uniting to work on a singular vision that was remarkably precise. Despite the myriad moving pieces on the massive construction site and primitive understanding of mathematics compared to the modern world, the engineers and workers would ultimately build a pyramid that was only 1.4 inches away from being a perfect golden triangle based on phi, also known as the Golden Ratio.

We find this calculation by simply plugging phi (or Φ) into a right-angle triangle using the Pythagorean Theorem and then creating an identical triangle that is the mirror image of the first. Together, the triangles form a full pyramid with the square root of phi as the height, phi representing the sides, and 2 representing the base. The height-to-base ratio in our newly formed pyramid would be 0.636, which is the exact ratio of the Great Pyramid's original dimensions (at least to three decimal places). Whether the Egyptians had some unexplained knowledge of phi or not, they certainly appear to have been operating on mathematical principles that are – at the least – very close cousins to the number we now commonly associate with the Golden Ratio.

Although not quite as exact, the pyramid built by Khufu's son, Khafre, also has a very similar height-to-base ratio, as does the one built subsequently by Pharaoh Menkaure to complete what we now call the three Great Pyramids of Giza.

The Egyptians didn't have direct knowledge of the Golden Ratio, but they did have a remarkable way of calculating triangles that would act as a precursor to later geometrical advancements. At the center of this, the Egyptian seked method was nothing short of ingenious for the time. Despite not understanding angles like we do today, Egyptian engineers were able to calculate a slope based on the height of one cubit, which was just a fundamental form of measurement in Ancient Egypt.

For every cubit in height of the main Great Pyramid, for example, the base would need to expand by 5.5 palms – another Egyptian measurement in which every palm equals one-seventh of a cubit. The method wasn't used universally at the time but was utilized for the main Giza pyramid and the ratio could be altered to fit different projects. Remarkably, dividing the height (1 cubit or 7 palms) by the base (5.5 palms) yields a ratio of 1.27, which is almost exactly the square root of phi.

When you talk about one famous irrational number (phi), ultimately the most famous of them all will need to have its say as well. For a very simple connection, you can take half of the perimeter of the Khufu pyramid (1,512 feet) and divide it by its original height (481 feet) and you – like magic – end up with almost exactly pi (*π*), which is commonly denoted as simply 3.14. If you do the calculation with either meters or Egyptian cubits, the number is nearly identical, pointing again towards pi being either an intentional or unintentional part of the equation.

Using a circle with a circumference (*π *x diameter) the same as the combined base of the pyramid in the phi experiment (4 x 2 = 8), we can also see a staggering connection with the dimensions of the main pyramid of Giza. Folding the semi-circle in half at a right angle so that the radius of the circle now represents the height of our paper pyramid, we again end up with a very similar height-to-base ratio as that of the Great Pyramid. Using this very basic paper model and extrapolating to the dimensions of the pyramid, the difference is actually less than six inches – a remarkable proximity given the 481-foot original height of the main Giza pyramid. Given that this original height is only an estimation, the true difference could even be smaller.

With calculations based on phi and pi coming so close to the original dimensions of Khufu's pyramid, the iconic Egyptian structure is an awe-inducing testament to the universality of both irrational numbers.

The Great Pyramids of Giza may just be the most famous example of the Golden Ratio, although they certainly aren't the only ones to incorporate its beauty. Another outstanding example is the Parthenon of Ancient Greece, which has a frame that is a picturesque 1.618:1 ratio that you can also find in many paintings and photographs. The Parthenon being a stunning example of the Golden Ratio of course makes sense, as it was the famous Greek sculptor Phidias who is the namesake for phi.

More than a 1,500 years after the Parthenon was finished, a different form of the Golden Ratio would create the frame of another world-famous building, the Notre Dame Cathedral in Paris. The western facade completed in 1200 shows a terrific example of the ratio, only the correlating rectangle is vertical instead of horizontal. Other buildings that have portions that appear dedicated to the Golden Ratio include the Taj Mahal in India, the U.N. Building in the U.S., and many, many more throughout the world as well.

The Egyptian engineers who were able to match Khufu's stunning vision of a grand pyramid couldn't have known about phi, pi, or sacred geometry in the way that we do today. But even though their knowledge was limited, the Egyptians were able to tap the same elegant beauty and universality of sacred geometry that continues to withstand the test of time. As long as Khufu's masterwork and the neighboring pyramids built by his successors remain iconic international spectacles, visitors the world-over will continue to witness stunning examples of sacred geometry at its most awe-inspiring.

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