Evidence of mathematical influences in art is present in the Great Pyramids, built by Egyptian Pharaoh Khufu and completed in 2560BC. Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments. They note that the length of the base edges range from 755–756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet.[6] If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. This would also indicate that half the cross-section of the Khufu’s pyramid is in fact aKepler’s triangle. Debate has broken out between prominent pyramidologists, including Temple Bell,Michael Rice, and John Taylor, over whether the presence of the golden ratio in the pyramids is due to design or chance. Of note, Rice contends that experts of Egyptian architecture have argued that ancient Egyptian architects have long known about the existence of the golden ratio. In addition, three other pyramidologists, Martin Gardner, Herbert Turnbull, and David Burton contend that:
Herodotus related in one passage that the Egyptian priests told him that the dimensions of the Great Pyramid were so chosen that the area of a square whose side was the height of the great pyramid equaled the area of the triangle.[7]
This passage, if true, would undeniably prove the intentional presence of the golden ratio in the pyramids. However, the validity of this assertion is found to be questionable.[8] Critics of this golden ratio theory note that it is far more likely that the original Egyptian architects modeled the pyramid after the 3-4-5 triangle, rather than the Kepler’s triangle. According to the Rhind Mathematical Papyrus, an ancient papyrus that is the best example of Egyptian math dating back to the Second Intermediate Period of Egypt, the Egyptians certainly knew about and used the 3-4-5 triangle extensively in mathematics and architecture. While Kepler’s triangle has a face angle of 51°49’, the 3-4-5 triangle has a face angle of 53°8’, very close to the Kepler’s triangle.[9] Another triangle that is close is one whose perimeter is 2π the height such that the base to hypotenuse ratio is 1:4/π. With a face angle of 51°50’, it is also very similar to Kepler’s triangle. While the exact triangle the Egyptians chose to design their pyramids after remains unclear, the fact that the dimensions of pyramids correspond so strongly to a special right triangle suggest a strong mathematical influence in the last standingancient wonder.
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