Great Pyramid of Khufu and Euclid’s ‘The Elements’

초록

W. Flinders Petrie (1883) proposed a theory related to the golden ratio (?) (or apothem slope 14/11) and pi (π) based on his measurements of the Great Pyramid of Khufu. His hypothesis, the π-theory, was also addressed in the debate between Robert Paler and Martin Bernal that took place from 1992 to 1994. In recent years, specifically in 2010 and 2011, the Moscow Mathematical Problem No. 10 (MMP 10) was raised by Leon Cooper to find the surface area of a semicircular cylinder. According to his claim, the ancient Egyptians approximated the circumference of a circle using the perimeter of a square, which increased the plausibility of Petrie's π-theory. This paper argues that Petrie's π-theory is not considered independent of the golden ratio but somewhat dependent on it. Therefore, it is believed that the ancient pyramid builders could calculate the circumference of a circle using the height of the Great Pyramid as its radius, either during its construction or afterward. This paper presents a geometric interpretation of the golden ratio or π, which was claimed by Petrie and various other archaeologists to be embedded within the Great Pyramid of Khufu, based on Euclid's Elements (Book II, Proposition 11 and Proposition 14). Of course, this claim does not suggest that Euclidean geometry was directly influenced by ancient Egyptian mathematics. Nevertheless, the ancient Egyptians incorporated highly sophisticated mathematics regarding the golden ratio and π into their architecture. Hence, the possibility of an influence on ancient Greece is reasonable. Finally, this paper proposes a geometrical method to apply the quadrature of the circle to the ellipse presented in the Great Pyramid, initially suggested by Col. R. S. Beard in 1968 and Ludwig Borchardt in 1896.