Ratio of In-Sphere Volume to Polyhedron Volume of the Great Pyramid Compared to Selected Convex Polyhedral Solids

摘要

The architecture of the Great Pyramid at Giza is based on fascinating golden mean geometry. Recently the ratio of the in-sphere volume to the pyramid volume was calculated. One yields as result R _( V ) = π ⋅ Φ ~(5), where is the golden mean. It is important that the number Φ ~(5) is a fundamental constant of nature describing phase transition from microscopic to cosmic scale. In this contribution the relatively small volume ratio of the Great Pyramid was compared to that of selected convex polyhedral solids such as the Platonic solids respectively the face-rich truncated icosahedron (bucky ball) as one of Archimedes ’ solids leading to effective filling of the polyhedron by its in-sphere and therefore the highest volume ratio of the selected examples. The smallest ratio was found for the Great Pyramid. A regression analysis delivers the highly reliable volume ratio relation , where nF represents the number of polyhedron faces and b approximates the silver mean. For less-symmetrical solids with a unique axis (tetragonal pyramids) the in-sphere can be replaced by a biaxial ellipsoid of maximum volume to adjust the R _( V ) relation more reliably.