Abstract
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 RV = π ⋅ φ5, where is the golden mean. It is important that the number φ5 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 RV relation more reliably.
Highlights
The golden mean architecture of the Great Pyramid at Giza was investigated with a surprising result [1]
In the appendix of reference [1] the area ratio was carelessly calculated violating this proved fact. This flaw was already corrected in the prepublication [2]. In this contribution the in-sphere to polyhedron volume ratios for selected convex solids such as Platonic solids [3] respectively Archimedean ones [4] were compared to the result of the Great Pyramid
Known elementary techniques of Euclidian coordinate geometry in 3 were used to determine area, volume, in-sphere volume and the in-sphere to polyhedron volume ratio for selected convex polyhedra, using the universal number of the golden mean when dealing with polyhedra of icosahedral symmetry
Summary
The golden mean architecture of the Great Pyramid at Giza was investigated with a surprising result [1]. 2 when calculating the respective areas ratio This connection is valid for all regular convex polyhedral solids. In the appendix of reference [1] the area ratio was carelessly calculated violating this proved fact. This flaw was already corrected in the prepublication [2]. In this contribution the in-sphere to polyhedron volume ratios for selected convex solids such as Platonic solids [3] respectively Archimedean ones [4] were compared to the result of the Great Pyramid. For solids with reduced symmetry such as pyramids the in-sphere has to be replaced by a biaxial ellipsoid with maximum volume to adjust the relationship between volume ratio and number of faces more reliably
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