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

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.

Magic Numbers of the Great Pyramid: A Surprising Result

Recently attention has been drawn to the frequently observed fifth power of the golden mean in many disciplines of science and technology. Whereas in a forthcoming contribution the focus will be directed towards Fibonacci number-based helical structures of living as well as inorganic matter, in this short letter the geometry of the Great Pyramid of the ancient Egyptians was investigated once more. The surprising main result is that the ratio of the in-sphere volume of the pyramid and the pyramid volume itself is given by π⋅φ5, where φ = 0.618033987⋅⋅⋅ is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with φ5, also ingenious ancient builders have intuitively guessed its magic before.

胡夫大金字塔外形设计图新探

论文设想大金字塔设计图是最简单的正方形外接椭圆几何图。在研究了椭圆形在古埃及墓葬建筑等方面有丰富的文化遗存后,着重探讨了此蛋形设计图是怎么画、为何如此画和如何计算等问题。提出设计制图过程包含司绳法作图、尺规作图和格网设计三部分。司绳法作正方形外接椭圆也是尼罗河的礼物之一。巧合的是,此几何设计图可以用比司绳法更精确的尺规作图,并辅以既方便画图又方便测量的14格网,用于制作塔修造图和修造尺。大金字塔当前倾斜角值为51°50'34",而非精确的正方形外接椭圆几何图计算值51°49'38"(黄金直角三角形),误差的根本原因就是格网测量的局限性。这使我们重新认识到,本质上大金字塔毫无疑问是按黄金比设计的。