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 <em>R</em>&...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 <em>R</em><sub><em>V</em></sub> = π <span style="white-space:nowrap;"><span style="white-space:nowrap;">⋅</span></span> <em><em style="white-space:normal;">φ</em></em><sup>5</sup>, where <img src="Edit_83decbce-7252-44ed-a822-fef13e43fd2a.bmp" alt="" /> is the golden mean. It is important that the number <em>φ</em><sup>5</sup> 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 <em>Platonic </em>solids respectively the face-rich truncated icosahedron (bucky ball) as one of <em>Archimedes</em>’ 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 <img src="Edit_79e766ce-5580-4ae0-a706-570e0f3f1bd8.bmp" alt="" />, where <em>nF</em> 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 <em>R</em><sub><em>V</em></sub> relation more reliably.展开更多
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 <i...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 <i>Fibonacci</i> 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⋅<i>φ</i><sup>5</sup>, where <i>φ</i> = 0.618033987<span style="white-space:nowrap;">⋅<span style="white-space:nowrap;">⋅</span><span style="white-space:nowrap;">⋅</span></span> is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with <i>φ</i><sup>5</sup>, also ingenious ancient builders have intuitively guessed its magic before.展开更多
文摘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 <em>R</em><sub><em>V</em></sub> = π <span style="white-space:nowrap;"><span style="white-space:nowrap;">⋅</span></span> <em><em style="white-space:normal;">φ</em></em><sup>5</sup>, where <img src="Edit_83decbce-7252-44ed-a822-fef13e43fd2a.bmp" alt="" /> is the golden mean. It is important that the number <em>φ</em><sup>5</sup> 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 <em>Platonic </em>solids respectively the face-rich truncated icosahedron (bucky ball) as one of <em>Archimedes</em>’ 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 <img src="Edit_79e766ce-5580-4ae0-a706-570e0f3f1bd8.bmp" alt="" />, where <em>nF</em> 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 <em>R</em><sub><em>V</em></sub> relation more reliably.
文摘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 <i>Fibonacci</i> 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⋅<i>φ</i><sup>5</sup>, where <i>φ</i> = 0.618033987<span style="white-space:nowrap;">⋅<span style="white-space:nowrap;">⋅</span><span style="white-space:nowrap;">⋅</span></span> is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with <i>φ</i><sup>5</sup>, also ingenious ancient builders have intuitively guessed its magic before.
