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 num-ber-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 inves-tigated 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 5 ϕ π⋅ , where 0.618033987 ϕ=  is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with φ 5 , also ingenious ancient builders have intuitively guessed its magic


Introduction
Recently, attention has been drawn to the more than expected occurrence of the fifth power of the golden mean in different disciplines of science and technology [1]. The golden mean or golden ratio is an omnipresent number in nature, found in the architecture of living creatures as well as human buildings, music, finance, medicine, philosophy, and of course in physics and mathematics including quantum computation [2] [3] [4] [5]. It is the most irrational number known and a number-theoretical chameleon with a self-similarity property. When writing down its continued fraction representation, which is the simples of all (see Equation (1)), one can catch an idea of why nature works so effectively, so that Olson et al. [4] arrive at the statement that the whole universe functions as a golden supercomputer.
In this contribution we use the golden mean φ as defined in Equation (1) The seminal E-infinity (ε ∞ ) approach developed by El Naschie years ago and applied mainly to cosmological questions based on this simple principle of nature, too [8] [9] [10]. It originates from a fractal Cantorian set theory [11] as a number-theoretical route of physics for explaining the dualism between particles and waves that can help solving cosmological mysteries such as dark matter and dark energy [12]. A probabilistic quantum entanglement calculation delivers effective quantum gravity formulas for the cosmological mass (energy) constituents. The baryonic matter amount e M resulted according to El Naschie and scientific fellows in Seeking for a more geometry-based interpretation of the fifth power of the golden mean, the present author has found a surprising result that will be formulated in the next chapters revealing more secrets of the Great Pyramid of Giza.
The numerous contributions about the mysterious geometry of this pyramid can't be quoted here. They would only contribute marginally to the subject.
However, reference is recommended again to the work initiated by Olson [2] [3] [4]. A short summary about the appearance of the fifth power of the golden mean is given in Chapter 2. Turning then to Chapter 3, the volume of a special quadratic pyramid (Great Pyramid) is compared to that of its in-sphere volume.

Appearance of the Fifth Power of the Golden Mean
In the following overview scientific fields were summarized according to [1], where the fifth power of the golden mean is involved indicating that this number is frequently observed all around the world: 1) Phase transition of the hard hexagon lattice gas model [16] [17]; 2) Phase transition of the hard square lattice gas model [17] [18]; 3) One-dimensional hard-boson model [19]; Topic 10 of this general overview will now be considered in detail in the following chapters pointing to a not expected beautiful result. Whether also the fifth power of the golden mean, apart from the golden mean, can be identified in the DNA genetic code is under discussion.

Relation between the Volume of the Great Pyramid and Its In-Sphere Volume
We were intuitively interested in the relation between the volume of the Great Pyramid of Giza and its in-sphere volume. Whereas in case of a general tetrahedron, for instance, a contiguous derivation of radius and center of the in-sphere was not solved as yet, very recently such derivation was delivered from P. P.
Klein of the TU Clausthal [24]. Contrary, in case of a quadratic pyramid the in-sphere can be quite simply derived from the given data explained in Figure 1, in which a mid-plane triangle cut down the apex of the pyramid is displayed together with the projected in-sphere in yellow color.
One can further derive that the radius of the in-sphere yields In connection with the Great Pyramid, mention may be made of the vision of the great inventor Nicola Tesla to harvest free energy from what we now characterize as dark components of matter respectively energy. The reader can find more information about this discussion in the references [15] and [25].
If one approximates π by a φ-based relation according to [26] the result is expressed solely by φ giving Another approximation is curious regarding the angle of α/2 (see Equation (4) and In addition, the surface ratio between the pyramid and its in-sphere is calculated. The result would better fit into the next chapter but is given in Appendix.

Journal of Applied Mathematics and Physics
In Appendix also another geometry-based approximation of the fifth power of the golden mean is derived for comparison.

The Fibonacci Number 13
Among the numbers of the Fibonacci series [28] [29], the numbers 5 as well as 13 seem to be most important. Whereas number 5 is involved in the definition of the golden mean, number 13 is found as helix repetition number for instance in tubulin protein, thought to be the location from where our thinking and consciousness originates [1] [4] [30] [31]. This number is very special because its square root can be approximated by In addition it yields

Conclusion
This contribution may be of general interest because it connects results of different disciplines of science and technology with the beauty of the mathematical knowledge of the ancient Egyptian builders of the pyramids. The present author has recently shown that besides the golden mean its fifth power φ 5 is also most important in describing different problems in science and technology such as phase transformations of microscopic to cosmic scale. Nevertheless, it was a surprise to elicit the Great Pyramid of Giza another secret by geometrical investigation of its golden ratio architecture. The ratio of the in-sphere volume of the pyramid to that of the pyramid itself turned out to be beautiful simply yielding 5 ϕ π ⋅ . It is a further step towards understanding our world and states once more that the golden ratio is the great numerical dominator of our world leaving also a fingerprint in the DNA genetic code [32]. It is under discussion whether also the fifth power of the golden mean can be identified in the genetic code [33].

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.