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The aim of the present paper is to explain and accurately calculate the missing dark energy density of the cosmos by scaling the Planck scale and using the methodology of the relatively novel discipline of cosmic crystallography and Hawking-Hartle quantum wave solution of Wheeler-DeWitt equation. Following this road we arrive at a modified version of Einstein’s energy mass relation E = mc2 which predicts a cosmological energy density in astonishing accord with the WMAP and supernova measurements and analysis. We develop non-constructively what may be termed super symmetric Penrose fractal tiling and find that the isomorphic length of this tiling is equal to the self affinity radius of a universe which resembles an 11 dimensional Hilbert cube or a fractal M-theory with a Hausdorff dimension <img src="Edit_83705130-2a13-4103-aa8f-3af0c09b9886.bmp" alt="" />where<img src="Edit_0b585bd6-670f-4314-a71c-b248b912f689.bmp" alt="" />. It then turns out that the correct maximal quantum relativity energy-mass equation for intergalactic scales is a simple relativistic scaling, in the sense of Weyl-Nottale, of Einstein’s classical equation, namely EQR = (1/2)(1/<img src="Edit_b2e78758-82e7-4b44-b315-a715ab83b1e4.bmp" alt="" />) moc2 = 0.0450849 mc2 and that this energy is the ordinary measurable energy density of the quantum particle. This means that almost 95.5% of the energy of the cosmos is dark energy which by quantum particle-wave duality is the absolute value of the energy of the quantum wave and is proportional to the square of the curvature of the curled dimension of spacetime namely <img src="Edit_6729d2ac-fbc0-438f-a57a-ab62ad3c408f.bmp" alt="" />where <img src="Edit_efa027fb-db97-421f-8985-5f8d00770940.bmp" alt="" />and <img src="Edit_a4b9ea26-a4fe-4d45-9c34-ab17beae748b.bmp" alt="" />is Hardy’s probability of quantum entanglement. Because of the quantum wave collapse on measurement this energy cannot be measured using our current technologies. The same result is obtained by involving all the 17 Stein spaces corresponding to 17 types of the wallpaper groups as well as the 230-11=219 three dimensional crystallographic group which gives the number of the first level of massless particle-like states in Heterotic string theory. All these diverse subjects find here a unified view point leading to the same result regarding the missing dark energy of the universe, which turned out to by synonymous with the absolute value of the energy of the Hawking-Hartle quantum wave solution of Wheeler-DeWitt equation while ordinary energy is the energy of the quantum particle into which the Hawking-Hartle wave collapse at cosmic energy measurement. In other words it is in the very act of measurement which causes our inability to measure the “Dark energy of the quantum wave” in any direct way. The only hope if any to detect dark energy and utilize it in nuclear reactors is future development of sophisticated quantum wave non-demolition measurement instruments. </p

Dark Energy Explained via the Hawking-Hartle Quantum Wave and the Topology of Cosmic Crystallography