We apply a discrete quantum walk from a quantum particle on a discrete quantum spacetime from loop quantum gravity and show that the related entanglement entropy drives an entropic force. We apply these concepts in a model where walker positions are topologically encoded on a spin network. Then, we discuss the role of the golden ratio in fundamental physics by addressing charge and length quantization and by analyzing the ratios of fundamental constants−the limits of nature. The limit of minimal length and volume arising in quantum gravity theory indicates an underlying principle that we develop herein.

]]>Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians. Recently, particular interest has been given to packings of three-dimensional tetrahedra. Dense packings of both crystalline and semi-quasicrystalline have been reported. It is interesting that a semi-quasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic simulation approach. However, this packing is not perfectly quasicrystalline and the packing density, while dense, is not maximal. Here we suggest that a “golden rotation” between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline packing. Using this golden rotation, tetrahedra can be organized into “triangular”, “pentagonal”, and “spherical” locally dense aggregates. Additionally, the aperiodic Boerdijk-Coxeter helix (tetrahelix) is transformed into a structure of 3-or 5-fold periodicity—depending on the relative chiralities of the helix and rotation—herein referred to as the “philix”. Further, using this same rotation, we build (1) a shell structure which resembles a Penrose tiling upon projection into two dimensions, and (2) a “tetragrid” structure assembled of golden rhombohedral unit cells. Our results indicate that this rotation is closely associated with Fuller’s “jitterbug transformation” and that the total number of face-plane classes (defined below) is significantly reduced in comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of tetrahedra which is both dynamic and dense. The golden rotation that we report presents a novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.

]]>A first principles theory of everything has yet to be achieved. An E8 derived code of quantized spacetime could meet the following suggested requirements:

(1) First principles explanation of time dilation, inertia, the magnitude of the Planck constant and the speed of light.

(2) First principles explanation of conservation laws and gauge transformation symmetry.

(3) Must be fundamentally relativistic with nothing that is invariant being absolute.

(4) Pursuant to the deduction that reality is fundamentally information-theoretic, all information must be generated by observation/measurement at the simplest Planck scale of the code/language.

(5) Must be non-deterministic.

(6) Must be computationally efficient.

(7) Must be a code describing “jagged” (quantized) waveform – a waveform language.

(8) Must have a first principles explanation for preferred chirality in nature.

This paper shows that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of RN, the sums of the squares of the original and projected edges are in the ratio N/M.

]]>Inspired by the Sum of the Squares law obtained in the paper titled “The Sum of Squares Law“ by J. Kovacs, F. Fang, G. Sadler and K. Irwin, we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.

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