The Curve of Least Disorder Formation of an optimum surface in j-space |
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10^{th} December 2018 |
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"There are at present fundamental problems in theoretical physics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before."
- P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 1931. "It has been suggested that the human visual system uses a curve of low energy when completing a contour.'" - B. K. P. Horn in "The curve of least energy", ACM Transactions on Mathematical Software, Vol. 9, No. 4, December 1983, Pages 441-460. |
We have described earlier the constraint of existence in 3-D in the macroscopic observer's, Obs The formation of the least-energy or least-disorder surface, is important as it provides us with the knowledge of the geometric structure of the infinitesimal volume element dV The energy corresponding to a curve passing through two neighbouring points with the arc length ds, is given as: E = ∫ κ where κ is the curvature of the arc, and it is related to the radius of the arc ρ as, κ = 1/ρ. Therefore for a pure point, the curvature is infinite and the radius is zero. Similarly for a precise straight line the curvature is zero, and hence the radius is infinite. The simplest case to consider, is when a curve is completed using a single arc. The curve of least-energy, is a circle and the corresponding least-energy surface is the surface of a sphere. In ideal discrete measurement space, no two measurements are alike. Consequently each ds, is of different curvature and different radii. If we plot a curve using same origin as reference, we run into a rather serious problem, as shown below:
The curve on left, is formed using a single arc. In this case when arcs are connected with respect to the same origin, a smooth curve i.e. a circle is formed, which is the curve of least-energy or the curve of least-disorder in this case. A real life example, is the surface formed by a soap bubble. However the situation in Planck's domain, is likely to be a little bit more complicated. As shown on the right in the picture above, if we try to connect arcs of varying curvatures using the same origin, a "smooth" curve can not be formed. An approach to solve this problem, is as following:
We connect two arcs ds
These shapes appear in various phenomena in nature, however which one of them corresponds to the curve of least-disorder when the number of arcs approaches infinity, is not very obvious. To determine the nature of the infinitesimal volume dV π/2. As the number of arcs is increased, the amount of energy stored in the curve can be shown to be decreasing. These values are shown below^{2}:
As it can be seen neither of the known curves i.e. Circle, Ellipse, and Euler's Spiral, correspond to the curve of least-energy. Furthermore as the number of arcs is increased towards infinity, the energy contained within the curve keeps decreasing until it hits the limit of approximately 91.39% of the energy, π/2, stored in a semicircle. These values are calculated using the multi-arc approximation. Analytically, the In discrete measurement space, the appearance of the Gamma function is always a good sign, because of the presence of the term ds, defined in General Theory of Relativity as:^{2}ds ^{2} depends upon the metric tensor g_{μν}. The curve of least-disorder, provides a method to estimate ds itself. Since ds is equivalently the curve of least energy, it will be an invariant. The requirements for the invariance of ds, are much more strict than those for ds^{2}.
We note that in j-space we are measuring the information space itself, with no assumption made about the nature of the information, such as known physical objects or any other known phenomenon. We are discussing here, the formation of a manifold from which the metric in observer's measurement space is derived. The relationship between the topology, the manifold, and the metric is shown here: The topological space in this case is defined by δ The curve of least-disorder provides a solution for the problem of the origin in the discrete measurement space or j-space, in a most satisfactory manner. ^{2}. More importantly, ds brings us closer to the underlying topological space, which governs the formation of the structures at cosmic as well as Planck scales. __________________
1. One way to think about it is that, if we assume that pure information is being fed to a black hole, the output of the black hole will be some energy which essentially, is the disordered information. In similar fashion, we can feed disordered information to a black hole, and in this case nothing would come out except for some occasional burps. 2. The numerical values and the subsequent discussion on Cesaro form and Whewell equation, are from the publication by B. K. P. Horn, "The curve of least energy", ACM Transactions on Mathematical Software, Vol. 9, No. 4, December 1983, Pages 441-460. |
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Möbius & Lorentz Transformation - II
Möbius & Lorentz Transformation - I Knots, DNA & Enzymes Quantum Comp - III Nutshell-2017 Quantum Comp - II Quantum Comp - I Insincere Symm - II Insincere Symm - I Existence in 3-D Infinite Source Nutshell-2016 Quanta-II Quanta-I EPR Paradox-II Chiral Symmetry
Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-Field & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity |

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