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[4] viXra:2208.0109 [pdf] submitted on 2022-08-19 18:37:38
Authors: Harish Chandra Rajpoot
Comments: 41 Pages. Original Research Work
This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.
Category: Geometry
[3] viXra:2208.0063 [pdf] replaced on 2022-08-15 00:49:44
Authors: Peter Cameron
Comments: 21 Pages.
Dark matter has two independent origins in the impedance model: Geometrically, extending two-component Dirac spinors to the full 3D Pauli algebra eight-component wavefunction permits calculating quantum impedance networks of wavefunction interactions. Impedance matching governs amplitude and phase of energy flow. While vacuum wavefunction is the same at all scales, flux quantization of wavefunction components yields different energies and physics as scale changes, with corresponding enormous impedance mismatches when moving far from Compton wavelengths, decoupling the dynamics. Topologically, extending wavefunctions to the full eight components introduces magnetic charge, pseudoscalar dual of scalar electric charge. Coupling to the photon is reciprocal of electric, inverting fundamental lengths - Rydberg, Bohr, classical, and Higgs - about the charge-free Compton wavelength $lambda=h/mc$. To radiate a photon, Bohr cannot be inside Compton, Rydberg inside Bohr,... Topological inversion renders magnetic charge `dark'.Dark energy mixes geometry and topology, translation and rotation gauge fields. Impedance matching to the Planck length event horizon exposes an identity between gravitation and mismatched electromagnetism. Fields of wavefunction components propagate away from confinement scale, are reflected back by vacuum wavefunction mismatches they excite. This attenuation of the `Hawking graviton' wavefunction results in exponentially increasing wavelengths, ultimately greater than radius of the observable universe. Graviton oscillation between translation and rotation gauge fields exchanges linear and angular momentum, is an invitation to modified Newtonian dynamics.
Category: Geometry
[2] viXra:2208.0049 [pdf] replaced on 2022-08-13 21:48:01
Authors: James A. Smith
Comments: 8 Pages.
To help fill the need for examples of introductory-level problems that have been solved via Geometric Algebra (GA), we show how to calculate the angle through which two unit vectors must be rotated in order to be parallel to each other. Among the ideas that we use are a transformation of the usual GA formula for rotations, and the use of GA products to eliminated variables in simultaneous equations. We will show the benefits of (1) examining an interactive GeoGebra construction before attempting a solution, and (2) considering a range of implications of the given information.
Category: Geometry
[1] viXra:2208.0021 [pdf] submitted on 2022-08-05 00:36:25
Authors: Thomas Halley
Comments: 1 Page.
Complex Variables has a link to general geometry in placing the geometry of squares. Given is a problem in geometry where a short-cut is taken to solve what the angle is in the given situation.
Category: Geometry
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