The electron is found at discrete energy levels within the atom, transition between these levels is considered to involve a `jump' rather than via a continuous motion. If we simulate the transition in the H atom as a series of individual steps, with each step the frequency of the electron, we can map a semi-continuous transition (from n=1 to n=2 requires about 1887860 steps, transition period a function of the photon wavelength). Plotting the electron from n=1 to ionization traces a (...) hyperbolic spiral, the significance being that at specific spiral angles, the angle components cancel returning an integer value for the radius (360°=4r, 360+120°=9r, 360+180°=16r, 360+216°=25r ... 720°). If we set initial radius r = Bohr radius, then we can use the spiral perimeter to calculate the transition frequencies at these angles. As these spiral angles correlate the energy level Bohr radii directly with pi, we may ask if quantization of the atom has a geometrical origin. Furthermore, the closer the vicinity of the electron to the proton, the greater the divergence from the idealized value suggesting the electron maybe causing a distortion within the proton internal structure. (shrink)

The Simulation Hypothesis proposes that all of reality, including the earth and the universe, is in fact an artificial simulation, analogous to a computer simulation, and as such our reality is an illusion. In this essay I describe a method for programming mass, length, time and charge (MLTA) as geometrical objects derived from the formula for a virtual electron; $f_e = 4\pi^2r^3$ ($r = 2^6 3 \pi^2 \alpha \Omega^5$) where the fine structure constant $\alpha$ = 137.03599... and $\Omega$ = 2.00713494... (...) are mathematical constants and the MLTA geometries are; M = (1), T = ($2\pi$), L = ($2\pi^2\Omega^2$), A = ($4\pi \Omega)^3/\alpha$. As objects they are independent of any set of units and also of any numbering system, terrestrial or alien. As the geometries are interrelated according to $f_e$, we can replace designations such as ($kg, m, s, A$) with a rule set; mass = $u^{15}$, length = $u^{-13}$, time = $u^{-30}$, ampere = $u^{3}$. The formula $f_e$ is unit-less ($u^0$) and combines these geometries in the following ratio M$^9$T$^{11}$/L$^{15}$ and (AL)$^3$/T, as such these ratio are unit-less. To translate MLTA to their respective SI Planck units requires an additional 2 unit-dependent scalars. We may thereby derive the CODATA 2014 physical constants via the 2 (fixed) mathematical constants ($\alpha, \Omega$), 2 dimensioned scalars and the rule set $u$. As all constants can be defined geometrically, the least precise constants ($G, h, e, m_e, k_B$...) can also be solved via the most precise ($c, \mu_0, R_\infty, \alpha$), numerical precision then limited by the precision of the fine structure constant $\alpha$. (shrink)

An orbital simulation program is described that uses a geometrical approach to modeling gravitational and atomic orbits at the Planck scale. Orbiting objects A, B, C... are sub-divided into points, each point representing 1 unit of Planck mass, for example, a 1kg satellite would divide into 1kg/Planck mass = 45940509 points. Each point in object A then forms a rotating orbital pair with every corresponding point in objects B, C... resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs. (...) Each orbital pair rotates 1 unit of Planck length per unit of Planck time at velocity c in hypersphere space co-ordinates, the results are then summed and averaged to give the new co-ordinates, the program then repeats. When these rotations are mapped over time on a 2-D plane (representing 3D space), objects A, B, C... appear to be orbiting each other. The basic simulation uses the fine structure constant alpha as an orbital constant to simulate gravitational orbit parameters. As each orbital comprises only 2 points, 1 at each orbital pole, information regarding the objects A, B, C... ; momentum, size, center of mass, barycenter etc ... is not required, instead only the start (x, y, z) co-ordinates of each point are defined. Each point, by having a mass of Planck mass, is itself a construct of multiple particles, and so we can also form individual particle to particle orbital pairs. The simulation uses only geometry, no dimensioned constants (G, h, c ...) are required, although the results can be measured in Planck units for comparison. Points are physically linked together by a unit of momentum (a graviton), the rationale for this is described in a subsequent article on atomic orbitals. (shrink)

Outlined here is a simulation hypothesis approach that uses an expanding (the simulation clock-rate measured in units of Planck time) 4-axis hyper-sphere and mathematical particles that oscillate between an electric wave-state and a mass (unit of Planck mass per unit of Planck time) point-state. Particles are assigned a spin axis which determines the direction in which they are pulled by this (hyper-sphere pilot wave) expansion, thus all particles travel at, and only at, the velocity of expansion (the origin of $c$), (...) however only the particle point-state has definable co-ordinates within the hyper-sphere. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally (in hypersphere co-ordinate terms) between particles and so this hypersphere expansion cannot be directly observed, relativity then becomes the mathematics of perspective translating between the absolute (hypersphere) and the relative motion (3D space) co-ordinate systems. A discrete `pixel' lattice geometry is assigned as the gravitational space. Units of $\hbar c$ `physically' link particles into orbital pairs. As these are direct particle to particle links, a gravitational force between macro objects is not required, the gravitational orbit as the sum of these individual orbiting pairs. A 14.6 billion year old hyper-sphere (the sum of Planck black-hole units) has similar parameters to the cosmic microwave background. The Casimir force is a measure of the background radiation density. (shrink)

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation. Outlined here is a method for programming relativistic mass, space and time at the Planck level as applicable for use in Planck Universe-as-a-Simulation Hypothesis. For the virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven (...) by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle motion may be dispensed with. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this 3-D space and the hyper-sphere co-ordinates, relativity resembling the mathematics of perspective. (shrink)

Following the 26th General Conference on Weights and Measures are fixed the numerical values of the 4 physical constants ($h, c, e, k_B$). This is premised on the independence of these constants. This article discusses a model of a mathematical electron from which can be defined the Planck units as geometrical objects (mass M=1, time T=2$\pi$ ...). In this model these objects are interrelated via this electron geometry such that once we have assigned values to 2 Planck units then we (...) have fixed the values for all Planck units. As all constants can then be defined using geometrical forms (in terms of 2 fixed mathematical constants, 2 unit-specific scalars and a defined relationship between the units $kg, m, s, A$), the least precise CODATA 2014 constants ($G, h, e, m_e, k_B$...) can then be solved via the most precise ($c, \mu_0, \alpha, R_\infty$), with numerical precision limited by the precision of the fine structure constant $\alpha$. In terms of this model we now for example have 2 separate values for elementary charge, calculated from ($c, \alpha, R_\infty$) and the 2017 revision. (shrink)