Articles

This is a geometrical model, Each article uses only the fine structure constant alpha, pi, an expanding universe and a geometrical guide-rail. The articles 1 to 4 have been updated (Oct 2025) so that they can be read in sequential order (article 2 follows on from article 1...). Article 5 is pending. They have also been partly transcribed onto wiki pages as this is a familiar format. The original articles described only the theory behind the model without linking to modern physics. In these articles I used AI to bridge the gap, saves a lot of time. There will be errors however, but as the AI written sections relate to mainsteam physics they should be easy to pick up.

• The mathematical electron
  
• 1. Planck unit scaffolding correlates with the Cosmic Microwave Background
  
• 2. Relativity as the mathematics of perspective in a hyper-sphere universe
  
• 3. Gravitational orbits from n-body rotating particle-particle orbital pairs
  
• 4. Geometrical origins of quantization in H atom electron transitions
  

 Mass, length, time, ampere as geometrical objects embedded in the mathematical electron

Cite:"Programming Planck units from a virtual electron; a Simulation Hypothesis"
Eur. Phys. J. Plus (2018) 133: 278. https://doi.org/10.1140/epjp/i2018-12094-x

1. Planck unit scaffolding correlates with the Cosmic Microwave Background

In this article we compare the parameters for a hypothetical Planck unit universe (sans particles) with the Cosmic Microwave Background. The model postulates a Planck unit scaffolding upon which the particle universe resides and supposes that within the CMB parameters can be found evidence of this non-baryonic background. The model uses only Planck mass and Planck length as the primary structures and a spiral geometry as the `rule set'. We begin with the peak frequency of the CMB to establish an age of the universe in Planck time units and use this as our sole variable, nevertheless from this we can derive estimates for the radiation energy density, the CMB temperature and a cold dark matter mass density that are shown to be consistent with current observational values. Intriguingly this suggests that dark matter may be predominantly non-baryonic. The Casimir force equation reduces to the equation for radiation density implying that the universe has finite boundaries, albeit these are expanding at a constant rate. This article is part of a Planck scale Simulation Hypothesis project that attempts to demonstrate that the universe could in sum total be dimensionless, relying on geometrical artifice to create actual physical structures.

Cite: "Programming relativity for use in Planck scale Simulation Hypothesis modeling".
download: doi.org/10.2139/ssrn.3333513

• Cosmic microwave background in a Planck unit universe
  wiki: Planck unit universe
  

2. Relativity as the mathematics of perspective in a hyper-sphere universe

In this article we look at relativity as a translation between 2 co-ordinate systems, our relativistic 3-D space-time residing on a non-relativistic Planck unit lattice background within an expanding 4-axis hyper-sphere. The hyper-sphere expands in discrete steps (the universe is spatially finite (a closed 4-sphere), but it is not a static system, as it expands with every Planck time step), and at each step Planck units of mass m_P, length l_p and time t_p are added, thus forming a scaffolding for the particle universe. As for each unit of Planck time there is a unit of Planck length, this Planck framework is expanding at a constant rate (the speed of light c = l_p / t_p). As the hypersphere expands, it also pulls particles with it (at the speed of light), and so all particles and objects are traveling at, and only at, the speed of light (in the hyper-sphere frame of reference there is only 1 velocity, c). However, if we consider 3-D space as the surface of the hyper-sphere then motion between particles is relative. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally across this surface (in 3-D space) and so this incremental hyper-sphere expansion at velocity $c$ cannot be observed directly via the electromagnetic spectrum, relativity then becomes the mathematics of perspective translating between the absolute albeit expanding hyper-sphere background and the relative motion of 3D space.

Cite: "2. Relativity as the mathematics of perspective in a hyper-sphere universe".
download: doi.org/10.2139/ssrn.3334282

• Relativity as the mathematics of perspective
  wiki: Relativity_(Planck)
  

3. Gravitational orbits from n-body rotating particle-particle orbital pairs

We present a geometric model of orbital mechanics in which gravitational and atomic orbits emerge from time-averaged networks of rotating point-to-point orbital pairs. The model discretizes macroscopic objects into Planck-mass points, each forming independent orbital pairs with all other points in the system, creating a universe-wide $N$-body network. Despite using only dimensionless rotating circles governed by the fine structure constant alpha and pi, the model reproduces Kepler's law and anomalous orbital precession. The simulation operates at the Planck scale, with each orbital rotating through one Planck length per Planck time (velocity c in hypersphere coordinates). Crucially, the model treats particles as oscillations between an electric wave-state (duration: particle frequency) and a mass-point state (duration: one Planck time), thereby replacing 2 abstract forces with 2 distinct states through temporal averaging. We demonstrate that when the gravitational coupling constant alpha_G is inverted, gravity becomes the dominant force at unit (Planck) time, with its apparent macroscopic weakness arising statistically from the rarity of mass-point states. The model uses only geometry, alpha, pi, and Planck units for dimensional conversion.

Cite: "3. Gravitational orbits from n-body rotating particle-particle orbital pairs".
download: doi.org/10.2139/ssrn.3444571

Gravitational and atomic orbitals
wiki: Gravitational orbital

4. Geometrical origins of quantization in H atom electron transitions

We present a novel geometric model of atomic electron transitions that derives quantum energy levels and transition frequencies from first principles using only the fine structure constant ($\alpha$), $\pi$, and the (proton+electron) Compton wavelengths. The model treats atomic orbitals as physical rotating structures that evolve through discrete angular steps during photon absorption. Unlike standard quantum mechanics, which postulates energy quantization, our approach shows that discrete energy levels emerge naturally from geometric stability conditions. The model achieves high accuracy for hydrogen transition frequencies and correctly predicts angular momentum-dependent transition dynamics without invoking wavefunctions or the Schrodinger equation. We demonstrate that photon absorption for the Lyman-$\alpha$ transition occurs via a series of steps, with each step corresponding to one Compton-wavelength oscillation. This work suggests that quantum mechanics may be an emergent description of underlying geometric dynamics.

Cite: "4. Geometrical origins of quantization in H atom electron transitions".
download: doi.org/10.2139/ssrn.3703266

Gravitational and atomic orbitals
wiki: Gravitational orbital