All atoms are composed of protons and electrons. If the universe consists entirely of Planck spheres (PSUs), each with a radius equal to the Planck length, then 1) the proton is a large composite sphere that equilibrates the PSUs in its volume with the PSUs on its surface, and 2) the electron has a toroidal form that equilibrates the PSUs along its circumference with the PSUs on its surface.
There are 3 physical constants related to Planck units: 1) the speed of light c, which represents a maximum rate of travel, 2) Planck’s constant h (also as h-bar = h/2π), which represents a minimum (electromagnetic) action and 3) the gravitational constant G, which relates the attractive potential between 2 massive bodies. These 3 empirically determined constants can be combined to yield a minimum length of space called the Planck length ℓP, as well as a particular mass called the Planck mass mP:
We’ll call a sphere with a radius equal to the Planck length a Planck spherical unit (PSU), a concept proposed by Nassim Haramein in 2012. If PSUs fill space, then we can calculate 2 numbers with respect to the proton: V is the number of PSUs in the proton volume and S is the number of PSUs on its surface.
We find that the proton charge radius rp forms a sphere such that the PSUs in its volume have a total mass equal to the Planck mass mP and the PSUs on its surface have a total mass equal to the observed proton mass mp, with all PSUs carrying the same fractional mass.
This leads to a proton radius equal to 0.8412 fm, which is in perfect agreement with its current CODATA value of 0.8414(19) fm. It is by no means obvious that the size and mass of the proton would have this precise relationship with Planck units without incorporating any extraneous coefficients whatsoever. It implies that if space is filled with PSUs, then the proton radius simply equilibrates the PSUs in its volume with the PSUs on its surface to produce the Planck mass and proton mass, respectively.
The next question is clear: does a similar condition apply to the electron? If so, then it would take the form of mP/V = mp/S = me/N, where N represents the number of PSUs on the electron surface, just as S represents the number of PSUs on the proton surface. Using the modern ring electron models of David Hestenes and Oliver Consa as a guide, we can proceed with the assumption that the electron at rest is a point particle moving in a toroidal circuit at the speed of light. Its circumference is the electron’s Compton wavelength λe, where λe is equivalent to a photon with the same energy as the rest mass energy of the electron. Since this torus represents an electron at rest, we’ll designate the number N as T0.
So T0 is the number of PSUs on the surface of a torus which has a circumference of λe and a thickness of 2πX. These PSUs have a total mass equal to the observed electron mass, with each PSU carrying a fractional mass of me/T0, the same as in mP/V = mp/S.
With the electron traveling in a circular orbit constrained by λe, the thickness 2πX may be attributed to a small precession that retains the electron’s center of mass at the center of the electron torus. According to the electron PSU model then, successive deformations to the PSUs along this wobbling path generate the electron.
Now the key question: is 2πX a purely theoretical construct or does it correspond to an actual physical limit? Two facts convince me it’s the latter. First, if T0 represents an electron with minimum velocity v=0, then there must also be an equation representing an electron with maximum velocity v≈c. To be clear, this is the translational velocity of the entire electron ring which moves in a direction orthogonal to its intrinsic Compton wavelength motion. Since the electron travels at the speed of light around its circumference λe, then in the time it takes to complete one orbit (λe/c), the maximum orthogonal distance it can traverse is also equal to λe. The surface area generated by this motion can be expressed in terms of the number of protons appearing on its surface as Tc:
Incredibly, we find that X, which determines the extent of the electron’s precession and thus the thickness of the electron ring at rest, is simply rp/Tc.
Like the proton PSU relation, the electron PSU relation includes no extraneous coefficients or numerical factors. So this coordination between the circumference of the electron torus and its thickness is striking. The thickness of the electron torus with minimum velocity and minimum surface area is given by the proton radius reduced in exact proportion to the number of protons that would fit on an electron torus with maximum velocity and maximum surface area. The electron’s Compton wavelength, which is determined by its rest mass (λe = h/me*c), not only constrains the electron’s circumference but also limits the extent of its precession. In this way, the electron torus equilibrates the PSUs along its Compton wavelength circumference with the PSUs on its surface to produce the electron mass.
Beyond the fractal nature of the electron ring’s thickness, the second extraordinary fact here is that 2πrp/Tc predicts a minimum photon wavelength, corresponding with a maximum photon energy. As David Hestenes has argued, if the electron has intrinsic motion along its Compton wavelength circumference, then photons may also travel along a similar circumference. The energy of photons along this path must be quantized, as well as its wavelengths, so for a photon to be transmitted by an electron the photon’s wavelength cannot be less than the thickness of the electron ring. Since this photon limit must be absolute, if a photon with a wavelength smaller than 2πrp/Tc were ever detected, then that would effectively falsify the toroidal electron PSU model proposed here.
In fact, the shortest photon wavelength ever detected, corresponding with an energy of 450 trillion ev, arrived here from the Crab Nebula in 2019. Despite journeying across 6500 years of intervening space, the photon is less than 3/2 longer than the minimum predicted by the electron PSU model. The fact that this model makes a testable prediction in such close agreement with observations, and which no other model can reproduce with a similar degree of simplicity and self-consistency, lends great credence to the entire PSU theoretical framework.
In my previous article, I showed a simple proportionality that links 3 essential atomic ratios: 1) the ratio of proton and electron rest masses (mp/me), 2) the ratio of the bound electron’s most stable radial distance to the proton’s charge radius (a0/rp), and 3) the fine-structure constant (~1/137) which is the classical ratio of the electron’s speed in this bound state to the speed of light (cα/c). Now, in light of its pivotal role within the electron torus, we should also include the electron’s reduced Compton wavelength (λe-bar = λe/2π) in this proportionality.
Despite the conspicuous absence of numerical factors in the PSU relations, mp/S = me/T0 actually reduces to this proportionality. Though the relation only holds for the integer part of mp/me, no other proportionality relates these essential atomic parameters with such an economical use of terms. The implication here is that the complementary components of the simplest atom, proton and electron, have core attributes such that their PSUs require this proportion of 135*136/20. I’ll reserve speculation about this for a separate article but for now it’s enough to just acknowledge its connection to the PSU relations we’ve been discussing.
One additional piece of evidence in support of a toroidal electron needs to be mentioned here. A high energy collision can cause the electron to temporarily transition into a high-mass state known as a muon. Because the magnetic moments of electrons and muons are so similar (the muon-electron g-factor ratio is 1.000006…), and because the 2 particles are otherwise identical, their difference in mass is extremely significant. Fortunately, both rest masses can also be measured with the highest degree of precision. So the muon-electron mass ratio provides us with a unique opportunity to study in detail the nature of mass at the most basic level. What we find is astonishing.
There are 3 parts of this relation to consider: precision, geometry, and numerical proportion. First, this relation produces a value (0.10565833 GeV/c^2) that agrees with all but the last significant digit of the observed muon mass (0.1056583755(23) GeV/c^2). The minuscule corrective factor can be attributed either to an inherent eccentricity or a kind of systematic measurement bias, but either way the precision here is extraordinary.
Second, the geometric factor of π*π indicates a toroidal transformation of the electron mass. With the electron’s intrinsic toroidal motion, the temporary mass increase being toroidal in nature follows logically. (Or conversely, if the electron had no relationship to toroidal geometry, then the muon mass would appear to be a total mystery.) This seems to imply that high energy collisions can temporarily disturb the normal toroidal motion of the electron either by distorting its normal dimensions or by adding an extra toroidal motion on top of its normal toroidal path.
Third, the rational number 20+19/20 has highly unusual coordination between integer and fraction. The number is 1/20 less than a whole integer greater than 20. Far from being an arbitrary coefficient, the factor of 20 here is also the denominator of the proportionality 135*136/20 that links the proton-electron PSU relations in their reduced form. (For further analysis and interpretation of this as the natural PSU oscillation, see my previous article.) Taken altogether, the muon-electron mass ratio presents strong supporting evidence of an electron with intrinsic toroidal motion.
Nassim Haramein’s PSU model is based on the simple idea that the Planck length is a fundamental unit of space with spherical geometry. The model describes the proton as a large sphere in relation to the PSU and successfully calculates its radius. With the aid of electron ring models by David Hestenes and Oliver Consa, I’ve shown here that the PSU model also describes the electron as a torus with a thickness that predicts a minimum photon wavelength in close agreement with observations. It does all this while employing only basic geometry and without incorporating any extraneous factors. The proposed intrinsic toroidal motion of the electron is also supported by the unambiguously toroidal nature of its temporary high-mass state. So the model is not only elegant in its simplicity, but useful in assimilating a range of physical phenomena. The conclusion that protons and electrons are made of Planck spheres seems inevitable at this point, and the nascent PSU model is certain to yield more insights as it is developed further.