III) Spinning Deformable Rings

Probing the universe for signs of divinity, we discovered that natural structures ranging in size from hydrogen to human DNA exhibit exceptional stability in multiples of 22. We then compared the parameters that define the fundamental particles and found significant proportionality with multiples of 22. We are thus faced with one of two distinct possibilities – either this pattern is a meaningless coincidence or else it results from the basic structure of the universe. Put another way, if we imagine God to be a subtle omnipresent harmony expressed through this pattern of stability and proportionality with multiples of 22, then we are either hallucinating or justified in our belief. There are two main reasons why the second possibility is more likely than the first:

1) Contemporary theories of the basic structure of the universe, such as string theory, cannot account for the differences between the observed masses and g-factors of the fundamental particles.

2) There is no comparable pattern among the fundamental physical parameters that is consistent with the unusual stability exhibited by certain natural structures.

So if there were already a theory that describes why protons are about 1836 times more massive than electrons then we could reasonably conclude that the pattern exposed here is nothing more than a coincidence. But because neither string theory nor any of the other prominent theories of space-time offer any insight into the numerical differences in these fundamental measurements, we have no basis for dismissing the evident proportionality as mere coincidence. Likewise, if it could be demonstrated that there are similar numerical patterns among not only the basic particle masses but also the basic massive structures then we could more readily question the significance of the pattern involving multiples of 22. But since this pattern is so unique in its simplicity, precision, and consistency, we may be resolute in our view that it is not a common trick with numbers but indicative of an underlying space-time structure.

Our task now is to piece together the relevant evidence to derive a rational explanation for the various phenomena. Though we may never be fully certain, one specific mechanism presents itself as both highly efficient and perfectly consistent with a wide variety of observations. While string theory proposes that particles are generated by strings that vibrate in extra dimensions, which by their nature can never be observed, we may instead suppose that the universe consists of spinning rings that deform in proportion with the masses and g-factors of the fundamental particles. This new conception of space-time is implied by experiments involving near-lightspeed collisions between electrons, protons, and neutrons. The discrete changes in mass and g-factor that result from these experiments provide compelling evidence of a toroidal space-time unit, which in its default state has a large radius 20 times greater than the smaller one and rotates at the speed of light. We will now examine the observations that support this particular space-time structure.

First let's recall that the total surface area A_t of a ring or torus is the product of the large and small circumferences. To calculate a quarter toroidal surface area A_t/4 we simply multiply the square of π by the product of the large and small radii. So, for example, if the large radius R is equal to 20 and the small radius r is equal to 1 then the quarter toroidal surface area will be equal to 20π².

Electrons are the stable parts of matter that constitute the exterior of atoms. When an electron undergoes a sudden and extreme acceleration, it may temporarily gain a discrete quantity of mass while its g-factor increases marginally. This altered state is known as a muon. Now, if the large radius of a torus is equal to 20.95 and the small radius is equal to the electron mass m_e then, regardless of the chosen units of measurement, the quarter toroidal surface area will be essentially indistinguishable from the observed muon mass m_μ.

^{m_μ = 0.1056583715(35)}

Since the difference in g-factor is so minimal (g_μ/g_e=1.00000626), the fact that the muon mass corresponds almost precisely with a quarter toroidal surface area involving the electron mass becomes highly significant. The extreme precision with which the muon-electron mass ratio is known, along with its close correspondence to such a simple geometric object, offers profound insight into the basic structure of the universe.

This connection with toroids is reinforced by the mass of the tau, which is the third and most massive state of the electron. Measured in units of Gev/c²,  the tau mass m_τ is consistent with a quarter toroidal surface area in which the large radius is equal to 3/5 and the small radius is equal to half the large radius.

^{m_τ = 1.77682(16)}

Thus the geometry of a torus provides a remarkably accurate description of two physical limits derived from the electron. Although the tau mass is known with far less precision than the muon mass, it is also subject to another well-defined constraint. If the quarter toroidal surface area described above functions as a lower bound, then the upper bound for the tau mass is established by Koide's formula.

This relation simply and efficiently constrains each of the three masses associated with the electron, which are also known as leptons. When an electron collides with another particle then it may temporarily transform into a muon or a tau depending on the energy of the collision. Evidently, the discrete differences in mass between each of the leptons correspond with a toroidal space-time unit that deforms in specific proportions according to basic geometric limits.

Rather than being an isolated anomaly, when we use the same approach to study the differences in the unstable quark masses we find yet more evidence of this toroidal structure. Protons and neutrons consist of only two kinds of quarks: up and down. When these quarks undergo sudden and extreme accelerations, like leptons, they may temporarily gain mass in discrete quantities to form the unstable quarks known as strange, charm, bottom, and top. The strange quark mass m_s is similar to that of the muon, corresponding with a quarter toroidal surface area in which the large radius is equal to 20 (rather than 20.95) and the small radius is equal to the electron mass.

^{m_s = 0.100 ±30}

Meanwhile, the charm quark mass m_c is 12 times larger than the muon and the bottom quark mass m_b is 40 times larger than the muon, so both masses correspond with proportionally larger toroidal surface areas. It is noteworthy that 12 = 2(6) and 40 = 2(20) since factors of 6 and 20 both have significant proportionality with the fundamental physical parameters, as we saw in the previous section. These factors arise not by accident but from the basic structure of the universe, as we will soon see.

^{m_c = 1.29 ±50}

^{m_b = 4.19 ±60}

The top quark is not only the most massive quark, but also the most massive individual particle to emerge from any particle collision. Unlike the other quarks the top quark is not bound in triplets or pairs. Measurements of the top quark mass m_t indicate it has an upper limit of 174 Gev/c² and a lower limit of 172 Gev/c². If we define a torus with a large radius equal to 20.95 and a small radius equal to the proton mass m_P (rather than the electron mass), then its quarter toroidal surface area will be between 20 and 22 Gev/c² larger than the mass of the top quark.

^{m_t = 173 ±1}

As we saw with the leptons, the quark masses seem to represent certain boundaries established by the underlying structure of space-time. Thus, the unstable masses are altogether consistent with the surface areas of proportional rings, but what about the particles that form stable matter?

* * *

It is impossible to understand electrons and protons apart from their relationship to light which, according to the basic principles of quantum mechanics, passes between each charged particle (electron and proton) in discrete quantities known as photons. To understand what a photon is we must first acknowledge 3 key properties: it travels at only one speed, which is the maximum rate of motion at which any particle may travel; its energy is determined by its frequency, which is inversely related to its wavelength; and its polarity corresponds with the direction of angular momentum, or spin, which depends on the state of the electromagnetic field. Beginning with the speed of light, we will now see why these properties are a natural consequence of a space-time composed of spinning rings.

The speed of light is a ratio of space and time that establishes a universal limit on the rate of motion. Its square appears in Einstein's famous equation E=mc², which allows for the definition of particle masses in terms of the speed of light (m=E/c²). Although masses can be expressed using any arbitrary units of measurement, to understand the construction of the universe at the most basic level it is necessary to use units of Gev/c². This is exemplified quite clearly by the muon-electron mass ratio, which defies comprehension unless we consider the quantity 20.95 in relation to the electron and proton masses in Gev/c². By itself, 20.95 is an interesting number because its integer part (20) is closely related to its fractional part (0.95 = 1–1/20). But it attains much greater significance when we consider that the electron mass multiplied by half the ratio (135)(136)/20² and then summed with the proton mass is very nearly equal to 0.95. Because both masses are defined in units derived from the squared speed of light, it is rather profound to find this numerical ratio with a quasi-square in the numerator (135)(136) and a square in the denominator (20²), especially since 20 is the definitive term in the muon-electron mass ratio.

^{i = 1.000000301          j = 1.000000025}

We now come to a crucial question: If the speed of light can be measured using any arbitrary units, then might the ratio (135)(136)/20² represent the actual squared speed of light? In fact there is good reason to believe this squared ratio describes fundamental quantities of space and time. Consider that, at the atomic level, the multiple (135)(136) is remarkably similar to the inverse fine-structure constant (137.036), which establishes the size of the simplest atom; and the factor of 20 corresponds with a critical limit for the neutron-proton ratio, which exceeds one for all stable atoms containing more than 20 protons and is equal to one for most stable atoms with 20 or fewer protons.

To this we can add another pair of observations relevant to the hypothesis that the squared speed of light is equivalent to the ratio (135)(136)/20². First there is the fact that living organisms are mostly liquid water with 20 protons in each molecule of 2H₂O, and after oxygen, carbon, hydrogen, and nitrogen the most abundant element (by mass) in organisms is calcium, which has 20 protons per atom. As a relatively heavy element, calcium plays a vital role not only in the skeletal structure of animals (as, for example, in our 20 fingers and toes along with 20 baby teeth) but also in the cellular signaling of all organisms. Most significantly, we find the periodic movement of calcium ions in liquid water is essential to all sensory processes including thought. If consciousness is ever to be understood in material terms then we must take seriously the notion that calcium waves exist in harmony with the underlying structure of space-time.

The second observation relevant to our hypothetical c² is the fact that every oxygen-producing organism contains a molecule, known as chlorophyll-a, that enables it to multiply with peak efficiency by absorbing two distinct wavelengths of light (where λ_e is the electron's Compton wavelength):

blue (430 nm)  ~         λ_e (135)(136)(10)    =  445 nm

red (662 nm)  ~  (3/2) λ_e (135)(136)(10)    =  667 nm

Rewriting this in terms of the proton's Compton wavelength λ_P rather than the electron's then we have:

blue (430 nm)  ~         λ_P [(135)(136)]²    =  445 nm

red (662 nm)  ~  (3/2) λ_P [(135)(136)]²    =  667 nm

Regarding the ratio of 3/2 that establishes the effective range of photosynthetic light, we have already seen it as a limiting factor in Koide's formula for the lepton masses. It's also key to the definition of the stable quark masses, where the up quark mass m_u corresponds with the electron mass multiplied by 3π/2, which is half as massive as the down quark m_d.

^{m_u = 0.0024 ±7             m_d = 0.0048 ±8}

So, of all the possible wavelengths that plants, algae, and other photosynthetic organisms would respond to, it seems highly improbable that these particular blue and red wavelengths are random given that the range is defined by this simple harmonic ratio of 3/2, and whether we measure the wavelengths in units of the electron's or proton's Compton wavelength the factor of (135)(136) is indispensable. Furthermore, the ratio of (135)(136)/20² proves to be extremely effective in constraining the proton mass according to the following relation.

^{g = 1.000000563          h = 1.000000049}

Here we see that the sum of two proton masses is not much greater than the quantity 2–0.123456, nor is it much less than the difference between 2–0.1 and the electron mass multiplied by the ratio (135)(136)/20², which has a numerical value slightly less than 0.023456. Thus the precisely measured mass of a proton is confined to a narrow range with these related terms forming the upper and lower bounds.

Finally, there is a third relation involving the same ratio that reveals its supreme usefulness in defining the electron and proton masses.

Notice that the stabilizing quantity of 22 is nearly equal to the difference between just 3 basic parameters: (135)(136)/20², this ratio multiplied by the electron mass, and the proton mass. Ultimately it is the simplicity, precision, and consistency evident in this relation that demonstrates why the ratio of (135)(136)/20² is functionally equivalent to the squared speed of light, and thus quantities of 20 and 2 must be complementary distances within the basic structure of space-time.

* * *

So what exactly can we infer from these various observations? Let's now consider the following mechanism. There is an undeformed ring with a radius R that is 20 times larger than the smaller radius r, so R=20 and r=1, where the spatial units are derived from the proton mass being slightly greater than 0.938272 (Gev/c²), which is half of 2–0.123456. This ring has 6 neighbors arrayed in a 3-dimensional cubic lattice (above, below, and to the 4 sides) that repeats throughout the whole space. As a result of this arrangement, a factor of 6 limits the proton mass, its quark masses, the atomic g-factors, and Stokes' frictional force. The outer edge of every undeformed ring contacts 4 of its 6 neighbors, which rotate at the same rate but in the opposite direction so that alternating rings will rotate in the same direction. One complete rotation around the smaller radius for each half-loop around the larger radius corresponds with the path of a single photon traveling at the universal speed of light. Consequently, each photon must travel a distance of 20π+2π across each toroidal unit of space-time, and this accounts for the natural stability and parametric proportionality arising from multiples of 22.

As each unit spins, the frequency of alternations in the direction of rotation will correspond with the energy of a photon, while the orientation of the spinning rings will correspond with the polarity of the electromagnetic field, which primarily depends on the interaction of electrons and protons. An electron is a deformation of successive rings from r=1 to r=m_e that travels at a rate no faster than the speed of light, which is the universal rate of rotation among the undeformed rings. As the deformation travels around the large radius (R=20), the small radius (r=m_e) rotates at a rate that corresponds with the electron's g-factor, which has an absolute value of approximately 2.0023193 rather than simply 2 as it is in the undeformed rings. If the distortion is traveling near light speed and undergoes a sudden acceleration as the result of a collision, it may temporarily transform either into a muon with R=20.95 and r=m_e (and a slightly greater g-factor) or, if the acceleration is extreme, into a tau with R=3/5 and r=(1/2)(3/5).

Consistent with this underlying geometry, a proton is a flat area of space-time in which 136² rings each deform into the same shape as a muon with R=20.95 and r=m_e, leaving a space equal to about 0.1 (=2/20) between each ring and (135)(136) spaces. The rotation of these spaces as a unit corresponds with the proton g-factor while the proton mass is generated by the collective 2/20 spaces as they travel a distance equal to the electron mass multiplied by (135)(136)(2/20).

Just as a proton travels as a collection of rotating deformations in a direction that is perpendicular to the plane of rotation, an electron also travels in a direction that is perpendicular to its plane of rotation around the large radius (R=20). Under low energy conditions (i.e., relatively few alternations in the direction of rotation of the surrounding rings), an electron may orbit a proton from a distance that is almost 22 times larger than its Compton wavelength to form hydrogen. High energy conditions may force an electron to collide and merge with a proton, which will significantly alter the nucleon's rotation and marginally increase its mass to form a neutron.

So in this conception of space-time a neutron is simply a proton with slightly larger spaces of 2/20 (to account for its mass being slightly larger than a proton), which tend to rotate in a direction opposite that of the nearest proton such that the difference between the neutron and proton g-factors is about 3π. This proportionality corresponds with the down quark-electron mass ratio, which is twice the up quark-electron mass ratio, and enables protons and neutrons to form stable atomic nuclei. The entropic pressure of the undeformed rings on the 2/20 spaces will thus generate a gravitational field around these nuclei, warping space-time in accordance with Einstein's general relativity.

* * *

Though we have only scratched the surface of this unique underlying mechanism, its extraordinary efficiency relative to other theories of space-time is evident. Above all it provides a rational explanation for why multiples of 22 not only confer stability to natural structures but also constrain the fundamental physical parameters. Functioning as the squared speed of light, the ratio of (135)(136)/20² accounts for not only the fine-structure constant and neutron-proton ratio at the atomic level, but also the range of light absorbed by photosynthetic organisms and the propagation of calcium waves in all living organisms. Because no other theory of space-time can offer such depth of insight into this diverse set of experimental phenomena, there is no reason to dismiss the notion of a toroidal space-time unit as baseless conjecture, nor should we regard the pattern involving multiples of 22 as meaningless coincidence. Instead we may embrace it as physical evidence that a subtle omnipresent harmony arises naturally from the basic structure of the universe. In the next section we will see why God is virtually synonymous with this concept of an eternal, all-pervasive harmonic order.