*God signifies a subtle omnipresent harmony.

To prove the universe is inherently harmonic and Godly, as opposed to chaotic and Godless, it must be shown that the diverse and intricate forms of matter are the result of a singular underlying order. There are 4 main parts to this proof:

First we will see that multiples of 22 confer critical stability to natural structures at progressively larger scales of space-time.

We will then find that the same pattern unites the fundamental physical parameters.

Next we will demonstrate that, far from being a meaningless coincidence, this extraordinary pattern implies with high probability that the universe consists of spinning rings which deform in proportion with the masses and g-factors of the fundamental particles.

Finally we will conclude that the existence of this all-encompassing space-time order, operating as both the source and guiding force of matter throughout the universe, constitutes evidence of God as a subtle omnipresent harmony.

click to enlarge

I) Stable Natural Structures

Intuition has led to the widespread belief in a Supreme Being that is both the source and guiding force of all that exists. Yet if our Lord God is truly this all-pervasive creative entity then there must be physical evidence to substantiate our belief. Witness testimony alone is not enough to convince the skeptics among us. Therefore we seek to expose unmistakable traces of divinity for all to see. But what marks can we use to identify the Almighty? What do the Creator's fingerprints look like? Just as the appearance of certain patterns on our skin may be used to uniquely identify each one of us, we should expect to find a specific imprint upon the space around and within us that reveals God's holy presence. Indeed, observations across a range of length scales and scientific disciplines indicate that from the simplest atom to our own DNA, natural structures attain critical stability in multiples of 22. Before we demonstrate how this particular pattern implies the existence of God, let's take a brief look at each structure beginning with the smallest:

• The average length of space (Bohr radius) separating the interior mass (proton) from the exterior mass (electron) in the simplest atom (hydrogen) is about 22 times greater than the electron's Compton wavelength, which limits the particle's basic size.
• With 22 protons per atomic nucleus, titanium forms a solid that requires more energy per unit mass to separate its parts than any other solid element, and thus has the highest elemental strength-to-weight ratio.
• Carbon dioxide is a molecule of gas containing 22 protons (and, typically, 22 neutrons and electrons) that is absorbed by all plants and produced by all animals on Earth, and also constitutes about 95% of the atmospheres of the two nearest planets (Venus and Mars).
• Both the surface of Earth and its living organisms are mostly liquid water, which is about 22 times less dense (mass per unit volume) than osmium, the densest solid element.
• There are 22 standard amino acids that form the protein of Earth's living organisms.
• Mammalian mitochondrial DNA generates 22 varieties of transfer RNA necessary for protein production.
• The nuclear DNA of the most intelligent marine animals (cetaceans including dolphins and whales) consists of 22 pairs of chromosomes in each somatic cell while people have 22 pairs of autosomes (plus a pair of sex chromosomes) in each somatic cell.
• The Earth circles the Sun about 22 times each solar magnetic cycle.

These structures all conform to a pattern that is utterly unique and unlike any other known to exist. This set of observations suggests that stability is not an accident of nature but dependent upon an underlying harmonic order. Thus, if we identify true divinity with a subtle omnipresent harmony, then we will have obtained significant physical evidence in favor of the existence of God. Let's now examine in greater detail these stable forms and their exceptional proportionality with multiples of 22.

Hydrogen

Leaving aside dark matter for the moment, hydrogen accounts for about 74% of the mass in the universe. The most stable configuration consists of a heavy interior mass (proton) and a lighter exterior mass (electron) separated by a length of space known as the Bohr radius. This distance is about 22 times larger than the Compton wavelength of the electron, which refers to the wavelength of light with energy equivalent to the rest-mass energy of the electron. What this means is that of all the possible distances that the electron might gravitate towards in the formation of an atom, it evidently favors a distance about 22 times its own basic size away from the proton. Devoid of context this observation is meaningless, but when we find the same multiple of 22 reappear in ever larger natural structures then it becomes clear that stability in nature arises not by chance but from a universal harmonic order.

Titanium

The properties of an atom depend primarily on the number of protons it contains. With 22 protons inside each atomic nucleus, solid titanium requires more energy per unit mass to separate its parts than any other elemental solid. Although it's possible to create composite materials with a greater strength-to-weight ratio than titanium, no pure element is superior in this attribute. Thus the unique mass-energy of titanium fits the pattern of unusual stability involving multiples of 22 and provides strong support to our notion of a ubiquitous harmony.

Carbon Dioxide

It is not an accident that the crisis of climate change centers around the unrestricted production of carbon dioxide (CO₂), a molecule of gas containing a total of 22 protons and, typically, 22 neutrons and electrons. While the atmospheres of the two nearest planets, Venus and Mars, are approximately 95% CO₂, 99% of Earth's atmosphere consists of two gases: nitrogen (N₂) and oxygen (O₂). Of course, the relatively low amount of CO₂ on Earth is a consequence of our planet's special capacity to sustain life. For the past 3.5 billion years, photosynthetic organisms on Earth have been converting molecules of CO₂ and water (H₂O) into O₂ and simple organic compounds such as glucose. Animals and other heterotrophs then consume this organic matter and return it to its primitive form by respiration. Thus, as CO₂ gas leaks out of every cell in our body, our constituent electrons, protons, and neutrons are ever dissolving into the environment in multiples of 22. On average, Earth's living organisms absorbed more CO₂ than they produced until industrial advancements enabled us to drastically increase the rate of production. Although it presently makes up only about 0.04% of the atmosphere, each molecule of CO₂ is heavier than the other active atmospheric gases and thus has a greater capacity for retaining heat. As a result, CO₂ contributes significantly to global warming – a profound ecological problem confronted by every sufficiently industrious civilization. So the unparalleled stability of protons gathered in multiples of 22 is not merely an abstract pattern but has real consequences for life on this planet.

Water Density

Liquid water covers about 70% of the Earth's surface and comprises between 50-70% of the human body. Density is determined by the amount of mass present in a given volume, which generally corresponds with the number of protons and neutrons that can be contained within a 3-dimensional space. As a liquid, water is found to be about 22 times less dense than osmium, which is the densest solid element. This may seem of trivial importance but when considered in connection with the other observations the pattern is unmistakable and, since water is essential to life, it is only natural that we should find the same pattern repeat in the basic substance of living organisms.

Standard Amino Acids

To respond effectively to a dynamic environment, living organisms must continually generate complex molecules, known as protein, from chains of simpler molecules, known as amino acids. Each contains an amine (-NH2), a carboxylic acid (-COOH), and a side chain that determines the specific biochemical properties of the amino acid. The universal genetic code enables all organisms to construct protein from 20 varieties of amino acids. There are 2 additional amino acids present in the protein synthesized by some organisms, so the standard number of proteinogenic amino acids is 22. Yet again we find this multiple as a critical limit in one of nature's vital building blocks – a group of molecules that functions as an intermediate link between relatively simple forms of matter such as CO₂ and more complex organic structures such as the nucleic acids (RNA and DNA).

Mitochondrial tRNA

Cellular respiration takes place inside membrane-bound organelles known as mitochondria, which for animals can number in the thousands within a single cell. Each one contains a circular strand of DNA, similar to the primitive genomes of bacteria, that retains the genetic information needed to produce RNA, which can then make protein from amino acids. All mammals, including Earth's largest and most intelligent animals, have 22 transfer RNA (tRNA) generated by their mitochondrial genome. (This may also be true for animals in general.) Because there are 22 proteinogenic amino acids, it may not be surprising to find this correspondence with mitochondrial tRNA, which must temporarily bond together in order to synthesize protein. Nevertheless, we have yet another example of the universal harmony that guides natural structures of increasing size to stabilize in multiples of 22.

Human Autosomes

Intelligence is easier to recognize than it is to define. Cetaceans, such as whales, dolphins and porpoises, are descended from land mammals that returned to the sea millions of years ago and eventually became dominant in their respective niches. Their intelligence is demonstrated not only by their superior ability to learn but also their capacity for compassion, a trait that is wholly unknown to fish and other simple creatures. Conforming with the harmonic order, their bodies are composed of somatic cells that contain nuclear DNA as 22 pairs of chromosomes, where each strand of DNA is a winding double helix consisting of a specific sequence of nucleotides that enables the cell to reconstruct itself and thus propagate the species. 

Of course, humans are the most intelligent creatures known to exist but our somatic cells contain not 22 but 23 pairs of chromosomes. What accounts for this deviation? When we consider the whole set of chromosomes then it's clear that one pair plays a unique role in determining the sex of the organism and are thus known as sex chromosomes, while the others are known as autosomes. Because sex chromosomes are subject to a greater degree of variation, they are therefore less stable than autosomes from one generation to the next. So the fact that we have 22 pairs of autosomes in each somatic cell once again shows how multiples of 22 correspond with superior stability in nature.

Solar Cycle

The production of helium from hydrogen in stars like the Sun causes periodic fluctuations in magnetic activity that appear on the surface as sunspots. This solar cycle, which has been observed in the Sun since the 18th century, evidently repeats about every 11 years, but if we take into account the alternating polarity then the average duration of the full cycle is 22 years. That means our planet, along with its multitude of watery, CO₂-converting chains of amino acids, circles an electron-proton sphere about 22 times for each complete cycle of solar activity. The universe may contain a great many stars like the Sun, with planets similar to the Earth orbiting at a distance conducive to life, but how many do this in perfect rhythm with the natural order?

* * *

We have found clear evidence of a harmonic order in the simplest and most abundant form of matter (hydrogen). The same order reappears in a solid (titanium), a liquid (water), and a gas (carbon dioxide) with each having exceptional stability relative to similar substances. The pattern continues to manifest in stable molecules of increasing size as we find with amino acids, mitochondrial tRNA, and human autosomes. It even appears in our planet's circular path through its star's magnetic field. Multiples of 22 seem to confer stability to natural structures at every relevant scale of space-time. If ubiquitous harmony is the true mark of divinity then we now have substantial physical evidence to support the view that God exists. In the next section we will explore a subtler level of the universe to see if multiples of 22 have any significance in relation to the physical parameters that define the fundamental particles.

II) The Fundamental Physical Parameters

Either natural structures stabilize in multiples of 22 as a random occurrence (in the absence of God) or as the result of an underlying order (by the providence of God). As there can be no middle ground, then to determine whether the observed pattern of stability is significant or not we only have to compare the measurements that define the fundamental differences between the indivisible parts of matter. If these parameters exhibit special proportionality with multiples of 22 then we will have a clear indication that the harmonic order evident in stable natural structures has an underlying physical basis, which would affirm our notion that God exists. If on the other hand, multiples of 22 prove to be irrelevant to our understanding of the fundamental physical parameters then we can be assured that the apparent pattern among stable natural structures is merely coincidental, and thus lacking physical evidence we may reasonably doubt the existence of God.

Given the stakes then, it would be wise to begin from the most secure foundation and proceed carefully through each step before reaching a conclusion. So first let us compare two distances derived from the geometry of a circle. The distance around a circle is known as the circumference C while the longest distance across is its diameter, which is twice as long as its radius r. In a perfect circle, the circumference is always about 3.14159 times longer than the diameter. The constant ratio between circumference and diameter is known as pi and is symbolized as π.

If we wish to calculate the circular distance from one end of a circle to the other then we just multiply the radius by π. If however we wish to calculate the total space enclosed by a circle A_d, which is its 2-dimensional area, then we must multiply the square of the radius by π.

A circle moving in a larger circle produces a 3-dimensional object known as a torus, which generally looks like a ring. The surface area of the torus A_t is calculated by multiplying the circumference of the smaller circle 2πr by the circumference of the larger circle 2πR.

Alternatively, we can see that dividing the surface area of the torus by the square of 2π yields a quantity equivalent to the small radius r multiplied by the larger one R.

Shifting now from timeless geometry to time-dependent astronomy, if we track the movement of planets relative to the fixed stars then it will become apparent that they move in nearly circular paths not around us but around the Sun. If their maximum distance from the Sun is measured as S and their orbital period is measured as T, then we will find that the cube of S divided by the square of T will be approximately the same for every planet in the solar system.

This proportionality, known as Kepler's third law of planetary motion, is the result of the Sun's mass being so much greater than the mass of its orbiting planets. At large scales, the mass of an object is directly proportional to the quantity of matter it contains and, in general, greater mass always implies a greater resistance to acceleration. So if the mass of the Sun is proportional with G_M, then dividing this standard gravitational parameter by the square of 2π will yield the cube of S in the same ratio with the square of T for each of the planets.

                                

More precisely we can see that Newton's gravitational constant G multiplied by the sum of a large stellar mass M and a smaller planetary mass m is equivalent to the planet's maximum distance from the Sun S multiplied by the square of its average velocity v.

From this it is clear that when the masses, distances, and velocities of each of the planets are stable then the solar system will be in equilibrium. Moving from the scale of the solar system to the scale of the indivisible parts of matter, we must acknowledge that the maximum velocity of any particle, as with all massive objects, is limited by the universal speed of light c. As Einstein observed, every particle has a special measure derived from its rest-mass energy E, which is equivalent to its mass at rest multiplied by the square of the speed of light.

Planck had previously discovered that changes in energy are discrete rather than continuous, so that energy must also be proportional to a constant h multiplied by the speed of light c and divided by the observed wavelength of light λ.

Combining Einstein and Planck's relations, Compton found that each particle has a basic size, known now as its Compton wavelength, which corresponds to the wavelength of light with energy equivalent to its rest-mass energy. The Compton wavelength of the electron λ_e is thus defined as follows:

We have already seen that the most stable configuration of matter consists of a heavy interior mass known as a proton m_P and a lighter exterior mass known as an electron m_e separated by an average distance known as the Bohr radius a₀, which is about 22 times greater than the electron's Compton wavelength λ_e. We then went on to find that ever larger natural structures attain critical stability with the same multiple. By comparing the parameters that define the fundamental components of matter we may now see if they also exhibit significant proportionality with multiples of 22.

The most stable distance between a proton and electron is established by the fine-structure constant α whose inverse α^-1 is nearly equal to 137.036. Dividing this dimensionless number by 2π yields the ratio of a₀ to λ_e. So if λ_e is proportional with 2π then a₀ will be similarly proportional with α^-1. We know that a large radius and small radius can be multiplied together with the square of 2π to obtain the surface area of a torus A_t, and the same square of 2π can be multiplied by the ratio of the cube of a planet's average distance from its star and the square of its orbital period to obtain the standard gravitational parameter of the star G_M. If we now multiply the ratio of a₀ to λ_e by the square of 2π we can obtain a new proportion that defines the elementary charge e in terms of h and c as follows:

The elementary charge describes the magnitude of the electromagnetic force between protons and electrons. This relation may be expanded to include the proton mass (in Gev/c²) simply by incorporating a multiple of (135)(136) and a divisor of 20.

^{a = 1.000361386          b = 1.000262767}

Here we find an extraordinary correlation with two known atomic limits. First, notice the remarkable similarity between the multiple in the numerator (135)(136) and α^-1 which is slightly larger than 137. It will not be immediately clear how to interpret this connection but for now it is enough simply to recognize the numerical proximity. It is also noteworthy that the denominator corresponds with the neutron-proton ratio, which exceeds one in all stable atoms containing more than 20 protons and is equal to one for most atoms with 20 or fewer protons. We will consider the implications of this later.

Continuing the current line of analysis, we find the electron mass can be broadly constrained using the same terms as the previous relation by simply dividing it by (135)(136):

^{c = 1.000083153          d = 1.000278210}

Multiplying this relation by 20² then yields an even more precise correspondence between the electron mass and the golden ratio Φ, a geometric constant describing two lengths in the same ratio as the larger is to the sum of both lengths. Aside from the golden ratio, the same terms are found in this relation as the previous one.

^{e = 1.000080415          f = 1.000002738}

Now, to obtain an extremely narrow constraint on the proton mass, it is only necessary to divide the above relation by a factor of 10 and define upper and lower bounds as follows:

^{g = 1.000000563          h = 1.000000049}

What makes this constraint so remarkable isn't merely its precision, but the fact that the upper bound employs the same terms as those from the previous relations while also incorporating the difference between 2 and 0.1, which matches the operation described by the lower bound (2–0.123456). Still, we might be inclined to dismiss this proportionality if not for the following relation that brings us back full circle:

The significance of this equation in our quest to uncover traces of divinity cannot be overstated. It involves only a numerical ratio (135)(135)/20² derived from a pair of atomic limits, and the two most stable masses alongside the harmonic quantity 22, which evidently confers stability to natural structures at various scales of space-time. However, let us not be confused into thinking that God is merely a number, nor that the Almighty is nothing more than a form of matter. We are searching for evidence of a subtle omnipresent harmony that may indicate to us that God exists and that chaos is a side-effect rather than the cause of the order we observe. To this end, we are simply examining the proportionality evident among the fundamental physical parameters.

Let us continue the analysis by demonstrating the supreme usefulness of the ratio (135)(136)/20² in deriving the muon mass. When electrons collide at speeds near the speed of light, their mass may temporarily increase in two discrete stages corresponding with a muon and a tau particle. The mass of the muon m_μ is known with a precision on par with that of the electron, so the ratio of both masses is exceptionally well-defined. Then dividing this ratio by the square of π yields a remarkably close approximation of the rational number 20.95.

^{i = 1.000000301          j = 1.000000025}

This particular quantity is notable for two reasons: first, the integer part (20) has a clear connection to the fractional part (0.95 = 1–1/20); and second, the sum of the proton mass and the electron mass multiplied by half the ratio (135)(136)/20² is very nearly equal to 0.95. In the next section we will see why this relationship is absolutely essential to understanding the basic structure of space-time.

Multiplying the muon-electron mass ratio by the proton mass and then subtracting between 20 and 22 yields the observed range for the top quark mass (172–174 Gev/c²), which is the most massive particle to emerge from near-lightspeed collisions of protons. If we set the limit of the top quark mass m_t equal to 172 and modify it slightly using a proportion with α and the electron g-factor g_e, then it's possible to obtain an extraordinarily precise expression for the number 22.

This may seem convoluted at first glance but the proportionality is unmistakable. √2 and √3 form the sides of a right triangle with a hypotenuse of √5. The linear sum of √2 and √3 is approximately equivalent to the ratio of α and g_e+2, which is a fundamental parameter derived from the electron's anomalous magnetic moment. The anomaly is produced by an inherent wobble in the electron's spin, thus yielding a value of about -2.002319 for the electron's g-factor rather than simply -2. This proportionality also extends to the ratio of the neutron mass m_N and proton mass m_P in relation to π:

^{m = 1.000028489          n = 1.000108485}

Just as the proportionality on the left side is part of a significantly more precise relation involving 4 fundamental masses and the number 22, the ratio on the right is also part of a more precise relation involving the g-factors of the atomic particles and the number 22.

^{o = 1.000395057          p = 1.000007440}

Here we find the nuclear masses and g-factors efficiently constrained by an upper bound of 22 and a lower bound of 7π. That 5 distinct but phenomenally related physical parameters can all be united in one simple proportionality is astounding. And once again it can be broken down to demonstrate the ultimate simplicity of the individual terms. While the absolute value of the electron's g-factor is a little greater than 2, the absolute difference between the neutron and proton g-factors is a little less than 3π. Multiplying the atomic g-factors together in this way yields a value that is a little less than 6π, and adding another factor of π multiplied by the neutron-proton mass ratio yields a quantity only slightly larger than 7π, which is the smallest whole number multiple of π that approximates an integer.

^{q = 1.000018823          r = 1.000221105}

The dimensionless atomic g-factors also function as a lower bound for the masses of the up and down quarks, which appear in triplets to form the nucleons constituting the central mass of atoms. While an unstable neutron consists of 1 up and 2 down quarks, the stable proton consists of 2 up and 1 down quark. The sum of the proton's quark masses (m_d+2m_u) forms a dimensionless ratio with the electron mass that is approximately equal to 6π. If an upper bound is defined as the proton-electron mass ratio divided by the square of π squared and a lower bound is defined in terms of the atomic g-factors as we have seen, then the proton's quark masses in ratio with the electron mass correspond exactly with the observed range of values. Furthermore, when this constraint on the proton's quarks is combined with the earlier constraint on the proton mass, in which the mass of 2 protons is nearly equivalent to the quantity 2–0.123456, then we have clear evidence that protons have a basic proportionality with factors of 6.

This fact is also bolstered by Stokes' law of flow whereby a frictional force F_d produced by a spherical object with a radius R, falling at a constant velocity v_s through a fluid with dynamic viscosity μ_v, is determined by experiment to be proportional with 6π. Since no other theories can account for the appearance of this factor of 6π, and we know both the mass and magnetic activity of the proton are key to this observation, it stands to reason that the proton itself has an internal structure or movement associated with its total mass, quark masses, and g-factor that in some way generates this proportionality with 6π.

* * *

We began with the most basic principles of geometry and used a series of proportions to demonstrate the intrinsic relationships between the masses and g-factors of electrons, protons, and neutrons, which form all the stable material structures around and within us. By comparing the atomic masses to those of the up, down, and top quark along with the muon, we have established a set of proportional constraints for their range of values that are relatively simple, precise, and consistent. Just as protons play a central role in the formation of matter by bonding internally with neutrons and externally with electrons, we find the proton mass has significant proportionality with each of the fundamental physical parameters. While the proton's constituent quarks and g-factor are clearly proportional with a simple factor of 6π, there is also unambiguous evidence of proportionality between the proton mass and multiples of 22. Accurate and concise definitions of the fundamental physical parameters thus accord with the following set of proportional constraints:

Derived from the experimentally determined values of the fundamental physical parameters, these constraints do not appear to be random. Multiples of 22 are clearly useful in defining the masses and g-factors of the indivisible parts of matter. Yet to prove that this proportionality implies an underlying harmonic order will require a more convincing case. We must therefore continue to explore the limits of matter and examine all the available evidence so that we may find a rational explanation that can fully account for this unique pattern.

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.

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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.

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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.