Whether by coincidence or not, certain particle mass ratios, in physics, happen to be nearly equal to certain geometric ratios in simple patterns. These patterns are somewhat analogous to "close packing" of spheres. This article correlates some of these particle mass ratios with some volumetric ratios in simple patterns.

DESCRIPTION: (All spheres intended perfectly "round" and touching.)

Geometric Pattern (Centers of all spheres coplanar)	(See Pattern) Volumetric Ratio	("Important" Particles) Ratio of Masses	"ave" Mass Ratio
case "A" R/r = 6.4641/1	1 large sphere to 1 small sphere (centered in the pattern): 270.10/1	pion+ or pion- to electron: 273.13/1, piono to electron: 264.14/1	270.13/1
case "B" R/r = 9.89898/1	3 large spheres to 3 small spheres (all 3 smaller spheres also same size) 970.00/1	Kaonso or KaonLo to electron: 973.92/1 Kaon+ or Kaon- to electron: 966.04/1	969.98/1
Close Up Views   case "B" and case "C" R1/r =9.89898/1   R2/r =13.9282/1	3R13/6r3 + 3R23/6r3 = 1836.00/1, i.e. 6 equal small spheres with radius r, 3 intermediate size spheres with radius R1 as in case B, and 3 large spheres with radius R2 (case C)	Proton (or antiproton) to electron: 1836.15/1 Neutron (or antineutron) to electron: 1838.68/1	1837.42/1

In some simple geometric patterns, such as when three large touching spheres surround one or three small touching spheres, certain volumetric ratios arise, (See Table above).

When comparing the masses of certain important particles (pions, kaons, and protons, with electrons) certain particle mass ratios also arise, (See Table above).

In some cases (shown above), the geometric volume ratios and average mass ratios are nearly equal.

The centers of all the spheres are co-planar in all the above patterns.

Any readers who find the above Table rather self explanatory, may just "scan" the three
paragraphs below, or skip down to the addendum.

The first two patterns (case A and case B) consist of three large spheres, in a triangular pattern, surrounding one and three small spheres, respectively. The volumetric ratios (large sphere to small sphere) are compared to the mass ratios of so-called "semistable" mesons to electrons. (The "semistable mesons" above are generally more stable than most mesons, the less stable ones being discovered later, historically.) Case A compares "pions" to electrons, and case B compares "kaons" to electrons.

The last "pair" of patterns shown in the table, case "B and C ", involve six equal small spheres. It also involves three "intermediate" size spheres (as in case B), and three "larger" spheres (shown in case C). The packing, in case C, is less efficient than case B, as each large sphere is touching only one small sphere instead of two. The average volumetric ratio (last pair of patterns) is the three "large" spheres plus the three "intermediate" size spheres divided by six small spheres. This volumetric ratio is compared to a mass ratio consisting of the average mass of a proton, antiproton, neutron and antineutron, to the mass of an electron. ((If the neutron and antineutron were ignored, the ratios comparison would be in better agreement. The proton is a stable particle, but the mean life of a neutron outside of a nucleus is about 12 minutes.))

From data in various books,^1-4 or perhaps more recent sources, one may calculate or "check out" the approximate ratios found in the above Table. (R and r denote the radii of large and small spheres, respectively.) ((The volumetric ratio (large sphere to small sphere) varies as the cube of their radii, i.e. (R/r)³.))

REFERENCES AND NOTES:
(1) Dalitz, R. H.; Goebel, C. In McGraw-Hill Encyclopedia of Science & Technology, 7th ed.; McGraw-Hill,Inc.: New York and other cities, 1992; Vol.10, "Meson". P 662
(2) Handbook of Chemistry and Physics, 73rd ed.; CRC Press: Boca Raton, FL, 1992; Section
1-2, Table 2, The 1986 Recommended Values of the Fundamental Physical Constants".
(3) Semat, H. Introduction to Atomic and Nuclear Physics, 4th ed.; Holt, Rinehart & Winston: New York, 1962; Chapter 15, p526.
(4) Note, the simplest case (three spheres surrounding one) is also addressable using Descartes' "Circle Theorem". Later mathematicians greatly expanded the form and scope of this theorem.
(5) Note, some aspects of sketches may remind one of Tait's "Dynosphere". Some of Tait's work is described in V. B. Ginzburg's Unified Spiral Field and Matter, pp. 385-387, Helicola Press, Pittsburgh, PA (1999).

ADDENDUM: Speculative Thoughts and Miscellaneous Comments

We now attempt to explain why particle mass ratios and geometric volume ratios are nearly equal, as described in this article.  I doubt if it is just coincidental. 

Historically, Huygen visualized a space filled with ethereal spheres for his effective treatment of light’s behavior.  It appears that Maxwell and Peter Tait also toyed with a notion of ethereal spheres in space. (I doubt that well defined, small spherical electrons actually dwell neatly between the large spherical nucleons in the nucleus, itself.)  But I think that small and large ethereal spheres do likely exist in ethereal space!  (Or something equivalent.)  And that the large energized ethereal spheres have larger energies than the smaller ethereal spheres between them, and in proportion to their greater size.  I believe the following occurs, (or something like it):

Digest:  There exists in most of space, spinning vortices (or the like) of ultra low density matter, rotating at ultra high speed.  Perhaps they are spherical spinning balls of aether, (about the size of ‘the Bohr hydrogen atom’).  These ultra high energy spinning ethereal balls help provide the ultra high ethereal pressure in space.  Those ethereal balls have great spin -- roughly a Planck’s constant amount of angular momentum, despite their low density.  That causes prospective long-life particles (like the proton) to develop roughly a Planck’s constant worth of angular momentum.  Gross particles, such as the proton, must exhibit that much angular momentum to be compatible to the ethereal spinning spheres nearby, and thus survive.  (Particle spin may also aid stability.)

Nuclear matter has approximately the highest density that compact matter can have in the universe.  Protons (and electrons in the nucleus) are made of nuclear matter; and, therefore, they have very high density. (That concept is consistent with the Bohr ‘liquid drop model of the nucleus’.)  The interaction, between the ultra high pressure aether and the very high density nuclear matter, leads to, roughly, ‘the velocity of light’ as being the maximum speed that nuclear matter can obtain. 

So the prospective stable proton is encouraged to form with these attributes:   It has, roughly, the highest density of matter possible -- but, roughly, also exhibiting a ‘Planck’s constant’ worth of angular momentum as it spins.  And it spins at roughly, ‘C’, the highest speed possible.  And, physically, the proton maintains a nearly minimum spreading-out of itself through space -- while still exhibiting that much angular momentum. 

There exists in space -- small balls and very small balls of energized aether.  These tend to form in patterns, as pictured above; and therefore such ethereal balls are more stable than otherwise.  The small and very small balls fit between the crevices of larger ball arrays, etc.  Small balls of aether interact with the prospective proton.  Some of those aether balls are somewhat larger and some are somewhat smaller, in size and energy, -- compared with the proton.  But the average energy of those aether balls (i.e., some likely bigger and some likely smaller than the proton) roughly equals the energy of the proton.  So an ‘equipartition’ of energy interaction occurs, and the proton helps promote that.

Thus, the spinning proton causes much of space to form patterns consisting of small and very small sized ethereal spheres.  Thus, these ethereal spheres contain small and very small “quanta’s” of energy, respectively.  Those are illustrated in the pattern shown above.  Then, those ethereal ball arrays, in turn, help to maintain the stability of the proton (by ‘feedback’), and the stability of the electron (a particle much less massive than the proton).  And also some stability of some other important particles in physics.

Thus, the small and very small energies of the small and very small aether balls, respectively, help stabilize protons, electrons and other particles too.  ((The rather non-concentrated electron, has to spread out, (perhaps like a spinning doughnut) to roughly generate a Planck’s worth of angular momentum.  Thus it would seem, at first, that the electron would be a poor candidate for stability.  But the many standard very small ethereal balls in space, that fit so well into the ethereal patterns in space, maintain the electrons’ stability, by sharing an equipartition of energy condition with it.  ((Incidentally, according to some theories; that supposed ‘doughnut-shaped’ ‘free’ electron is also like a twisted-dough doughnut.  I.e., it also rolls as it spins, (with say, a ‘clockwise’ roll if it is an electron, and ‘counter-clockwise’ roll if it is the mirror image of the electron, namely a ‘positron).))

Optional concluding remarks:  An equilateral triangle has been depicted, by the mathematician, Richard Courant, as exemplifying the simplest figure in two-dimensions from a structural or "combinatorial" point of view.  And some ancient Greeks regarded a sphere as the perfect form.  Those are like the patterns shown in my illustrations, above.  One of Courant’s associates (Ian Stewart) even wrote a book entitled, Fearful Symmetry, Is God a Geometer?

It is interesting to note that the non-spinning, non-charged (neutral) kaon particle tends to break up shortly into smaller particles that do spin!  And those particles ‘develop’ so-called ‘charge’.  And many of those, in turn, break up to form electrons, i.e., very stable elementary particles, with spin and so-called ‘charge’.  Consider this:  It seems very unlikely that the little mundane (non-spinning) type of kaon has a ‘standby’ miniature centrifuge inside it.  Nor something like an automated sugar coating dip-bath to ‘surface coat’ the evolving elementary particles with ‘charge’ (like ‘M & M’ candy’s hard surface coatings)! 

We, thus, conclude this:  It is the appreciable spin of the major spinning ethereal balls (in space) that causes non-spinning particles in our world to develop spin!  (Or break up into other particles that develop spin.)  Those various style spins of various types of particles continue to spin in that environment.  That spin and spinning environment is the cause of particles ‘attracting or repelling’ one another.  That is what humans have chosen to call ‘electrostatics’.  I.e., or Coulomb’s attraction and repulsion.  In other words, to cope efficiently with the ensuing ‘paradigm’; humans have concocted the abstract word, ‘charge’, namely ‘positive charges and negative charges’!  But ultimately, so-called ‘charge’ and charge behaviors are caused by Planck’s constant related spins of ethereal balls or vortices in space! 

If the reader wishes, he/she may click over to my article, ‘What We See and What We Don't See’, for estimates of the density, velocities and pressures of the ‘aether’ in space.  One also finds there a somewhat speculative basic discussion of ‘combinatorics’, regarding the ‘primary attributes’ that ‘happen’ to be in this world, including ‘angular momentum per volume’.

Optional, continued:  To put the subject of ‘electrical forces’ in perspective, compared to nuclear and gravitational forces; I add these conjectures:

In the case of the very strong ‘nuclear forces’; those strong forces arise because of the following factors:  ‘Bernoulli-related’ forces arise, associated with the flow of very high density nuclear matter.  That flow arises within the nucleus, itself; and the flows is roughly at the speed of light.  That (and the ultra-high external aether pressure) causes the strong nuclear forces to develop, i.e., the strong so-called ‘attractive’ forces.

In the case of electrical forces; electrical forces are also strong forces, but not as strong as nuclear forces.  That is likely because the electric forces arise due do lower mass particles (or only part of more massive particles).  Let us compare the forces required, say, to pull a proton apart compared to pulling a ‘nuclear electron’ away from a neutron.  Here is an analogy:  Von Guericke’s many horses could not pull his two large hollow ‘Magdelburg’ hemispheres apart, but could have easily pull a small cork off a bottle, having a similar vacuum interior. 

Particles, such as the ‘free’ electron, (despite their often ‘puffed-up’ volumes) are not as able to fully harness the pressures of space, to create as much total force resultant -- compared to what the compact higher-mass particles accomplish.  But the high pressures that those ‘electric’ particles harness are due to much faster-than-light circulations of ‘thin aether’, subtly directed.  (Some more details are provided in my other articles.)

Now for Gravitational forces:  They are very weak forces compared the above forces, because they arise between two or more particles likely due to weak ‘Bernoulli flow-related’ effects.  In that case of gravity; the weak forces arise because the average aether velocity is relatively small.  It arises only because, say, two ‘ponderable or gross’ bodies are vibrating or have some motion at roughly the speed of light, C.  And that causes a resulting constricted aether flow between them; and that flow involves very low density aether in the region.  I.e., that flow is at very roughly ‘C’ -- ‘more or less’. 
In other words, when two particles are very near each other; weak gravitational forces develop and very strong nuclear forces can develop. Here is an analogy:  If two identical propellers both rotate at ten times per second, one in high-density water, the other in low-density air; then the one in water will cause much greater ‘undertow’ force than the one in air.  I.e., that principle makes it risky for kids to wade in the deep water off ocean beaches!   (That said, however; I still do not know all the details regarding the interaction between the aether in space and gross matter -- regarding gravity, etc.)    

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