Particle
Mass Ratios, and Similar Geometric Volume Ratios
by Carl R. Littmann (revised 9-2-2007, and second table added 11-1-2009)
The first Table below informally shows relationships in an earlier article by me and published in the Journal of Chemical Information and Computer Sciences, 1995, 35 (pp. 579-580)
DESCRIPTION: (All spheres in article intended perfectly round, touching and coplanar.)
Geometric Pattern |
(See Pattern) |
(Important . .Particles) |
ave. Mass Ratio |
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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 |
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|
3 large spheres to 3 small spheres (all 3 smaller spheres also same size) |
Kaonso or KaonLo to electron: 973.92/1 |
969.98/1 |
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Close Up Views
|
3R13/6r3 + 3R23/6r3 = 1836.00/1, i.e. |
Proton (or antiproton) to electron: 1836.15/1 |
1837.42/1 |
Click frames to show patterns faster.
Whether by coincidence or not, certain particle mass ratios, in physics, are 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.
ADDENDUM (11-1-2009) – Important:
Happily, I have recently discovered a previously over-looked pattern ratio and nearly matching particle mass ratio, (the ‘Muon’ to Proton mass ratio). Thus, we can now attach the below ‘addendum Table’ to previous (first) Table, ‘figuratively speaking’. The most notable difference is that the medium sized spheres shown below are not merely surrounded by the larger spheres – they are entirely inside of them!
ADDENDUM TABLE (for appending to previous table)
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Geometric Patterns … continued …..
. 
Case ‘D’ and Case ‘E’
r1 / R = 0.5/1.0 r2 / R = 0.46410/1
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Volumetric Ratios ((below, we assume the vol. of each large sphere is 1836.15.
That corresponds to the proton mass equaling1836.15 ‘electron masses’, (‘me’).
Next, we calculate the ave. vol. of the shaded spheres in terms of electron masses)):
1836.15 [ (r1)3 + (r2)3 ] / 2 (R3) = 206.54 me est. for muon vs. 1836.15 me for proton.
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Important particles and their empirical Mass Ratio:
Muon– (or Antimuon+) vs. Proton: 206.77 me actual for muon vs. 1836.15 me for proton
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Let us start by commenting on the first table presented, not the last just shown.
In some simple geometric patterns, such as when three large touching spheres surround one or three small touching spheres, certain volumetric ratios arise. When comparing the masses of certain important particles (pions, kaons, and protons, with electrons) certain particle mass ratios also arise. That first table correlates the cases where the geometric volume ratios and average mass ratios are nearly equal – for basic patterns where spheres are always outside one another.
And the second (addendum) table shows an additional case where the geometric volume ratios and average mass ratios are nearly equal, i.e., the muon to proton case. The second table is somewhat like the first; except in the second table, the sphere patterns are entirely inside the larger spheres, and one of those enclosed set of spheres consists of only two spheres. (In fact, I would like to correspondingly show two large spheres surrounding one small sphere, but that, of course, would not quite ‘close the horse corral’. Nor would that result in structures having ideal symmetry, simple combinatorics, or fine basic structure.)
Had the large sphere we used - to calculate the muon’s mass - been based on the Neutron (with mass equivalent to1838.68 electrons) instead of the Proton (1836.15 electron masses); our estimate would have been still better; although landing on the ‘high side’.
Our estimate does not “hit the exact center of the bull’s eye”, but is remarkably close. And let us remember this: The geometric patterns, used in all the comparisons, are rather similar, and so very basic! And they had been previously used to make impressive estimates, in the first table, before attempting another satisfactory use of them. And, indeed, by using a related pattern again -- another good comparison was achieved! (See second table). Thus, I think that all the above together -- provides continuing, connective and escalating evidence that the corresponding relationships shown are not likely just coincidence!
Again we stress that the centers of all the spheres are co-planar in all the above shown patterns and Tables.
Important: In the first table; the four-sphere pattern in case ‘A’ (the pion case) would exactly fit into each of the three very large spheres shown in case ‘C’. ((Case ‘C’ is the larger of the two co-acting patterns shown in case ‘B & C’. The spheres from those two co-acting patterns, averaged together, represent our proton The full extension of the co-acting pattern shown in case ‘C’ extends beyond the small portion of it shown in the first table.).)) Click to view optional “Fig. 3, Visual Aids for other relationships”. By comparison, the proton sphere is relatively small. The fraction by which the co-acting pattern in case ‘C’ (the ‘9-Pion’ pattern) exceeds the relatively modest proton’s radius – roughly represents the extra extension of ‘nuclear forces’ beyond a proton’s or a nucleon’s surface. (Yukawa also used his own conception of ‘pion action’ to account for the extended range of nuclear forces.) All this paragraph may relate to the magnetic field of the proton being several times greater ‘than expected’.
Any readers who find the above Tables rather self-explanatory, may just ‘scan’ the four
paragraphs below, or just skip them and scroll down to the next addendum.
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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 the vast number of particles that have been discovered since 1950. By today, so many different particles have been discovered, that many scientists consider ‘all the species’ as comprising ‘a particle Zoo!’ This article addresses the more basic ones: ‘Case A’ compares ‘pions’ to electrons, and ‘case B’ compares ‘kaons’ to electrons, and etc.
The last ‘pair’ of patterns shown in the first Table, case "B and C ", involves 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.))
The second Table features the Muon, an important non-stable particle discovered in 1936. But it has a longer half-life than most particles discovered later. The corresponding geometric patterns shown might seem to “turn the first table’s paradigm ‘inside out’.” In fact, for quite a number of reasons, the muon, historically, did turn the expectations of the middle 20th century Physicists ‘inside out’. I.e., the muon did not turn out to be the (pion) particle that Yukawa predicted – the muon did not even turn out to be a ‘true meson’ at all! And when the muon’s unusual characteristics came to Rabi’s attention, Rabi exclaimed, “Who ordered that!” (the muon).
From data in various books,1-4 or perhaps more recent sources, one may calculate or ‘check out’ the approximate ratios found in the first Table. ‘Wikipedia’ might be used as a preliminary source for the muon and most other particles in the ‘particle Zoo’; but even better, it may refer readers to more the specialized (original) sources.
(In the above Tables, 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).
(6) Note, the volume of any very large sphere in case ‘C’ is exactly equal to the volume of the very small sphere at the center as shown in case ‘A’ plus 10 times the volume of any modest-sized sphere also shown in case ‘A’. (Of course, that assumes, as intended, that each of the very small spheres shown in cases ‘A’, ‘B’, and ‘C’ are equal.) Note #6 is true even when the volume of each very small sphere equals 1 unit (a simple integer), and then the volume of each surrounding sphere in Cases ‘A’, ‘B’, and ‘C’ comes out as a ‘irrational’ number.
(7) Note, although not shown; imagine a tetrahedral array four large spheres surrounding a tetrahedral array of four very small spheres. Suppose that both tetrahedral arrays are similarly directed. (That inefficient packing is the ‘non-planar’ analogy to the ‘planar’ inefficient packing shown in case ‘C’, the three large spheres surrounding three very small spheres.) Assuming, as usual, that each of the very small spheres are equal; then the large spheres generated in that inefficient tetrahedral case would (surprisingly) have volumes equal to that shown in case ‘B’ instead of case ‘C’.
(8) Note, the possibility that aether consists of material spheres -- is mentioned in: Christiaan Huygens’s Treatise on Light (1678), English rendering by S. P. Thompson, 1912, University of Chicago Press, (Project Gutenberg eBook, book #14725, released 1-18-2005) http://www.gutenberg.org/etext/14725
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Addendum (9-2-2007): 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 low density, 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 for high density bodies. 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, see first table. 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.
((Optional: Incidentally, according to many theories; the ‘free’ electron is larger when it is outside the nucleus. And some detailed theories suppose that the free electron assumes a ‘doughnut-shape’ and that it is also like a twisted-dough doughnut. I.e., that is, it also rolls as it spins, (with say, a ‘clockwise’ roll if it is an electron, and ‘counter-clockwise’ roll if it is a ‘positron’, the mirror image of the electron). I have sometimes wondered if one layer of that ‘dough’ might be spinning, and a separate layer of ‘dough’ is rolling. Or even if the ‘stunt’ is being done mainly by the aether around the electron instead of the electron’s material, itself.))
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.
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’.
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Optional, continued: To understand the great differences in magnitude manifested by electrical forces vs. nuclear forces vs. gravitational forces - I add these few pages of conjectures (to put all that in perspective):
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 and stays close to the nucleus, itself; and the flow is roughly at the speed of light. That, and the ‘Bernoulli equation’ (or principle), and the ultra-high external aether pressure -- causes the strong nuclear forces to develop, i.e., the strong so-called ‘attractive’ forces – with their short distance limitation.
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 (i.e., the proton case), but the horses could have easily pulled a small cork off a large bottle, having a similar vacuum interior (i.e., somewhat like pulling a small mass electron from a neutron – in our analogy).
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 ethereal pressures that those ‘free electric’ particles harness (especially when separated by quite some distance from other ‘charged’ particles) 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 to the nuclear and electrical forces, because they likely arise between two or more particles due to very weak ‘Bernoulli flow’ effects. Let me try to explain it better than I once did.
First and foremost, gravitational ‘attraction’ occurs, say, between two particles, because of an aether flow between them. And that aether flowing is a very low-density flow. And let us now compare that low density flow to the high-density flow of nuclear fluid, say, between two nuclei that are rather close together, i.e., ‘nuclear forces’. We first consider the cases where the aether and the nuclear fluid are flowing at the same velocity: (The density of the flowing material is a major factor in Bernoulli’s equation that determines the forces arising.) Thus, when the velocity of the flows are equal, the gravitational forces will be lower by a magnitude proportional to the aether’s lesser density compared to thicker flowing nuclear fluid’s higher density. (Using a partial analogy, if you are hit by a 10 mph thin air breeze, you will be affected much less than if hit by a thick10 mph water stream!)
Of course, gravitational forces, although weak, can exert themselves at much greater distances than nuclear forces, because although the nuclear fluid flow is confined to close to the nuclei, the ethereal movements can extend out to very great distances through space! Of course, an ethereal ‘stream’ is less congested by two particles when the particles are widely separated – a factor causing the gravitational forces to be less when the two particles are very far apart. (And the ‘asymmetry’ in space is less, then, also.)
A final subtle point I wish to make is this: A quickly changing breeze can hit you at 10 mph in the face and then 10 mph in the back. But that does not mean that air molecules vibrate with only 10 mph speed, nor that the ‘speed of sound’ is only 10 mph. Instead, those underlying vibrations and speed are grossly greater -- closer to 800 mph.
Important: By analogy, I think that the aether in space is like a quickly changing cyclic air breeze; it has two vastly different velocities or movements associated with it. One is like the very fast (air) molecular or particle vibrations – the underlining speed. And the other is like the changing cyclic winds, say averaging a slow 10 mph, north to south and back again, say, in a second – the ‘over-lining’ speed. (Or like a continuous fast vibrating radio wave that can be slowly modulated by the broadcaster’s voice.)
Like continually stirred-up little storms; we can imagine aether’s normal ‘little’ storms have a minimum current of about velocity ‘C’, the speed of light. But when, say, two neutrons are near one-another, they constrict the storm breezes (flowing or undulating between them). And, thus, the speed of flow increases slightly, say, to a speed slightly greater than ‘C’ between them. And, thus, a (Bernoulli-equation) related gravitational ‘attraction’ occurs. But the underlining ethereal super-fast vibrations is like our air molecular analogy. It consists of miniature ‘cells’ or a substructure vibrating at velocities far greater than ‘C’ by many magnitude!
Gravity is a case of relatively slow ethereal cyclic drift superimposed on a much faster underlying ethereal vibration!
And I am not the first to say that sort of thing, nor have I likely said it better than some others. Yet, with all the above mentioned; I do not claim I know all the details regarding the interactions between the aether in space and gross matter -- regarding gravity, etc.
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Carl R. Littmann
(Readers’ comments
always welcome)