Gregory Stiles

We live on a planet made mostly of atoms-the small building blocks of matter. We know this is true because with modern technology we can see them (albeit blurrily), and because a great deal of consistent data has been collected concerning atomic properties. However, not all of this data has been made as consistent, clear, or coherent as it could be, because science as yet has failed to adopt Synergetic geometry as its multidimensional gridding system for visualizing the atom.

Today, almost all scientists rely on the Euclidean geometric system (and its cousin, the 3 axis, 90 degree Cartesian coordinate axis system) when they choose to visualize atomic structure and matter arrangements (Note 1). The geometry of Euclid, however, was conceived in ancient Greece at a time when the vast majority of humans believed the earth was flat-thus the majority of high school students today, more than 2,000 years later, still learn all about how to draw shapes on a flat plane (Note 2).

Nevertheless, we actually live in a multidimensional world, and it stands to reason that the best geometry (geometry is a term which comes from the ancient Greek for "earth measure") should best be able to help us measure the world as it actually is. For centuries, the geometry of Euclid stood alone, as the only geometry. But then, in the 19th century, three more geometries were introduced, all of which were as internally consistent as Euclidean-but their premises, or starting ideas, were totally different. A debate then followed as to which geometry was the best one for helping us to understand life as we experience it.

The debate was not settled in the 19th century-but we may safely settle it today.

R. Buckminster Fuller, hereafter referred to as Bucky, invented a fifth form of geometry in the 20th century, Synergetics, and unlike the other four, this one is internally consistent and a perfect match for life as we experience it.

The evidence supporting Synergetics has been on record for years, but the scientific community has stubbornly refused to acknowledge it thus far.

First I shall briefly describe the current atomic model in vogue, then describe the evidence which has been available for years suggesting the flaws of this popular model and Cartesian thinking in general, and then add my own team's breakthroughs which should, I hope, forever secure Synergetics as the ultimate geometric system for describing nature.

The atomic model presently used by virtually all scientists on earth is shown roughly in Figure 1. Science has agreed that atoms are made up of a nucleus that can hold supposedly "positively charged" protons or "uncharged neutrons" (both of which are relatively large at this small scale), and that these nucleii are orbited by relatively small structures referred to as electrons, which allegedly carry a negative charge. The nucleus is presented as being made up of spherical nucleons (a nucleon is a proton or neutron) packing together to approximate a larger sphere, with protons and neutrons evenly mixed (Note 3).

Despite the popularity of this model, there has been evidence available for quite some time that this paradigm, or commonly accepted set of concepts, is quite faulty, and that alternatives were readily available.

The evidence contradicting the established view has come in many forms. First, Bucky published Synergetics in 1975, and presented a different mathematical model of the structure of Uranium. In the Synergetic model, nucleons did not close pack to approximate a sphere, but rather formed a vector equilibrium, a structure made of 14 faces, 6 of which were squares, 8 of which were triangles. This model made far more spatial sense, and gave a precise visual demonstration of the atomic mass of Uranium.

In the Synergetic model of Uranium, nucleons obey strict and elegant geometric rules. While the physicists may tell us that Uranium has atomic number 92 (meaning that it has 92 protons) and atomic mass 238 (derived from adding the 92 protons to the additional 146 neutrons always found when uranium is in its most stable form), they can offer no visual explanation for this. Bucky does so easily.

For simplicity, let us start with circles. (You may use 7 coins of the same value from your pocket for this exercise). Consider a circle. How many circles can you close pack around one circle to surround it completely?

The answer is 6. The answer will always be 6, regardless of what scale of size we consider-whether it is the scale of skyscrapers or atoms. This idea is technically referred to as spatial constraints. Space, as Peter Stevens once wrote, is not a passive medium, but imposes limitations on the objects that manifest in its confines. The very same reasoning applies to spheres. Start with a sphere. Close pack as many spheres around that one sphere as possible. You will be able to fit 12. The answer will always be 12. Surround that layer again with spheres. You will fit 42 in the next layer-always 42. Surround that layer again, and how many do you get? 92, always 92.

We call the layer of 12 one frequency.

We call the layer of 42 two frequency.

We call the layer of 92 three frequency.

These numbers lead us to what we call the nuclear generator equation-for it shows how nucleii are generated: # = 10F^2 + 2

# of units in shell layer= (10 times (F (frequency) to the second power)) plus 2.

Though you started with a sphere, by close packing more spheres around it you start forming not a larger sphere, but a vector equilibrium-a polyhedral shape called the cuboctahedron by Archimedes. Bucky renamed this structure a vector equilibrium because not only are all the edge vectors the same length, but the distance from the center of volume to each vertex, or intersection point, is also that very same value. For this reason we may say that the vector equilibrium is one of the most beautiful shapes in the cosmos, and it lies at the heart of the atom.

What proof is there that such beauty lies at the core of the atom? You already have it at your disposal. 12+42+92=146, the exact number of neutrons in Uranium, while the 92 protons are simply face bonded to the external 92 neutrons. Thus neutrons always orient to the inside, protons to the outside, as approximately shown in Figure 2.

In reply to this awesome work, the official science community did absolutely nothing.

Second, in 1979, Bucky published Synergetics 2, in which he stated that the same reasoning that enables the visualization of Uranium (element 92 of the "periodic table") could also be used to visualize elements 12 and 42 (Magnesium and Molybdenum, respectively). Third, he also mentioned that a student of his, Philip Blackmarr, had used Synergetics to accurately guess the structure of protons as multipacked rhombic dodecahedra (rhombic dodecahedra are multisided figures with twelve faces, each face being a rhombus, where a rhombus is a square smacked sideways a little-see the bottom right polyhedron in Figure 2), for the ratio of a unit volume tetrahedron (a tetrahedron is a 4 sided figure with each face being an equilateral triangle) to a number of close packed rhombic dodecahedra was quite close to the mass ratio of an electron to a proton.

Once again, the scientific community did nothing at all.

Fourth, as a supplement to Synergetics, Harvard Professor Arthur Loeb showed that Synergetics could be used to visualize all known crystalline structures of geology.

Again, the scientific community did nothing whatsoever.

Fifth, a new form of carbon was discovered in the 80's. The mathematical shape of this structure was identical to the polyhedral shapes of Bucky-and still the scientific community refused to accept Synergetics.

Sixth, in January of 1998, scientist Donald Ingber published proof of Synergetics structuring in the biological structures of cells, the building blocks of life itself. He went on to reinforce Bucky's earlier ideas by noting that viruses, many cell vesicles, and many forms of undersea life all show Synergetics in their structuring.

Yet again, no word of acknowledgement came from the scientific community as a whole, no math textbooks have been rewritten to reflect and embrace this evidence, there has been no admission of the clear superiority of synergetic models of real phenomena in crystal texts, and this is but a fraction of the evidence collected.

This has gone on long enough.

If all of the heavy snow gently falling has been insufficient thus far to cave in the roof, perhaps the following avalanche can accomplish the task, to prove once and for all that Synergetics is indeed the most useful geometric system to be used in modeling the atom. Shall we begin?

Atoms are made of nuclei surrounded by shells of electrons; therefore we begin by studying my team's breakthrough offerings in the atomic nucleus, and then turn to our breakthroughs in electron shells.

The nucleus Nuclei may be modeled by the use of closest packed rhombic dodecahedra (rhombidodecs) (Note 4).

As yet, popular physicists stubbornly cling to modeling nucleons as spheres, yet this is folly for at least 7 primary reasons:

1) Subatomic physicists have shown that nucleons may be divided neatly into thirds as "quarks": spheres do not divide preferentially into thirds with any inescapable logic, rhombidodecs do. Rhombidodecs are composed of precisely 144 quanta modules (quanta modules being 1/24th tetrahedral subdivisions of a regular tetrahedron). We divide 144 by 3 and obtain 48 quanta modules. In shape terms, we therefore even have at least one suggested shape: the ditetrahedron, or two tetrahedra bonded on a face, as the regular tetrahedron is composed of 24 quanta modules. If nucleons were spheres that divided neatly into thirds, then quarks would have to be slices of orange floating around oddly, hoping to hook up with their own kind in order to form a whole sphere-a rather absurd proposition. But if the hypothesis stated elsewhere in this work is correct, namely that a proton may itself already be composed of 306 close packed rhombic dodecahedra, then dividing by 3 could be done by any grade school kid--a quark would be 102 rhombic dodecahedra.

2) Nucleons bond intimately with one another in the nucleus, and spheres would have the absolute worst possible shape for bonding; rhombidodecs, on the other hand, are much better, where in this case a higher value for surface area to volume would be desired. In simple terms, the higher the surface area to volume ratio, the more surface area available for face bonding-and stability. The rhombidodec wins again.

3) In order for a structure to be stable, it must have triangles in its skeleton, as triangles are the only self-stabilizing polygons. Spheres have no triangles; while rhombidodecs, derived entirely of quanta modules, which are themselves tetrahedra composed wholly of triangles, are completely triangulated.

4) Students of symmetry know that the rhombidodec is the "dual" of the vector equilibrium-all the more reason to suspect its presence in a vector equilibrium based nucleus.

5) Of all the completely triangulated polyhedra, only rhombidodecs close pack with themselves according to the actual data uncovered in subatomic particle physics without awkward leftover space. Rhombidodecs close pack exactly the same way as the spheres in Figure 2, according to the same nuclear generator equation. And again, the ratio of multipacked rhombic dodecahedra to the ratio of a unit volume tetrahedron matches the ratio of the mass of the proton to the electron.*

6) Rhombidodecs abound in nature, as any amateur mineralogist knows, appearing in copper, almandite, uvarovite, other garnets, and more, while there has never once been found a spherical crystal. Just visit your local mineral store and find out for yourself! Bucky even went so far as to say the rhombidodec is the most common allspace-filler found in nature, whereas spheres are nowhere to be found in nature (though sometimes natural objects appear superficially spherical).

7) While synergetics informs us that the rhombidodec has a dihedral angle (a dihedral angle is formed when two polygons are attached at a hinge) of 120 degrees, nature has already made a great many speeches regarding her love of 120 degree junctions. In flat planes, 120 degree junctions are formed by hexagons, and 3 hexagons wrapping around 1 vertex represent the minimum number of regular polygons necessary to accomplish this feat, as no regular polygon can have a vertex angle of 180 degrees or more. In the same way, in a multidimensional situation, polyhedra cannot have dihedral angles of 180 degrees or more, and rhombidodecs represent the minimum of four polyhedra necessary to wrap around one vertex. Both situations represent ideal economy, and both situations are used repeatedly in nature. Consider the reference Patterns in Nature by Peter Stevens for more on this concept. Suffice to say that when economy is an issue, 120 degree junctions are ideal, and one of nature's favorites.

Thus, rhombidodecs prove themselves by matching the data exactly.

Clearly, the nucleon is best considered a rhombidodec.

Using this model of the nucleus, we may also now understand vividly the reason why, in the energy generation of nuclear physics, nuclear fusion is used for all elements below iron, and fission for those above. For even as there are what I refer to as consistent vector equilibria shells (nuclear neutron shell layers where neutrons fill out shell layers all by themselves), there are also combination shells (where shell layers are filled by combinations of neutrons and protons). Iron is element 26: therefore we know that we have 26 rhombic dodecahedra acting as external protons, along with 29 rhombic dodecahedra acting as internal neutrons (deduced from the Fe atomic mass of 55). We may even begin to suggest the true difference between a neutron and proton. If we use the definition of a proton as 306 close packed rhombic dodecahedra, and note that the neutron has 1 percent more mass than the proton, this suggests the neutron has three more rhombidodecs in its mass, (derived from 1+12+42+92+162= 309), which likely affect its spin, and thus its lack of "charge" or attractiveness. More accurately, then, in iron we have neutrons arranged as: 1 rhombidodec, surrounded by 12 rhombidodecs, surrounded by 16 more rhombidodecs, leaving an incomplete rhombidodec shell by 26, as 42 rhombidodecs would be needed to fill this shell, and 42-16=26-the number of protons-which therefore neatly fill out the rest of the shell, and establish the energy links which mark the edge of the fusion transition. This presentation represents a "middle of the road" iron conformation, between the more popular abundances of 54 (the one most mathematically consistent with the nuclear generator equation) and 56 (the 56 being more popular than the 54). I have been asked about the abundances, and our response makes several points. First, Bucky's second frequency consistent element, molybdenum, only occurs at 16%, and it, too, is not the most popular abundance--the 98 conformation is more popular. If we talk about comparing overall percentages, carbon occurs more frequently at 12 (98%) than magnesium at 24 (78%). There has been no final word on abundances--but there is still room to note patterns considered significant.

Further vindication of this rhombidodec reality is demonstrated in the most significant of twentieth century alloys: steel. While steel is predominantly iron (90% or more iron, to be exact) the useful addition of a small amount of carbon for strengthening (2% or less for most high endurance uses) can now be demonstrated graphically.

Even as iron marks a combo shell completion of a two frequency vector equilibrium, carbon marks a combo shell completion of a one frequency vector equilibrium. Therefore, steel is nothing more than the union of one and two frequency combo shell atoms. You should be able to check the math for yourself at this point, and discover the three frequency combo shell element by studying the "periodic table."

Is it mere coincidence that the nitrogenase enzyme, used by bacteria to "fix" nitrogen into a more usable form, mysteriously contains both molybdenum and iron?

Even these notes do not reveal the full significance of the vector equilibrium in the nucleus. Two other broad categories also use the vector equilibrium model explicitly: the "noble" gases and isotopes. Let us consider each in turn.

With regards to the noble gases, the nuclear generator equation once again assumes command. Noble gases suggest a vector equilibrium link, although at times there may be subtleties involved. Perhaps the least subtle element is xenon--with a blatant vector equilibrium number of 54, (derived from 12 + 42 = 54). A coincidence, you say? Then how else could one possibly explain the isotopal variations that end mysteriously at the uncommon number of 146? Again, 12 + 42 + 92 = 146. It stretches credulity to the limit to assume this is mere accident, and that the awkward spherical model accepted presently by all the world (which model, by the way, is utterly unable to generate the accepted data of physics (by all means, double check this data in the most accredited references, such as the internationally acknowledged Handbook of Chemistry and Physics in your local library's reference section)) should be favored over the synergetic model.

A moment ago, I mentioned that there may be subtleties involved in vector equilibrium manifestations. As an example of the sometimes dangerously subtle reasoning we occasionally use, consider one of the landmarks for the noble gas krypton, where mass equals 84. Should we consider this some inexplicable mystery, or should we suggestively note that 84 can be derived from 42 + 42? The implication here is that polyhedral stability need not necessarily always involve completing shell layers, but also might involve the neat addition of external faces to existing ones. Dare we create such a precedent? Yes, we do, as we can dare much if the evidence truly supports our reasoning. But we need not even really believe we are establishing a new precedent here--not at all. Work that has already been done on metallic clusters (see the article in Scientific American, "Metal Clusters and Magic Numbers"), shows that nature has already used this face-adding strategy for quite some time! Is it really so far-fetched to imagine that the nucleus might be acting similarly to these metallic clusters? We think not! Next, consider radon, with protons of 86 and isotopal variations currently ending at 228. 86 is just two nucleons away from 84, and this is suggestive to us because of the possibilities presented by what we refer to as the "cap hypothesis". The design of the rhombidodec is such that there are two different kinds of vertices: 3 way and 4 way. If we get to two frequency rhombidodecs at 42, and consider the three way vertices at the top and bottom of the one frequency, we find three rhombidodecs at the top and bottom of the one frequency--providing a perfect nesting place for an additional rhombidodec. Might nature be taking advantage of this, and be placing caps along a central axis of two frequency conformations? If so, this would explain molybdenum's abundances at 98 as well as 96, iron's at 56 as well as 54, and radon's 86 (42+42+2, just like molybdenum) proton count--all cases involving two frequency rhombidodecs capped along a central axis. If radon seems suspicious, as if it is being forced to match the nuclear generator equation, consider that its isotopes end at 228--just two nucleons short of nuclear generator equation number 230 (146+42+42=230). Argon's isotopes end just three nucleons away from nuclear generator number 54. We have been watching these isotope numbers change over time--has the pattern been demonstrated? Hopefully computer models will shed further light on these trends.

The whole of the metal clusters article could be another paper in itself, but we shall not go into greater depth on these points here, except to make several brief observations. As the author of the metal cluster article notes, polyhedral shapes are more stable than irregular arrays because the energy involved in holding them in place is lower. Nature does indeed seem to like the economy of perfect polyhedra. The icosahedral and octahedral symmetries mentioned in the article are a consequence of specific uses of space. Symmetries are determined not only by the shapes involved, but also by their relative diameters. With respect to the initially mentioned coin exercise, coins of different value and varied diameter would lead to different symmetry patterns. Our point is that, with sufficient clues about the macro shape patterns involved, the micro patterns may be deduced--thus our recommendation of the nucleon as rhombidodec based on the patterns observed. Occam's razor asserts that the simplest, most powerful explanation should be favored over the more complicated one. The nuclear generator equation leads to certain patterns suggesting the rhombidodec--and thus we feel strongly that, on these grounds alone, the current atomic model should be replaced.

To sum up these observations thus far, what this work demonstrates is that a proposition elucidated by Harvard Professor Arthur Loeb has far broader application than previously realized. We here refer to the VEP, or vector equilibrium postulate (Note 5). The basic idea of the postulate is that atoms, ions, or natural polyhedra in general tend to arrange themselves in such a way as to have even distances amongst themselves. While Loeb helpfully tackles the tougher cases involving more than one type of atom or ion, we suggest that in a rhombidodec close packed nucleus where all the polyhedra are the same and allspace-filling, the VEP may also be considered to apply. Sometimes there appears to be some confusion about the VEP, and some assume that it really tries to assert that nature is constantly trying to construct vector equilibria--but this is not at all what the postulate states. 4 polyhedra orienting in a tetrahedral conformation would be doing so in accordance with the VEP, as the tetrehedron is a prime example of even distancing. So as long as any structures are arranging themselves with even distances, such as the atoms and ions Loeb treats with his postulate, whether or not 12, 42, or 92 items are involved is completely irrelevant. Actually, even these nuclear notes do not convey the full magnitude of the VEP, as we will later show its prevalence elsewhere in the atom.

Further still, it is tantalizingly poetic to note that two major kingdoms of life, the plant and animal kingdoms, have at their root completed vector equilibrium shell elements: magnesium and carbon (magnesium may be considered the essence of chlorophyll).

Electron Shells The preceding demonstrates the power of synergetics with respect to the nucleus, but what of electron shells? Can synergetics explain these as well? Once again, synergetics accomplishes its designated task with offhand ease.

Scientists tell us that the most stable atoms have 2, 10, 18, 36, and 54 electrons...but these numbers are mildly misleading for they mask the underlying reality that electrons occur in shells of 2, 8, 8, 18, 18, 32 and 32, and the former sequence may be derived from combinations of the latter numbers. Thus the challenge given to the follower of synergetics is to see if his mathematics can generate the sequence of 2, 8, 18, and 32 in a synergetic fashion. Had Bucky been here to witness the solution we propose, he would likely call our solution an instance of triangular accounting, as well as triangular projection.

Triangular accounting, as shown in Figure 3 tells us that we can accomplish the same numerical progression of second powering with triangles as easily as we can with squares...but Bucky often stressed that nature actually uses triangles in most cases in preference to squares. The case of electron shells is an excellent example.

Consider an icosahedron, a polyhedron made of 20 equilateral triangles for faces (shown also in Figure 3). Imagine such an icosahedron surrounding our rhombidodec/vector equilibrium nucleus at a given outward distance. If we were to project a triangle of the icosahedron outward until we formed a triangle twice as tall, that triangle could be subdivided into fourths, as shown in the triangular accounting/projection graphics. If we projected again, to a triangle thrice the height, we could make 9 triangles; once more, and we would get 16. Simple enough!

Yet this simple progression is all we need to create a new electron shell equation! All we have to add to get the equation is a simple hypothesis--that all electrons must have ten triangles of space in which to operate. We should note that the electron's influence is not necessarily presumed restricted to the vectors of our imaginary icosahedral shells, but the electron may be understood to exert a multidimensional repulsion and our model will be exactly the same. We should also stress that these icosahedral shells need not be real in the same sense as the rhombidodec nucleons, but may be considered as merely extremely helpful and convenient guides for the modeling of space. (Indeed, such models are highly suggestive to us that in many instances of second powering in our physics equations, some such similar triangular projection is likely occurring!). Thus, in the same spirit in which we affirmed the nuclear generator equation, we now submit what is, to our knowledge, the entirely original shell generator equation:

20F^2/10=# (obviously the equation simplifies to 2F^2=# but 20F^2/10 is initially more descriptive)

20 times F(frequency)to the second power, all divided by 10 equals the number of electrons in a shell layer. As you can see from Figure 3, four iterations of the equation yield the very numbers sought: 2, 8, 18, and 32, a perfect match, again made possible only in synergetics! (The repeat shells are tentatively presumed to feature electrons "offset" from the shells beneath. Fascinatingly, the offset shells are evocative of the offset forms of the nucleus as shown in Figure 8, where we see spherical regions in the next layer out offset from those within them!). Once again, the VEP manifests--only this time the forces are repulsive rather than attractive.

Not only does this model explain electron shells, but it also accomplishes for us an even more awesome task, the redesign of the "periodic table" into its true, multidimensional form--The Grand Icosahedron of the Elements. The original table constructed by Mendeleev took advantage of data that came mostly as a function of electron shells, so we may use our new shell model to transform the flat, awkward, rectangular "periodic table" into the multidimensional, elegant, triangulated Grand Icosahedron. Imagine placing a label for hydrogen on one triangle of the 1 frequency icosa, then place a helium label on the opposite side--you have just built the first period! The rest of the noble gases may then follow out in a linear sequence of labels from the first shell to the seventh! Periods become shells, and families become linear rays from the core to the periphery! A preliminary artistic conception of the shells without labeled elements we offer in Figure 4.

Our goal was to define the atom, made up of a nucleus surrounded by electron shells, in synergetic terms, and this we feel we have accomplished.

Thus we see that Synergetics, is, indeed, the true geometry of nature. For those trained exclusively in Cartesian geometry, Synergetics may open a door to a new world. To further explore this world, we highly recommend a visit to the R Buckminster Fuller Virtual Institute. Enjoy!

The insights of Fuller have been staring us in the face for more than 20 years now, and the vast majority of academia has twiddled their thumbs the entire time. Do you want this to change? If so, we urge you to get involved. Talk over these ideas with teachers, friends, parents, and strangers in the street.

In conclusion, I must repeat the sentiment of Bucky that Nature often reveals itself in patterns so rich that the working hypotheses of the scientists often seem crude in comparison, and as long as men persist in believing what they can imagine alone in their cloisters is more beautiful than what Nature can imagine, humanity shall remain a plodding enterprise.

In the view of this researcher, no mere human imagination can match the beauty of raw nature. Until next time...

Fire in!

Gregory Stiles has taken legal precautions to protect his team's priority claim to the new breakthroughs presented in this paper. You may direct inquiries to him at:

Polytope Press

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