A hypothesized structural model tests possible application to Energy Wave Theory.
By Bo Atkinson Last Edited: December 30, 2020 (First Uploaded November 2020)

Common 3d model formats follow at page bottom.

My exploration of Energy Wave Theory (EWT) inspired my thinking and 3d modeling, which grew more involved than initially intended, because some tetrahedral models began fitting EWT hypotheses, and demanded more and more rationalization. In my first posting around a month ago, a mistaken model needed correction, and is now corrected further below.

Precise geodesic spheres were used to define the spherically-cut semi-octahedral form to be found between four spherical forms, which represent the interaction of four spherical wave fronts. The internal structure of spherical standing waves will be hypothesized first.

The objective here is to apply computer modeled structure to the EWT concepts, but also to introduce a special tetrahedral form, in a verifiable 3d model, which is a helix in itself, all at the same time. This variable model will radically-reconsider spin as is commonly illustrated in physics. The following models offer variations of helically formed tetrahedra. EWT physics has guided the modeling as best it could be grasped, by a generalist, without physics specialization.

The following image depicts a trihedral focus, as an organized standing wave, adding vorticity or helicity, which is converged by four symmetrically aligned wave fronts, attaining the symmetrical trihedral helix, through an increasing radius explained forthwith.

Above: Granule wave fronts on the left are angled towards the viewer, which hides their half wave distinctions (to the left of image). More curiously, the tetrahedral framework is 'spun' in itself, as will gradually be explained. (See the above model animated here.) The sharper, trihedral, black wire shape was the surprisingly discovered type of helix, which was explored through it's wide range of variability; and finally, through EWT is here tested as a possible standing wave convergence.

In the formZ geometry app, there is provided, a powerful, generic, helix tool which enabled this discovery. Converting this same black, facetted helix into to smoothed-out closed-curvature, provides possible spin, directionality and chirality.

Geometrically speaking, white and black objects share the original source inputs, and were edited with optional control factors, (which the software can regenerate in a wide range of variations displayed herewith). The white is a smoothed out curve of many segments, originating from the black-segmented object which consists of just nine straight segments. The white curve is a smoothly converted copy of the black. The lower segmentation in the black wire represents control points for developing smooth curvature (such as the white object). Control points are the standardized, manual-shaping tools for curves, (found inside geometry software), and were here derived from the black object,as a minimal set of points to get the trihedral form, (which is the focus of this writing).

Below: The black straight (or facetted) wire helix was formed with two radii sizes, (which is the key focus of this geometric work). The function of these two cooperative radii generate a consequential trihedron product; which otherwise shares perimeter orbits around the first equilateral-triangular-perimeter; and furthermore in this work specifically, the second radius is typically much larger than the first radius.

Said otherwise, an unusual helix, (a trihedron), is generated by an orbiting center, instead of a stationary center, and is therefore a multi-axis-circuit, (as the equilateral triangle). Thankfully a geometry software tool achieves all these steps with just a few clicks, providing that the software user inputs the stated criteria, in the right order.

A versatile framework, whirling and yet faceted all at once, depicts a progressively adjustable radius of tetra-helical properties to hypothesise sub atomic particles types, from short lived particles, on through neutrino types, and through the more stable atomic matter, by enlarging a radius, (explained further below). The ten-part electron is hypothetically modeled below, by interconnecting trihedral (black) segments.

December Edit : Updating my model with EWT's electron– Quote: "In energy wave theory, the electron is formed from a collection of ten wave centers (neutrinos), expressed in the wave constant variable K=10..."

Diagram Above: EWT has hypothesized 10 composite wave centers, or what I assumed are fundamental particles. This work proposes geometric rules may fit relatable rules in physics. Several steps are needed to explain the sequential development of this helical trihedron, as follow, (where black trihedrons are re-colored orange and blue to show how the 3d model could connect two vertices symmetrically. Two identical trihedrons were moved around to find that a maximum of just two could match, per connection as sampled:

Such connections lead into finer abstractions, because there are other possible orientations for ten composite parts. Furthermore, this helical trihedron, has dual, parallel separations at edges; therefore, providing for too many connection variations, for the preliminary stage of this study, so that demonstrating one example is considered sufficient for now.

The diagram below shows a helix and a path, both of whose radii are made equal, and this ratio of one-to-one, generates a helix around the red equilateral triangle. The fact of two codependent radii instead of one radius is the proposed mechanism to vary particle tightness, both of particle form and also of particle stability. The four parts of diagram indicate top view on left, perspective view, top right, front view, bottom left and right-side view, bottom right.

This diagram above demonstrates dual orbits, (or rather the control points able to generate smooth orbiting curves). The primary orbit is an equilateral triangular path in red the secondary orbit is the black helix cycling three turns, with nine steps, while the yellow 3d section of the helix indicates just one 3d cycle. Formal geometry defines open and closed wires, segments, surfaces and solids, with potentially useful word applications for particle physics, which could be explored in these models.

Below are three displays, one of a close up model to the left, and two zoomed out views to the right of image, as the trihedral helix radius is increased, by sliding the arrow. The helix radius was steadily increased by manually dragging the arrow upwards. The tan color of the trihedral helix signifies the modeling app has the helix under control for increasing helix size, with the green arrow, to slide up or down, centering on on the progressive red-line axel-bearings, indicated by green dots, to left, because at those levels of increase, the red is too tiny to see. By starting close-up (on left of image), the display has to be zoomed way out to progressively see a trihedral tetrahedron (to right of image).

The simplest expected particles like neutrinos are suggested to result from loose edges, which suggests their energy can leak out. Resilient tightness of edges, might be a necessary rule for the physical property of stability. Too small a radius loosens the polyhedral seams. (This only refers to the secondary radius mentioned above). The natural rarity of observable neutrinos, so far, may be due in part, to this instability. The smallest ratios of secondary radii, which geometrically form cruder or broken looking enclosures, might well account for the unstable accelerator products as smashed particles, (the so called atomic zoo). Would tight enclosures well describe stable particles?

The different meanings in trihedra and tetrahedra are both applicable in this hypothesis, because both meanings find applicability for developing finer points of this study, and both definitions also overlap proportionally. Trihedra are the simplest enclosures, as defined by fewest parts, also having a triangulated stability, and furthermore as a helix may form the inherent spin, (which may furthermore offer polarity and force orientation, like electrical charge, through the uncovered base. Next, notice the seemingly sealed up trihedral edges, (due to very high ratio of radii, as noted with r1 and r2).

Another note on visual appearance is the 60º projection of a 3 point perspective, above to right, as compared with the close-up, 0º axonometric view to right above. Three sufficiently closed helical loops and one resulting open helical loop, bind together as a self stabilizing, trihedral helix. Very low ratios of trihedral helix radii, to the triangular axel path radius, might fit the nature of short lived particles in accelerator experiments. Could the standard model of physics curiously fit more clearly structural geometrical rules? The multi-view image below is all axonometric, (equivalent to zero perspective and this explains the flatter look of the angled view).

A 3d geometric stability might satisfactorily describe a 3d universe stability. Would two, codependent, internal orbiting paths clarify particle functionality, as compared with singular pivotal-spins? These trihedral models may be tested to explain the most fundamental particles as, a walled in "surface object", which may also explain the electron's directional charge. The parallel edges which are finely separated, proportionally to the infinitesimally smaller triangular axel, provide one infinitesimal crack on each of the three vertical faces, symmetrically angled at 120º from each other, and might suggest study for an additional spin effect.

The next level of joining together two "surface objects" into a geometrically defined enclosure, presents options to model evidence found in physics. Would atoms and protons demark the threshold equivalent to a 3d particle stability, in some way joining the "open bases"? The model's directional arrows were reversed to signify external stability, according to geometry's rules of well formed objects; however, there remains the open question based on clockwise helices versus counterclockwise helices, geometrically speaking, as a question for physics to analyze, for additional property effects.

Next, two fundamental trihedral bases were attached with reversed arrow directions, to also consider coercive stabilization of the proton.

Geometry offers multiple modes of assembly to consider and compare systems of the micro and the macro systems. However geometry needs experience of sciences to guide model making. The images above and below only indicate that variable alignment and chirality factors, might emulate electromagnetic coercivity for modeling different particle types.

The "whirls of granules", inside particles may be perpetuated by the longitudinal standing wave resonance close by, more symmetrically inside the atom than outside of it, and further away, at edges of molecules, possibly averaging out the longitudinal wave shapes opportunistically rather than symmetrically.

Here again EWT indicates that protons are expected to contain a greater number of composite particles, whereas my emphasis simply suggests that the open trihedral bases are connected and form a closure, within protons. ( Final models are not intended any of these images).

A "tiny hole" at trihedron apices is not visible at the above image resolution, but zooming in closely does reveal it clearly, (as in the next image below). It is close in size to the tiny triangulating-concentric-axel. Could it pass granules by connection to other particles? Is there a threshold for this radius, or not? The image below magnifies the hole with it's signature separation of parallel wires, which become increasingly parallel to higher degrees of accuracy, as the ratio of the two radii are increased.

Are granules shared externally or through particle bonding, or not at all? These geometric frameworks present multiple conceivable connection modalities, which must be rationalized with other scientific facts, before devoting more time to modeling. The basis of this model was discovered long before intensively attempting to detail it's EWT applicability. The possibilities seemed somehow conceivable, but the geometric work-out needed the push or inspiration which is graciously supplied by EWT.

The possible misattribution of the default helix option, as found in formZ, possibly is due to using the vertices instead of facet-midpoints, for it's control-points, and radius positioning. The currently resulting curve of the formZ smooth helix . Should mathematical helices always use vertices, and was that decided in the ancient past? (This issue is for another study).

The formZ, built in option of helix parameters offers this instant, optional spherical representation. It appears a physical wire strung around the vertices, but only in a polar fashion. So that the truer trihedral helix must instead be manually plotted from progressive states of facetted helices, (which formZ does representatively and as I have modelled above).

As of December 2020, another toolset, (known as smooth geometry for curve and surface editing), was tested for helical modeling. The following is a semi-transparent model based on the same three cycle faceted-helix. The value of this type of smooth object would be to further explore possibilities of particle-sphericity, trihedral spin and connective properties in 3D space. This type of smooth tool can be edited to fill spherical or other forms.



In the spirit of collaboration, I invite interested persons to discourse this subject matter, possibly at The Field Structure Institute, which promises to set up a discussion area. (Link will be added when their site is completed). Or, please invite me to your discourse location.

In the spirit of free software, here is a quick way to see the arrows in images above...

The formz free app, 0$ for Windows & Mac is downloaded here.

My 3d work is best seen with arrowed indicators of direction, as can be seen in the formZ Pro app. Arrows simplify analysis of direction and chirality. These are possibly key to discussing bonding mechanism, as the user can reverse direction of segments with one mouse-click. Whereas other software brands lack this feature, as most modelers ignore direction of segments and facets, which in formZ are a very compact code embedded deeply into the app, rendering quickly for navigation and simulations. To see these arrows first hand, immediately upon opening my model files, one may use formZ.

It is known among 3d modelers that importing and exporting 3d formats often looses model features as those displayed in these 2d images of this presentation. Whereas generic formats will display only plain lines and curves, at best. I will reply to questions on public forums if alerted, and emails are welcome, especially when the subject line includes special terms like EWT or helical, or trihedral, etc..

Simple Models in differing 3d formats follow:

Plain 100:1 Ratio Trihedral Helix:





Facetted trihedral helix with internally facet-rounded, trihedral helices, but only formZ will automatically display the arrows as an option already built into formZ. (This option can be turned on or off under formZ/Options/Display Options.)




The formZ format is named differently to avoid confusion:


A Geometrical Simulation Navigable In Virtual Reality And 3D Software (Animation Below)

The trihedral helix is a variably sized mathematical convergence, scaled to the point which observers will deem appropriate, (and will determine consistencies for each sort of sub atomic particle reaction), and possibly scaled in terms of a granule-unit-resolution, where it's radius (r2) can be taken as the distance between the trihedral face centers, to the centroid. Geometry can accept the physicist's call for a numerical value, input to the geometrical model, (which can be converted to a nominal, metric, multiple value, because currently available geometry apps have a 'bandwidth' of values limited under a billion and may or may not work at nano scales or smaller). Yet a scaling factor should handle those issues.

The physicist also must assess and declare whether the trihedral faces comply with their physics models, because the trihedral helix is slightly shorter than the equilateral tetrahedron and the faces are slightly offset from tetrahedral centers, and the apices are not fully converged, (closed); wherein, all these parameters first depend upon the resolution of unit values, to round off the value of r1, (these radii are described in further detail on the first page).

What is more, is that physics must assess if the very slight openings in the helical trihedron cooperate with energetic releases and uptakes, which is here proposed as the mechanical energizing hypothesized to operate something like pressure dependent valves or gates; and, all particles trim the appropriate amounts of ingress and egress of granules, possibly based on the radius sizes, (which are adjustable through sizing of r1 and r2).

The models given on the same first page are outlined by the trihedral framework, whose radius is here referred to as r2 measured at the sharp vertices, while the whirls of force in the models extend to the trihedral faces, and only adjacent to the vertices will represent the radius of r2. The trihedral face centers are closer to the center (or centroid), than are the vertices, which presents parameters more complex than simpler spherical wave fronts. Beyond two trihedra joined at the base, eight trihedra offer the next fully bonded base set.

Furthermore, physics might be at an inception point of 3d space definition only at the fully matured fundamental particle, which is proposed as the electron, while it's smaller radii versions are lesser particles, (as described on the first page). Whereas the granule is proposed as motion without mass, until incorporated in growing pre-particles whose r2 is not mature or stable, but do directly interact with fundamental particles, in the turbulence near large bodies of matter and it's energies. The critically important proton is proposed as two fully sized r2 electrons bonded together at the base, and further verifications of spin direction need verification by physics, (and not by geometry which is the content of this page). Opposed arrows are proposed to indicate closure of particles, after the definition method in solid geometry.

According to the dictionary, only the word helicoid describes a helix wound along an axis; however, the modifying word trihedral is necessary to describe the surface solid framework and enclosure, coincidently generated in the same formula! How is this possible? Mathematics must investigate this factor to understand why a helical formula and a surface solid formula coincide. (We are not describing an equilateral tetrahedron but rather a trihedral framework, and the word objectification could be added to emphasize the source of three dimensionality, by coincidental enclosure of space, by real boundaries, ( and all generated in one formula). The fact is that this writer doesn't know the associated formulae, except that a powerful, geometry, software app does indeed generate these parameters in one process.

As a preliminary step, it is suggested that the  classical electron radius given on, https://energywavetheory.com/equations/ might be taken for r2. This is is expected to fit into the scaling factors mentioned above, and the result would be plugged into calculations of particle energies, also supplied on the EWT page. The process of corroborating the geometries with the anticipated reactions is expected to begin with the sizing of r2.

Added December 30, 2020
Regarding the most fundamental particle as a spherical standing wave, what exists beyond the immediate spherical wave boundary? Could packed spheres touch without overlap according to wave mechanics and thereby enclose a separate spacial form? Could our physical universe categorically subdivide into spacial volumes as modelled here? These visualizations are meant to represent just an infinitesimal fraction of all existence, and as a thought experiment, to observe matter through structural models. If spherical wave crests remain consistently arrayed in a matrix, all touching the adjacent spheres, what is the space outside the touching spheres? Here just four spheres are modeled to carefully observe that resulting space form, which for lack of a traditional name, is here called a "quad-joint".

Quad-joints have four concave sections that we might call spheric faces, (or sphere-boundaries). These concave sections could be imagined as constituting one face each, and the remaining, four, purple faces, are cut flatly to differentiate from neighboring quad-joints. In this sense we count eight generalized faces in the quad-joint object. Therefore, once the spacial boundaries between spheres are compartmentalized equally with the adjacent sphere sets, the reulting forms are somewhat octahedral.

Initial consideration of four bounding spheres may predict a tetrahedral form should fit, but closer examination reveals that a full tetrahedron actually exceeds the boundaries between any four spheres. In case a smaller tetrahedron is substituted, then much of the space between spheres becomes empty, and our objective is to utilize all space. Therefore, once the spacial boundaries between spheres are compartmentalized equally, the unusull form apears more like an octahedron rather than a tetrahedron.

These models happen to use geodesic spheres as an initial experiment to observe the nature of four spheres touching. Geodesic spheres are here used to avoid use of high-resolution smooth geometry which is harder to manage for a preliminary experiment. Exploration of spherical types presents yet another experiment, which might continue gradually, where the experience of preliminaries provides successional observations, and thereafter allows renewed focus, more especially on the standing wave, model interiors.

Next above is modeled four additional quad-joint objects to view the contiguous nature of geometric quad networking, where five individual quad-joints are flatly connected into the network, which in the case of our physical universe may contain an uncountable number of standing wave spheres and quad-joint connections, but nevertheless provide a self satisfactory perspective.

Hypothetical spheres filling the entire physical universe would establish octahedral forms with concave surfaces between four touching spheres. Triangulated surfaces instead of smooth or shaded surface geometry, here differentiates the green spherical surfaces from tan spherical molds. The connecting flat-cross-sections, between neighboring octahedral forms, joins the octahedral segments omnidirectional, (to appear as slices of solids). Given that preconception, we may next visualize the network of quad-joints, but the purple face connections are hidden by the opaque, tan surfaces. The remaining models further clarify the nature of sphere packing, (and the un-perfected nature of preliminary modeling).

According to one possible rationale, these two representative forms, 1) a single standing wave sphere, and 2) a single quad-joint in between, could be combined, and possibly be called a voxel, with which to subdivide and fill all physical volume of our universe, like two repetitive units in a 3D physical universe. Today's most powerful geometry apps provide means to analyze just how boundaries could geometrically fit.

The concept of interactive voxels compares to a visual device displays, where each pixel changes intensity and color, to simulate images and motions. Likewise, a 3D volume filled up with interactive voxels is suggested to simulate virtual reality, (and without all the kinds of major aspects, of actual existence). Further analogies suggest exponentially more resolution and bandwidths might simulate spherical standing waves of physical matter, from the sub-atomic, and up through all forms of life, which sciences may yet strive to entertain and to discover.

The model on the left has been boolean-differenced from the model on the right. Considerable work would be needed to subdivide so many quad-joints to display all together, and this particular model has not yet optimally arranged the geodesic spheres, which is also a deeper study in itself, making two more studies. Add to that is the question of what geometrical form may best represent the "granule", which is only briefly mentioned in EWT, and which better directs attention back to the most essential basics; because, if granules have a true geometric form, then that specific form would influence the model of standing waves, their boundaries, textures and all space besides. The surface of spherical waves might not prove exactly smooth, in that case. So more and deeper hypotheses are expected.

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