Benjamin Gurule



Written by Benjamin Gurule,

Stephen James Taylor and Jesús Salvador Treviño

The system of Natural Order that we’re introducing here began many years ago strictly as an art project. But over the years it took on a life

of its own that forced Ben Gurule to intrude into areas of Chemistry and Quantum Physics that, as an artist, he might never have visited on

his own.

To start at the beginning, let us introduce you to the seven topological shells that constitute the foundation of the system. And we say

topological because none of the rings can be removed from the structures without first cutting them.

But the circular rings can also expand themselves in other directions or they can be constructed from a wide range of other materials as is

shown here. Whatever the configuration the rings finally assume, all the rings on any given shell must be identical and therefore

interchangeable with each other.

Later we shall see how these structures might be used as metaphors to interpret subatomic phenomenon. Now what all these families of

shells have in common is that they are all space containers. Each shell is able to capture its allotted mass and is able to differentiate this

internal captured volume from all of the emptiness that surrounds it on all sides. The internal captured volume is secured not by the rings

themselves, but by the collective densities that are sandwiched inside each two- way crossings where the rings overlap each other on the

interwoven surfaces.

Each successive shell represents a step up the quantum ladder and each step on the ladder represents yet another dimension in space up

to a total of seven dimensions.

In the system, the first circular ring barely hints at the impending importance of the points of containment that will eventually define the

captive sphere. These do not appear until after the second ring has been added. When the second ring is added, the first two points of

containment, shown here as an orange and blue dot, come into existence at each two-way crossing where the two rings overlap. Another

way of clarifying the difference between the captive sphere and the captive points of containment is to compare each two-way crossing

to a cheese sandwich in which the internal slice of captured space is represented by a square slice of cheese and the rings are made of

bread that surrounds the cheese top and bottom like two bread slices. If we now get rid of the bread, we end up with two square slices of

cheese positioned at opposite ends of the spherical surface.

As we continue to add rings, up to the number of seven, the number of cheese slices, or points of containment, or density functions,

multiplies. But the pattern of growth is not linear. Because on Level three, the addition of the third ring increases the points of containment to six.

On Level Four the number of points of containment increases to twelve points.

On Level Five the number of points increases to 20 points.

And then thirty points.

And then, finally, forty-two points.

At first glance the distribution appears to be random but no such thing because the precise placement of each point of containment is

determined by the exact location at which the rings intersect each other on the surface.

These points of containment can get larger or smaller, depending on the thickness of the rings.

As the rings get thinner, the squares of cheese also get smaller, but not so small that they disappear. To get rid of the cheese, all we have

to do is cut one of the rings and remove it from the shell. As all of the two-way crossings along the missing path lose one of their bread

slices, all of the sandwiches affected go out of existence, cheese and all.

If we now go back to Level 1 of the shells, where only one single ring is involved, we can see that this beginning ring can consist of

anything that is able to capture its own private circular path through space.

This beginning ring has a certain amount of freedom.

With nothing to restrain it, this single ring can get wider or it can get narrower. At first glance this single ring would seem to be a poor

space container because the space can go in one end and out the other, but when you see this happening, all it means is that the ring is

moving through empty space. At this point, if we cut the ring, the internal captive space loses its identity and becomes indistinguishable

from the empty outside.

On Level Two, where two rings are involved, the two rings find themselves linked together and they generate a pair of two-way crossings

that can be visualized as two square pieces of cheese at opposite ends of the sphere.

And what’s in the middle? Well, the internal captured space of course. At this point. If we increase the size of the internal captive sphere to

its greatest allowable volume, we discover that every two-way crossing will have a counterpart two-way crossing at the opposite end of

the shell.

On the next level, where three rings are involved, the rings intersect each other a total of six times. Because of this, the internal volume,

which we know to be roughly spherical, begins to resemble a cube.

Once three rings are involved, we can begin to appreciate the lateral and radial expansion properties inherent in each ring.

In order to explain the expansion process that is seen taking place on this page, let’s imagine the laterally expanded shell on the left

reducing itself down to its beginning, as shown in the center, and then expanding itself radially so that the three rings that start out

looking like three sleeve like cylinders are transformed into the three flat, disc-like ellipses that are shown here.

Once the three elliptical rings are interlocked in space and we mark each two-way crossing with yellow dots, we can see the yellow dots

taking up residence on the surface of the captive sphere that defines the internal captive space.

On Level Four there are a total of four rings involved.

The four rings intersect each other a total of 12 times and they create 12 two-way crossings on the surface.

If we reduce the shell back to its beginning and then expand out in a radial direction, look what happens. Each elliptical ring is transformed

into a triangle and the four triangles are able to interlock in space without interfering with each other.

On Level Five, a total of five rings are involved.

The five interwoven rings create a total of twenty two-way crossings but what happens to the shape of the rings when they are expanded

radially? Well, the rings are transformed into five perfect Squares.

On Level Six, six rings are involved.

The six rings intersect each other a total of thirty times and create thirty two-way crossings. And then as expected, when the shells are

reduced back to their beginning and then expanded radially, look what happens.

The six rings are transformed into six perfect Pentagons.

On the Seventh and final shell, the seven rings intersect each other 42 times and notice that the level 7 shell is the only one that supports

hexagonal vortexes.

And so we end up at the Seventh dimension. And just by looking we can begin to notice signs of surface interference as the shell is

distorted out of true spherocity by rings that are competing with each other for space at the hexagonal vortexes.

And on this final shell, each of the seven rings is transformed into a perfect Hexagon.

Let’s take a closer look at the three-ring cube. We will see that each of its eight corners has an opening and we’re going to call each of

these openings a convergence. Now the reason we call these openings convergences is because some of these openings appear to face

in one direction and others appear to face in the opposite direction. If we now insert a rod into the center of the shell so that it enters the

shell at an upper convergence and exits the shell at a lower convergence, we discover that every convergence, like every two-way

crossing, will always have a counterpart convergence at the opposite end of the shell, something like a North/South pole.

We also see that on any given pole, the two opposing convergences will always be mirror images of each other. This means that every

pole on the shell surface is bounded, top and bottom, by two convergences that are facing counter to one another.

Now these convergences, where three or more rings are interwoven around a common center can be triangular…

Or they can be Square, or four-sided…


Or Hexagonal.

Each top and bottom convergence, however many its sides, is a mirror image of the other. Let us consider one convergence as being “left-

handed” and the other as being “right-handed.”

Simply put, if you are traveling around a convergence in a clockwise direction and are stepping down and down around what appears to

be one of those M.C. Escher perpetual staircases, then the convergence can be said to be right-handed.

On the other hand, if you are traveling around a convergence in the same clockwise direction and find you are stepping up and up and

around what appears to be an ever-ascending staircase, then the convergence is “left-handed.”

In exploring the right and left handed properties of shell convergences, Gurule turned his attention to a Level 3 shell and noted that when

expanded laterally the Level 3 shell is transformed into a platonic solid, a cube. Gurule discovered that all seven shells would transform

into a known polyhedral solids. The reason for this is simple.

In this Level 5 shell, for instance, it is clear from the beginning that no lateral expansion can take place at the three way junctions, as

pointed out by the arrow, because you can see right away, there’s no place to go. The rings are already touching there.

The only place where any kind of lateral expansion can take place is into those areas of emptiness that are represented by the twelve

pentagonal holes, as pointed out by the top arrow.

As all surface pressures are equalized, the shell takes on a spherical identity when all thirty-two surface junctions find themselves

crowding each other into peaks that resemble the mountain peaks that form when tectonic plates collide.

In the end, all twenty spaces and all twelve peaks end up relaxing into a thirty- faceted, minimal surface structure that is called a rhombic

triacontahedron that fits comfortably inside the perfect spherical bubble.

We can now also see that the surface is made of shallow three and five-sided tents, each of which is a convergence on the original shell.

If we now locate all the five-sided convergences, and there are twelve of these, we can see that each convergence is shaped like a

shallow pentagonal tent. If we now flatten all twelve pointed tents, the shell is transformed into a Dodecahedron, with its twelve

Pentagonal faces.

If we return back to the minimal surface structure and we locate all the triangular tents, and there are twenty of them, and then we flatten

them, a shown on the drawing on the right, the shell is transformed into a Icosahedron with its twenty flat, triangular faces.

During the exchange, in which we see the surface f the shell is changed from a Dodecahedron into an Icosahedron, it is important to note

that the change has taken place without affecting the topological identity of the shell in any way.

Also note that the shells have not only exchanged Pentagons for Triangles, but the shells have also exchanged right- handedness for left-


The only physical changes that can be detected are the small alterations in the shape of the rings as they circle the shell.

When the rings are disassembled and mounted on the wall, we can see the tiny changes that are taking place on the rings are tantamount

to changes in amplitude and wave shape but the frequency remains the same.

The same transformation takes place on the Q4 shell when the shell relaxes into the minimal surface bubble in which each of its 12

rhombic faces, representing a two-way crossing and each of its pointed peaks, ends up inscribed comfortably within the perfect sphere.

Once again, each of the shallow tent-peaks turns out to be a three-sided triangular convergence or a four-sided square convergence. If

we start on the left and we locate all of the Square faces on the minimal surface Rhombic Dodecahedron, and there are six of these

Square faces, we can see that each Square face is sightly peaked like a shallow tent. If we now flatten all six of these Square tents, as

shown on the right, the shell is transformed into a cube with six Square faces.

Let’s stay with this cube for a little bit because the cube is something that we’re all familiar with. Right away we can see that the flat faces

of the conventional cube to the left have now been changed into the square convergences that we see on the dynamic cube on the right.

The next thing we see is that pointed vertex on the conventional cube has also been transformed into a convergence, but the

convergence that forms on the pointed vertices will always be facing in the opposite direction from the convergence that form on the flat


This means that the Euler formula for Polyhedra that tells us that the faces plus the vertices are equal to the number of edges plus two, as

shown on the left, can now be rewritten to tell us that the right-handed convergence plus the left-handed convergence are equal to the

number of two-way crossings plus two, as shown on the right.

And finally, we notice that all of the edges on the conventional cube are actually two-way crossings on the dynamic cube where two rings

are overlapping each other.

If we now go back to the same minimal surface structure on this same Level Four and we locate all the triangular convergences, of which

there are eight, and then we flatten the eight convergences, the shell is changed into the Octahedron shown on the right, with its eight

triangular faces. Just as happened on Level 6, on this Level. Level 4, the shift has been accomplished without affecting the topology of

the shell in any way.

Once again, it is important to note that the shift has not only exchanged Squares for Triangles, but it has also exchanged right-

handedness for left-handedness.

On the polyhedra shift for Level 4, we see the paths of the rings slowly evolving from the straight strip of the bottom cube into the angular

strip of the top Octahedron. The ring that comprises the minimum surface bubble can be seen relaxing at the center.

This progression is analogous to a gradual change in waveform and amplitude. The paper strip at the bottom represents an amplitude of

zero and becomes a cube when assembled with others identical to it. With each strip above it, the amplitude increases gradually by

positive increments until it becomes the triangular waveform at the top forming an octahedron when assembled.

The same duality exchange happens on Level 3. But here we run into a problem because we see a tetrahedron changing into another

tetrahedron. At first glance the two structures look very much alike. But if we take a closer look at where these two arrows are pointing on

these two tetrahedrons,, we can see that the upper edge on the left is lit up and the paper edge on the right is casting a shadow. This

means that the two triangular convergences are facing in opposite directions. By showing us how to differentiate between left and right

handedness, the system allows us to comprehend mathematical duality from a completely different point of view.

On the polyhedral shift of level 3, we see the straight strip of minimal surface cube positioned at the center, with the two angular paths of

the two Tetrahedral duals positioned at the extreme left and right of center.

This polyhedral shift results from the simultaneous growth in the width of all rings to a point of maximum lateral expansion which forces

the ring to change its waveform. As we saw before, a ring can only expand into the free spaces left open within the larger convergences.

Thus, the particular wavelike shape is largely determined by the fact the ring can only expand into an available opening.

Maximum lateral expansion creates convex 3,4,5, and 6 -sided convergences that are like tents. Pressing those tents down, flattening

them, is tantamount to subtly altering the amplitude and in some cases, the phase of the waveform and hence produces different

polyhedra. In so doing, for example, a 5-way convergence can be exchanged for a 3-way convergence of opposite chirality. In fact, in this

system it appears chirality is a function of wave phase. This association is similar to the chiral switching phenomenon currently observed

in experiments with metamaterials.

The underlying left and right-handedness that comes into play with the polyhedral shift resembles some of the symmetries and dynamic

properties of subatomic particles and their interactions. This suggests a metaphoric relationship to quantum mechanics. Especially when

the Gurule system is viewed not as a static phenomenon but as a system of seven shells whose component rings are autonomously in


What insights are gained when we imagine each ring spinning along its own orbital path, like trains moving in circles around an imaginary

common hub point at the center of the shell, crossing over and under each other in a frictionless polyorbital dance?

When each ring is viewed as being in motion, as an orbital pathway, we can see that what we have been calling convergences are really

vortices. Gurule discovered that these vortices had the appearance of clockwise and counterclockwise “spin” and that they obeyed

specific laws governing 1) how they can transform, 2) how the shells can interpenetrate each other 3) how the shells can bond and 4) how

the shells can change from one level to another through modulation of frequency or addition of single wavelengths. An examination of

these specifics requires a much lengthier discussion than is possible here and will be covered in a future video.

After pondering the polyhedral shift in combination with the orbital motion of each ring, what would happen if we were to use this as a

lens to look at subatomic particle interactions in a new way?

This may seem like an incredulous proposition until we take a close look at a set of very haunting correlation of numbers that arise

organically from the Gurule system.

There is a close link between the number of Gurule two-way crossings and the set of numbers well known to any student of physics

relating to electron shell capacity.

The maximum number of electrons allowed in each shell (k,l,m,n,o,p) form the set


{2,8,18,32,50, and 72}

In the Gurule system, the number of 2-way crossings for each shell (he named them Q2, Q3,Q4,Q5,Q6 AND Q7)


{2,6,12,20,30 and 42}

As it turns out, set (a) is the cumulative sum of consecutive pairs of set (b), that is:

2 + 6 from set (a) = 8 from set (b)
6 + 121 =18
12 + 20 = 32
20+30 =50
30+42 =72

Can this correlation between two-way crossings and electron shells just be



Q2 2 2 K
Q3 6 8 L
Q4 12 18 M
Q5 20 32 N
Q6 30 50 O
Q7 42 72 P


These numbers show up again in the nuclear shell model of the atom as depicted on the wikipedia page for Nuclear Shell Model. When

considering nuclear shell occupancy via “three dimensional harmonic oscillator” we get the array seen below. Notice the bold presence of

the Gurule numbers on the right. When considering off/even parity in Spin Orbit Interactions, we find the following odd/even parity.

In the Gurule system, we see the same corresponding even odd parity Can these correlations be only coincidence?

Another interesting intersection of nuclear physics and the Gurule system lies in the work of Dr. G.S. Anagnostatos, of the Center for

Nuclear and Particle Physics in Greece (author of close to 400 peer reviewed papers). He encountered a direct correlation between

polyhedra and quantum mechanics. In his 2014 paper, Polyhedral Symmetry and Quantum Mechanics, published in Natural Science, he

found that “symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total

angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average

particle position.”

In another of his papers published in the Journal of Modern Physics in 2013, he found closely packed polyhedral shells lead to a shell

clustering of the nucleus. Employing a harmonic oscillator potential for each shell, Anagnostatos was able to corroborate this polyhedral

shell clustering of the nuclear structure.

John Schlieman, in his paper Classical and Quantum Polyhedra, has also identified a link between the properties of polyhedra and

quantum mechanics.

Zilong Kong, a Chinese researcher, in a 2013 paper in the Journal of Modern Physics, has linked electron shell stability and motion to a

magnetic force octahedron similar to Gurule’s dynamic polyhedra. Kong’s diagram shows that the vertices of a cube are actually electron

vortices whose spin configuration exactly matches the topological chirality of Gurule’s Q3 woven cube.

In conclusion, it is our hope that someone with an extensive scientific background will come forth to determine if the metaphor (and that is

all we can claim it to be) offered by the seven Gurule shells in motion, might provide new insights into the underlying nature of matter and

quantum physics. In other words, is there anything here worthy of further study? Also, if you have any colleagues who might find this

interesting please pass along our link to this film. Thanks for watching.