A Fractal CVT, Wheel Bearing, Bearing Motor, and Electrical Power Generator

The following paper shows an approach for developing a rigid CVT, wheel bearing means , electric bearing motor and electrical energy generator from one novel design. It is highly efficient, compactand capable of providing standard gearing means. All this can be accomplished by using a geometrical object that is capable of forming arrangements that can provide means for an Inifnitely Variable, Continuously Variable power transmission system (IVCVT). I will show how to derive this object in an extremely simple way and that it is a Lie Group Geometry. The new geometry when formed in dynamic arrangements is the first ever visual representation of quantum field dynamics. And it is fractal in nature.

Hypothetically, this single design can replace the bearing and the power transmission. It can also beused to replace a gas engine with an electric engine using the same single design. In other words one single device installed within the wheel can replace the bearing, power transmission and engine.In the rest of the discussions here, to avoid ambiguity, when I say fractal gears I am talking about classical mechanics applications of the fractal forms, and when I say fractal gear fields(or just fields) I am talking about quantum mechanics theory.

Further, I will show that this geometry is strikingly similar to ancient geometry known as sacred geometry, implying that there was a human civilization who had previously discovered all of these things before. After deriving the final forms and showing some possible dynamic configurations I will connect the final geometry to an incomplete work of Leonardo Davinci and try to explain why Ibelieve he was reverse engineering this design from ancient artifacts.

Mathematical Derivation:

We will start with the simple cylindrical representation of a spiral R (Θ)=a+bΘ , where 'b' can be a constant or a linear function and 'a' is any number greater than zero. Later we will see that 'a' has to be non-zero and positive. The 'a' and 'b' we use can provide for us all the different possible spiral forms, i.e. logarithmic , Fibonacci, etc. Theoretically, the different forms can provide various power ratios within the transmission, but I have not tried all of them to verify that they do indeed work correctly.

The Cartesian representation is then: x (Θ)=(a+bΘ)cos(Θ) y (Θ)=(a+bΘ)sin(Θ)

This represents a 2D curve , but we want a 3D geometry. We use the rotation matrix to rotate this spiral in space tracing out a 3D open surface made from a 2D curve.

The rotation matrix multiplication taking a the 2D Spiral to a 3D surface is the matrix product AB of the rotation matrix A about the y-axis and the rotation matrix B about the Z-axis. For simplicity, we will make the two rotations about the two axes have a 1/1 ratio so then we have a single rotation variable α otherwise the equations will get messy. Any ratio other than 1/1 can be achieved by scaling of the final geometric form. Geometry is often easier to understand than pure mathematical formulas. If a picture is worth a thousand words , than a Geometry may be worth a thousand formulas.

A = cosα 0 sin α

0 1 0−sinα 0 cosα

B=cosα −sin α 0sin α cosα 0

0 0 1

AB = cos2

α −cosαsin α sin αsinα cos α 0

−sinα cosα −sin2α cosα

AB⋅(x , y , z) =cos2

α −cosαsin α sin αsinα cos α 0

−sinα cosα −sin2α cosα

(x (θ) , y (θ) ,0)

The result of that operation is the parametrized surface X=(x,y,z) given by:

(a+bΘ)∗(cosθcos α−sinθsin α ,cos2α sin θ+cosθ cosα sin α ,cosα sin θsinα+cosθsin2

α)

When you plot that in 3D you get something like this shown with front,side,and top views like an engineering drawing because of the complex geometry.

Figure 1

So this geometry is useful for any possible rigid contact continuously variable gearing by the initial assumptions. In fact there was a South African manufacturer of CVT transmissions that used this exact geometry as a machined slot and that allowed them to build a ratcheting CVT. I cannot find any news about them now, and I assume that their invention was not commercializable because it was too complicated with too many parts. But the important thing to note is that they marketed it as the only rigid CVT on the market. The open surface could be useful for other continuously variable mechanisms. I found a patent that uses the open surface in composition to make a wind power fan blade that is continuously variable and operates like asail. So by pulling at the tips of a fabric formed like Figure 1, one can control the torque.

Moving on, we want to build up a closed surface homomorphic to a sphere using the above geometry. Some CVT transmissions use balls to transmit torque like the Millner CVT and others use cone shaped forms combined with a belt system like that used in some Audi transmissions. This closed surface that we have here can be described the melding of a sphere and a cone, because it is homomorphic to the sphere, with an axially decreasing curvature like a cone. This project was an endeavor to solve a problem in mechanical engineering, the CVT gear , as well as an endeavor to answer a long standing mystery in quantum physics. So although the form we have is capable of CVT in a large number of ways, the ultimate goal to solve both problems is a CVT gear not a CVT transmission; one single form that can produce CVT, as impossible as it sounds.

Figure 2

Joining the open surfaces to make a solid is the logical next step. So by successive rotations and gluing together, we get a shape that lines up with this one nicely and closing it. If you consider that we are building up a quantum field geometry, then what happens if two of these solid forms come stuck together spinning around as they would naturally do. They would have to alter each other's shape integrally to preserve infiniteand continuous variability. It is important to discuss the quantum field aspects to illustrate that this mechanism is able to cross through it's surface in natural occurrences at the subatomic level. That helps practical engineers think about material elasticity, as it would often come into play. For anyone unfamiliar with field interactions, it is useful to think of an object made from jello in analogy. The quantum field aspects are also important when talking about electrical power generation.

From our current mechanical point of view, the above pic is the natural progression towards a CVT geometry, but I have not been able to make it do that as it is using that form alone. The next step then is to imagine that fields can penetrate each other; they are not pure solids. It is like a ghost going through a wall. It is important to note that unlike a ghost, when one of these forms goes through the boundary of the other, both forms are equally deformed into something else. Another way to simplify the problem is to ask, what solid form would allow for two of the forms to interact in a infinitely variable and continuously variable way.In other words, we need a form that can be used as an automobile transmission.

Here is the solid, closed form above warped as a result of interactions between two like forms of the above spinning in rotation about one another. It is important to note that this form would fill all of space without gaps; an ether as theorized in the beginning of the 20th century. It was later abandoned but replaced again with the so-called Higgs field; basically another ether field.

Figure 35X3 fractal gear (Front,Side, and Back Views)

The forms as an etherfield would warp to fill the gaps as they engage dynamically with one another. Here they are shown as a solid which represents the average shape of the fields. So if one could somehow take a statistical sampling of the field shape , one would find that this is the average shape. Below is how the field forms would arrange to fill the entire universe without gaps. One can see that the inner gaps are perfectly suited to plug more shapes in there. It is a fractal array.

Figure 4

Some might argue that there are still gaps, but that can be solved by fractalizing each form even further as shown schematically in 2D by the diagram below.

Figure 5Progressive method offractalizing a finitelength curve to one oftheoretically infinitelength

Figure 5 shows how a curve can be fractalized to a curve with theoretically infinite length and finite volume,and the a second degree fractalisation of the 3D form would look something like the below. Figure 6 is a representation but it is not true to the Fibonacci number system one needs to use to make fractal gears work. The actual geometry would look like this , but the number of points would be different. Figure 6 shows schematically how that would apply to a 3D fractal gear surface. We can safely say mathematically that the fractal density increases without bound while the volume converges to a finite value.

Again, surface area is maximized for a given volume by this form which we have stated before, and maximal

surface area would be useful property for any field interacting with it's environment in the most efficient

manner. Maximization happens as a fractal gear becomes more dense with peaks.

Figure 6A schematic for progressivelygenerating a fractal gear surfacewith infinite surface area andfinite volume

Some might argue that this is far detached from the modern physics of the day. That is because this form andits motions are new discoveries of a fundamental nature. There is experimental evidence derived from this theory's predictions that show a dynamic, rotating fractal gear arrangement as below that created electromagnetic fields using aluminum forms without any applied electric fields. That is something not predicted or explainable by any modern theory of physics other than this fractal gear theory described here and further described in the related academic paper at this link https://www.academia.edu/s/0656dc274f?source=link

Figure 7An arrangement that when spunby hand produces EM radiation

A video link of a prototype test is at https://www.youtube.com/watch?v=nxAH8exS5KU and shows audibly the strange EM field generation ability. I will explain how returning to an ether field and defining it geometrically could actually help resolve the impasse in quantum field theory. This form generates electromagnetic radiation while rotating because it fractally meshes, like gears, with the fractal gear field structure of matter. Figure 7 also explains by field structure the working principles of a bearing motor; an explanation that has eluded physicists for a long time. The forms you see here would exist as electric fields and induced magnetic fields around the spherical balls within a bearing in a bearing motor. By forming the balls in this form to begin with, we get another field structure around them that is more similar to Figure 6. The field is able to be electric and magnetic in nature simultaneously; electric field is at 90 degrees with the magnetic, and it is known in physics that either one naturally begets the other.

Another interesting property of the fractal gear forms is that they are a geometric maximization of the surfacearea/volume ratio. A property which is useful for any required flux interaction with an environment such as electromagnetism and temperature applications for example, as well as other quantum applications in solid-

state electronics.

Some Exemplary ApplicationsHere below we have an exemplary configuration of what I call a fractal wheel bearing and it can operate as many things individually and simultaneously, such as being a CVT , wheel bearing , electric bearing motor, electrical energy generator, robotics applications, and more.

Figure 8, anassemblydrawing of afractal wheelbearing

3) is Tri-Lobed disk (An ancient Egyptian artifact) …. 2) is a Center Shaft for input or output drive as desired … 18) is a 5X3 Fractal Gear from fractal gear creation diagram, Figure 9 below … and 10)Toroidal case

Fractal gear forms can do many , many things in terms of engineering solutions. That is because I am claiming that this geometry is at the heart of matter and energy so it is the source of all that exists. If that is the case, it not only is able to do many things, it is able to theoretically do everything that exists in nature given the correct configurations. In all practicality it would not be suitable to do everything, but many things; some previously considered impossible. Here we see the basic fractal wheel bearing and CVT (primary best uses) in it's working configuration.

Figure 9, FractalWheel Bearingand CVT

Mechanical Working Principles of the Fractal Wheel BearingWhat are the mechanical working principles? Why does this thing work? Does it cut into itself? Does it make noise? All good questions. As you know from the description and the other posts thatthis Fractal Wheel Bearing and CVT as well as Fractal Gear Theory cover a vast array of fields, from physics to electronics, crystallography and quasi-crystals, Egyptology and other ancient mysteries and more. But I have been neglecting to talk about the mechanical working principles. So here it is.....What are the mechanical working principles? Really the best way to get a feel for it is to have two in your hands and play with them. The gears self adjust by the extra degree of rotational freedom and sometimes they have to "skip and jump". The skipping and jumping is more frequent and with greater force for less fractally dense gears. Increasing the fractal density decreases the number of skips and jumps and decreases the force by which that happens. There might be an infinite number of ways to arrange these gears. I know of many different ways. Too many to discuss here but I willadd them in the new book "Fractal Gear Theory and its Industrial Applications". The CVT method is my favorite and I think the most useful and informative. The skipping and jumping works out because of the underlying mathematical equation behind the golden ratio which governs the construction of CVT forms. Non golden ratio conformant versions from the creation diagram can be used as low friction alternatives to standard gears.

Figure 9 shows how to create the gears. The horizontal and vertical axes are 1,2,3,5,8,13,etc in the Fibonacci sequence for generating approximations to the Golden Ratio by integers. All the Fractal Gears can be used for low friction, high load applications as replacements for standard gear forms. The ones corresponding to 3X2, 5X3,8X5,13x8, etc. (first number is horizontal axis , second is vertical) are used for CVT gearing mechanisms because of their ability to skip, jump and realign automatically.Does it make Noise? It depends on the material used and the fractal density of the Fractal Gear, andweather it is being used as a magnetic bearing or not. Fractal density is the biggest factor-fractal density just means the total number of peaks(points) on the Fractal Gear. A higher fractal density allows for greater frequency of adjustment skips during operation. As the fractal density increases the more that the gear will look like a sphere. There is a limiting point for the density where it would not be practical and would not provide any grip. And that would be determined by the particular application load requirements. The perfectly quiet arrangement would be a fractally dense magnetic bearing. The gears while in operation are able to find a number of Quantum "Steady States"-almost like gears in a gear box. In other words , under a certain "power input to load ratio" there exists a certain "steady state". If you change the ratio, the "steady state" changes. The "steady state" is harmonic in 3 dimensions and aperiodic.Will it cut into itself? No. Because of the added degree of freedom of the gears (they spin in 3d) , itis not possible for the cutting pressure to ever allow for cutting. Further, there is no "Cutting Relief" to allow for cutting as would be found in a drill or endmill. Will it heat up? No, unless the material is really soft like rubber. Because the gears are only able to contact at points as compared to lines and surfaces in standard gears. The formula for friction is (friction coefficient)*(Force)*(Surface Area of Contact) , but the Surface Area of Contact is zero or nearly

zero depending on the hardness of the material, so therefore the friction is zero , and friction is the cause of heat. A soft material can be compressed so that the point of contact will become a surface of contact and so therefore soft materials are not suitable for heavy loads or high speeds.Here we see a schematic representation for a robotic snake system or a new way for building a robotic limb. The interesting aspect is that by controlling only two of the joints one can control the entire length of the robotic snake precisely. With a little work , one could perceivably make this into a tank-like chain of wheels with very low friction; low enough to allow for a wheel system to travel on hard terrain or mountains.

The method for manufacturing fractal gears is radically new as well. Ferro-fluids form the shapes shown in Figure 9 instantly, by a magnetic field. A ferro-fluid with a hardening agent or any ferro-magnetized molten material such as plastic or steel can be formed around a spherical magnet. In other words, take a spherical magnet on a string and dip it into the ferromagnetic mixture, then pull it out and you will have an instantly formed fractal gear. Any of the forms shown in Figure 9 can be produced corresponding to the bottom left being a weak magnetic field and the top right being the strongest. The stronger the magnetic field , the more fractally dense the fractal gear will be. Too much fractal density (high magnetic field) and you start to have the form of a sphere with microscopic peaks. In practice , the sphere can be an electromagnet with the string being current carrying wire so that after formation the magnetism will be gone. Here below is an example ferro-fluid formation, but to really appreciate the phenomenon one would be better suited to visit youtube videos on the subject.

As with any magnetized fractal gear, the field density at the peaks is always higher than the rest of the surface , so field flux density would be highest at the peaks, particularly at the two poles (North and South asin the earth) where peak density is highest, i.e. the fractal gear by it's geometry has net electromagnetic polarity. The fractal array in Figure 6, can be formed by an array of spheres dipped in the ferromagnetic mixture.

Further, the geometry shown in Figure 7 , by it's revolutionary ability to alter the magnetic field by slow rotations, can provide the basis for instant manufacturing by applied magnetic field. The geometry can be made into a nano array and by using 3D Fourier series mathematics one can create any desired form by controlling each nano structure individually. It is analogical to an LCD display which can create any two dimensional picture. After creating the 3D magnetic form, adding a ferromagnetic molten material will allow the created field to materialize in solidified forms. This can be done again and again for mass production, and the created field can be changed instantly without any additional tooling or effort. This is basically instant 3D printing for the 21st century. The only thing missing for this application is the software control and algorithms. Further research is required to develop the software algorithms.

Redefining Quantum Field Theory to allow for a Unified Field Theory

It is important for practicing engineers who want to work with this new technology to understand some aspects of quantum theories and in particular Bohmian mechanics and the Schrodinger's equation. In quantum mechanics, Heisenberg's uncertainty principle asserts a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can not be known simultaneously while doing measurements on them.

Quantum mechanics has, in recent years, moved away from speaking of particles and focused more on fields.Fields are defined as a structured set of numbers that describe a geometry by which it is easy for us to visualize a complex set of laws. Geometry is at the heart of all the laws of physics whereby we are able to understand the complexity in a visually simple way. Einstein looked at gravity which he defined mathematically by a field and he noticed that it is interwoven to space and time and he realized that field is geometrically curved, so he made the intellectual leap , stating that space-time itself is a curved field and therefore it creates the illusion of gravity. In my opinion that is basically saying that the field itself is more fundamental than the particle. So by that example I would have to conclude that there are no particles , but only fields.

Bohmian mechanics is a deterministic method for redefining the old Quantum principles and it is well suited for this endeavor. Bohmian mechanics is a Pilot-wave theory whereby one can talk about hidden variables and a quantum equilibrium of the universe. Fractal gears are the hidden mechanism behind the hidden variables which are abstractly mentioned in Pilot-wave theory, otherwise known as Debroglie-Bohm theory. Advanced applications of this invention will require a practicing engineer to have a general idea of the subject, including the solutions to the Schrodinger's equation.

The two pictures below show, on the left a fractal gear creation array using Fibonacci number series for axes increments (1,2,3,5,8,13,...). All can be used to accomplish highly efficient standard gearing. The fractal gears capable of CVT gearing are those that lie on the diagonal corresponding to the successive Fibonacci numbers(i.e. (X,Y)= (5,3), (8,5), (13,8),... etc)

Figure 9fractal gear creationdiagram.

Notice also that the fractal gear arrangements array are in actuality a successive spherical mapping of all the trivial modes of the Schrodingers equation. If you were to take a fractal gear and map its surface onto a flat sheet you would get what is the standard mapping as commonly seen below. Corresponding to the poles of the fractal gears you can see that the mapping is more dense in the middle.

Figure 10Schrodinger'sequation mappedonto a plane surface

Connections tocurrent modernphysicsThere are at least four major connections to leading field theories of the day. They are mentioned here because to really define the complex motion of fractal gear systems for further analysis, it will be important to understand parts of the modern quantum mathematics and theory related to them. First is the amplituhedron of Nima Arkani Hamed which is a simplified way to represent particle collisions. It is shown here below.

Figure 11Amplituhedron is a geometric objectthat can replace the 100's of Feynmandiagrams used to explain “particle”scattering.

The amplituhedron represents two peaks on a fractal gear field, separately or within a dynamic array. It is safe to say that if two balls collide they do so at a point, and similarly that if two fractal gear arrays collide they do so at the analogous two peaks which can be one or two points together simultaneously. In the case oftwo balls one would only need to know the spin, velocity, friction coefficient and mass in order to determine the outcome. In the case of fractal gear field arrays we can remove the friction because it becomes a property determined by the spin and velocity and the mass is a property of the number of the total peaks within the entire array. Friction in the case of fractal gear fields amounts to the ability of fields to gear into each other, thus retarding the reaction time and total effect of the collision. Mass in this case can be directly derived from the total fractal gear peaks in a fractal gear array since we can assume that each peak corresponds to one Higgs field energy unit which can be converted into mass. In other words, mass and the structure of a fractal gear array are connected.

So in conclusion, thinking of the foundations of matter as energy fields in a fractal gearing structure can explain a lot of formerly unexplainable phenomenon, and it is consistent with modern probabilistic theories. Fractal gear theory is a deterministic model of quantum fields that can be used simultaneously with the current standard model and they both can borrow concepts from each other without breaking the logical consistency. Many of the modern theories provide quantitative answers to the questions that arise from this new theory and are helpful to practicing engineers. A prime example is that strings from string theory can beunderstood as a line of contact through time within a fractal gear field array. Another example is Schrodinger's equation which is exactly the equation of motion for any fractal gear system. All that is needed is a modified Hamiltonian along with a modified Planck's constant. This theory was devised with very little mathematics and a whole lot of 3D geometry. I believe that if a picture is worth 1000 words, then a geometry is worth 1000 formulas, but geometry is much easier to work with.

The lack of excessive mathematics in this theory is due to the current state of fractal mathematics. Nevertheless, engineers who want to work with this form will have to learn a little bit about Schrodinger's differential equation, the associated Hamiltonian, and maybe a few things from string theory. In the words ofStephen Hawkings, “Most likely the foundations of our universe are fractal in nature, but we just don't know how to do fractal mathematics.” And in the words of Richard Feynman, “If all of mathematics disappeared today, physics would be set back one week” The fundamental ideas of physics don't require a lot of mathematics to understand. We only need the mathematics to arrive at quantitative analysis so that we can test the theories and use what we have learned in a practical way as in engineering, chemistry, crystallography, quantum computing and the like. Simplified formulas for practicing engineers are not to hard to produce so that engineers don't have to become quantum physicists, but I cannot predict the full future of the invention and the extent of the formulaic representations for all possible configurations.