Twisted Donut Chain
Model of Space, Matter and Origin of Gravity

by Richard L Marker

Alternative webpage with many more general details not included in this webpage:


ThursdayThoughts webpage with sequential thoughts leading to the donut chain:


This webpage contains details that cover the gravity portion of the model. Click on the link above to go to the more general and speculative version.


rich's picture

"If you would be a real seeker after truth,
you must at least once in your life
doubt, as far as possible, all things."
René Descartes
Discours de la Méthode, 1637.

Donut example:

click on image for larger version with explanations on a separate page

sample donut

This donut example has 11 nodes that complete in two complete revolutions (because of intertwining).

Description :

The Discrete Donut Twisted Chain (DDTC) model provides a bottom-up explanation of space. DDTC answers what, why and how questions about forces and particles.

This is a more recent webpage that is designed to provide details of calculating the ggee ratio. If the reader is trying to reproduce calculations, it is necessary to download the file referenced in the summary section. The zip file contains source code, a Windows 32-bit executable, and a windows help file. The source code section applicable to the actual calculations is confined to a small portion of the code. It is identified as such and contains source code comments. The Windows help file (.chm) contains important documentation and help (F1) with individual input fields in the executable.

Bottom-up :

Discovering the Structure of Space

What if you could see the most fundamental structure of space?

Traditional physics relates to observable phenomena. In smaller dimensions, we can't observe phenomena. Hence, we conjecture(or model) and predict behavior in the observable world.

At yet smaller dimensions conjecture reaches its limits. Beyond those limits we enter the natural philosophical world.

Physics is a “top-down” approach. It starts with thoughts about observable physics and extends downward from there. Philosophy is a “bottom-up” approach. It starts with thoughts about fundamental nature and builds upward from there.

Both approaches involve consideration of unproven ideas. With a "top-down" approach we only expect conjecture to predict something in our observable world. With a "bottom-up" philosophical approach we expect to start with extremely simple beginnings and build on those logically.

Understanding the most fundamental workings of nature provides a natural foundation upon which to build. This subtle, yet huge, difference between natural philosophy and conjecture can elevate our understanding of much of physics.

The Discrete Donut Twisted Chain model of space and matter starts with two simple constructs, "something" and "nothing". These two constructs evolve into the world that we observe.

The model reveals the most basic building blocks. It naturally develops the physical cause and quantification of gravity. It also yields quarks and possibly a new model of the universe.

Main index :

a. Introduction
12. Chain Length
b. Preface
13. Guiding Principles
1. Question is Born
14. Chain Length Calc
2. The Quandary
15. Chain Length Results
3. The Quest
16. Timing
4. Building Blocks
17. Putting it Together
5. Simple Link
18. Standard ggee Calculation
6. The Chain
19. Model ggee Calculation
7. Chain Features
20. ggee Calculation Explanation
8. Donut Dance
21. Model ggee Formulas
9. Two Donuts
22. GR Formulas
10. Chains
23. Discussion
11. Charge
  Copyright Notice

What next? :

24. One Step Dance
26. New Model of the Universe   
25. New Tool
27. MOND Gravity

a.  Introduction to Origin of Gravity:

The Discrete Donut Twisted Chain (DDTC) model of space and matter emerges after decades of pondering the fabric of space as the interaction of many simple, physically real objects residing in a void. DDTC provides the physical cause of gravity.

The DDTC model directly yields an underlying pure number (dimensionless quantity). This quantity when combined with the fine structure constant predicts a ratio of the gravitational force to the electromagnetic force for two electrons (ggee ratio).

The 2014 Codata values produce a derived ggee ratio of 2.40053(11)E-43. The DDTC model produces a ggee ratio of 2.4007107842(12)E-43(first calculated in March of 2016).

The 2014 Codata value for the Gravitational Constant, G, is 6.67408(31) E-11 m3kg-1s-2. The DDTC model derived value for G is 6.6745816766(21) E-11 m3kg-1s-2.

Experimental validation of this value will prove challenging. The model predicts other than an inverse square relationship at large distances or in weak gravitational fields.

The calculation for ggee is an exact calculation that includes alpha squared. The uncertainty in ggee is driven by uncertainty in alpha.

The calculation for G is derived from ggee and CoData 2014 constants. The uncertainty in G is driven by uncertainty in the mass of the electron.

Note that the inclusion of alpha squared in the calculation for ggee represents undetermined relationships of the electron with the fabric of space. This produces a ggee ratio that is tantalizing close (+1.6 std dev) to the value determined from 2014 CoData constants. The resulting ggee only has the precision shown if alpha squared does correctly account for the behavior between the electrons and the fabric of space.

b.  Preface :

The DDTC model relies on the reader accepting that physics in a vacuum differs from physics in a void. This occurs because a vacuum differs from a void. A vacuum by its very nature consists solely of the fabric of space itself. A void consists of nothing, not even the fabric of space.

Physics in a void raises philosophical questions that include the meaning of extension (measuring distance) and the choice of coordinate systems. Philosophers in earlier times debated these concepts in their search for answers[1]. In those earlier times the line separating philosophers and physicists was blurred.

The DDTC model dodges many philosophical issues by performing calculations as if extension exists in the void. One does not know if this view passes muster unless and until meaningful relationships emerge that are consistent with reality.

The calculated result from DDTC relies heavily on repeating synchronized patterns of motion in the void. The determination of these patterns form the very core of DDTC.

If the reader is familiar with Lissajous patterns on an oscilloscope, the understanding needed is similar to understanding these Lissajous patterns. With a Lissajous pattern you have a time dimension and a voltage dimension. With the fabric of space you have a time dimension and multiple distance dimensions as we will see later.

The development is presented as a chronology and sometimes in a story format. Hopefully, this helps the reader to follow the logic and possibly reach their own conclusion. A healthy dose of skepticism is welcome.

Most people are not looking for this answer. Indeed, they are likely unaware that a question even exists! This story tells how a seemingly simple beginning can flourish and grow into an elaborate and beautiful understanding of the building blocks of the universe!

My wife admits she would rather read the back of a cornflakes box than read this. Likely, she is not alone.

1.  Star Gazing: The Question is Born

Many have gazed into the night sky with wonderment. How many stars are there? Do they go on forever; or, are they limited? Where do we fit in this picture? How insignificant we seem when viewing the vast night sky.

One of my early memories dates back to an age when afternoon naps were the norm. Sitting on the front porch of my Uncle's wheat ranch in an isolated part of central Washington, I gazed skyward at the panorama of stars feeling certain at the time that the stars must go on forever.

Vividly burned into my mind is the memory of waking the next day from a nap with the view of the ceiling drywall held in place with bare nails not yet spackled over. A fleeting thought flashed through my mind. The thought vanished instantly as they so often do. However, the conclusion remained: "That is the answer." I somehow understood that the question was about the nature of space. What was the fleeting answer? Who knows, it was gone. Maybe it never existed anywhere other than in a child's imagination.

2.  Prejudices: The Quandary

We who have interest in the nature of space are, of course, guided by our prejudices. Lucky prejudices can guide us to marvelous places. Unlucky prejudices can mire us in a bog that is difficult to overcome.

The DDTC model is based on two heartfelt prejudices. One prejudice is that space consists of something. The other complimentary prejudice is that communication of anything requires either an intervening medium, or direct contact. There is no magic.

The two prejudices created a quandary. This view of the fabric of space, or ether, seemed to be impossible given the well-established results of Einstein's Theory of Special Relativity (SR). This nagging conflict would not leave.

3.  Dispel the Prejudices: The Quest

In 1978 the Special Relativity (SR) conflict was addressed. The approach was to simply demonstrate that a stationary ether conflicted with SR. In so doing the conflicting prejudices would be proven wrong.

The demonstration turned out to support an ether. To accomplish this one needed to assume that an object moving in a stationary ether would seek the same electromagnetic (em) equilibrium in motion that it had at rest. Objects are held together by em forces, so this assumption seemed natural.

It turns out that SR is compatible with a stationary ether. Indeed, it is a NECESSARY result of having a stationary ether. This seems totally counter-intuitive. For those who may be interested in a simplified view of the details and the split of SR transforms between real and observational phenomena, click on this link: download pdf document on SR.

Lorentz Ether Theory (LET) established in about 1905 that SR could exist in an ether. A discussion of the equivalence of LET and SR may be found in a paper by Pablo Acuña[2].

4.  The Journey Begins: Find the Basic Building Block

To solve a problem it helps to know what the problem is and believe that one can solve it. Then, all that remains is to figure out how!

Determining the interaction of components of the space fabric has seemingly insurmountable constraints if one avoids magic. Without magic we are stuck with only physical contact interactions.

Reality requires a pulling force of some form. How is this possible with only contact interactions? Bumping contacts don't seem to support a pull. A "sticky" connection would requires a magical force unless an unlikely hooking mechanism mediates the interaction. This creates a pulling force conundrum unless we believe in magic. However, the DDTC model does not allow magic.

The solution to the pulling force conundrum emerges from better understanding motion in a void. Recall that motion in a void differs from motion in a vacuum.

The DDTC model evolves from only two physical concepts. Those physical concepts are: "nothing" and "something". "Nothing" simply labels the void which contains numerous "somethings".

In the DDTC view "nothing" means a total void that contains nothing, not even the fabric of space itself. A vacuum contains the fabric of space. The concept of "nothing" differs from a vacuum. It will be helpful to the reader to appreciate there is a difference.

If "something" is surrounded by a total void ("nothing"), then, in the absence of magic, the "something" has no communication pathway with the rest of the universe. This means that the "something" has no way of knowing whether or not it is stationary.

This leads us to the non-existence of a preferred coordinate system in the void. Indeed, under these circumstances a circle is a stable path of motion! A circle upon a infinitum is a stable path as well. This means motion relative to other "somethings" was initially chaotic.

How can the fabric of space be built upon chaotic motions of "somethings"? We look around and see order. This view of chaos seems contrary to the world we see around us.

A solution to our "pulling conundrum" emerges from the stable circular path of motion. Now, we can pull by simply pushing from behind. The "pulling conundrum" relates to the very fabric of space itself. The fabric needs some way to hold itself together.

6.   You Guessed It: The Chain

What happens with chain links? That's correct! They form a chain. The exact process of forming the chain isn't known and doesn't really matter. It is the resulting configuration that matters.

How can one know that the links form a chain? That comes from finding meaningful implications of the resulting chain. It also comes from knowing order exists. How does one get order from chain links? Putting them into a chain seems like a reasonable way to proceed.

7.   24 Carat Gold: The Chain That Sparkles

How useful can a chain be? Unless, of course, it sparkles!

Boredom warning... Unless the reader is seriously studying this model, the author suggests cheating and skipping forward to the Let's Dance section. This section just lists some donut chain relationships.

The donut chain complexity may surprise one. After all, the donut link path consists of a simple circle upon a circle. An understanding of the donut path provides a vital platform for advancing the donut model to fruition!

Individual donuts are either left or right handed. In order to connect with neighboring donuts both donuts must be of the same handedness. Thus, the connected universe must be completely of the same handedness. This is illustrated by Beta decay of the neutron into a proton, electron and neutrino. The neutrino always has the same handedness[3].

Donuts rotate a quarter of a turn from one donut to the next. It takes multiples of four donuts to get back to the original twist alignment (in an untwisted chain).

In order for a donut particle to contact an adjacent donut particle each must be at the point of contact at the same time. The point of contact lies along a straight line between the center points of the two donuts. Contact occurs on the inside of each donut ring just like a chain link in a chain.

In order for the donut particle to make contact, it must be in the correct radial position on the main circular path and be on the inside position of the outer (ring) circular path simultaneously.

Motion of the donut particle at the point of contact occurs at an angle of pi/4 radians (45 degrees) relative to the major donut plane.

8.   Let's Dance: In Step, of Course

A donut that tries dancing out of step doesn't get a stubbed toe. Indeed, it gets nothing for the connection was missed. The donut model produces useful results only from understanding the dance needed to achieve contact.

Let's view the spiral around the donut ring as a repeating spiral staircase. Repeating means to immediately start climbing again at the bottom upon reaching the top. However, there is a very important caveat. You must start at the same circular position at which you just finished.

What happens if the previous climb ended in a different circular position than it began? Remember, these are imaginary stairs. The next climb may start in a different circular position than the previous climb started.

Eventually, after a sufficient number of climbs one will get back to the original starting circular position. This occurs if, and only if, the offset from one climb was some integer fraction of a circle.

Look for a moment at the integer fraction. If after one climb we end up 1/5 of the way around the circular stairwell, then it takes 5 climbs to get back to the starting point.

If the integer fraction is 1/6, it takes 6 climbs to get back to the starting point. However, if the integer fraction is 2/6, it takes only 3 climbs. Any factors that occur both in the numerator and denominator cancel out because the second set of 3 climbs simply duplicates those of the first set.

Now, consider the pitch of the stairs. Look at this from the standpoint of how many circles does one need to make to get to the top.

Assume it takes 2 and 1/3 circles (or, 7/3 circles) to get to the top. From previously, we know it takes 3 climbs to get back to the starting point. During these three climbs we will have completed a total of 7 circles. If the circular stairwell were instead a donut, then there would be 7 nodes of contact available on the inner side of the ring. These 7 nodes would only each be available 1 of every 3 major revolutions.

On first glance, it would seem that the denominator alone determines the number of climbs needed. There is more to the story. When there are two donuts, the numerator of one becomes the denominator of the adjacent one. This leads to some interesting synchronization rules.

9.   Did You Forget: This Isn't a Solo Dance

Previously, we looked at one stairwell. Donuts only exist because of their interaction with each other. Perhaps we should look at two stairwells.

Take a second circular stairwell and position it beside the first stairwell. This second stairwell will have the same handedness as the first. It will be positioned perpendicular to the first such that the direction of 'upward' travel will be the same. This is analogous to the connection between two donut chain links.

Some conditions emerge from this arrangement.

First, the two donut particles climbing their respective circular stairwell must occasionally make contact with each other. Otherwise, there would be no relationship between them.

Second, the two donut particles must be traveling in nearly the same direction at the moment they connect with each other. Otherwise, they would 'bump' each other. Recall that opposing motions simply cancel.

Third, the two donut particles must be traveling at nearly the same speed at the moment they connect with each other. Otherwise, one would again 'bump' the other.

Fourth, we postulate that the relative times to climb each stairwell are equal. This may not be an absolute condition, but a helpful one. Stability improves when the same pattern can be repeated simultaneously for adjacent donuts on each main pass.

Formula warning... If the reader prefers skipping formulas they should advance to the Size Matters section.

Define the following relative quantities for the donut:

R, the main radius, is the distance from the center of the donut hole to the inside edge of the outer ring (tube);

r, the outer radius, is the distance from the center of the outer ring (tube) to the edge of the outer ring (tube);

Ω, the main relative angular velocity, is the rotational rate of the donut particle position with respect to the center of the donut hole;

ω, the outer ring relative angular velocity, is the rotational rate of the donut particle position with respect to the center of the outer ring (tube).

n, is the number of donuts in a chain segment. Note that only one of the ends is counted so this is also the number of connections between adjacent donuts in a chain segment.

The conditions above lead to the following, where q is the integer number of outer ring nodes in circular pattern:

ω = q · Ω   Note that the donut particle must be positioned on the inner edge of the outer ring for adjacent donuts in order for them to make contact.

r = R / q   This is necessary in order for the donut particles' speeds and directions to be aligned. The resulting path is a pi/4 (45 degree) angle to the axis of each adjacent donut.

We can now better understand that all of the donuts must be either left-handed or right-handed. Opposite handed donuts cannot connect with one another.   (Author's note: This could easily mean that both a left-handed and right-handed version of the universe coexist. It is highly unlikely that they would be mirror images. There is a vast amount of space in the donut path structure when compared with the donut particle traveling that path. Fabric of space collisions between universes would be quite infrequent. At times and places where the intertwined universes did happen to collide there could be some very unusual sources of neutrinos; or, possibly, some contribution to dark matter.)

10.   All Tied Up: Chains from Links

The donut particle path resembles a chain link. Perhaps not surprisingly more than two donuts can interconnect. Assuming they are not branching out then this is forming a chain segment. Chain segments terminate by branching into more than one chain segment.

It is useful to know that it takes four donut links connected together to get back to the original position on the fifth. (In other words, each donut link is at an angle of pi/2 to the previous donut link.)

11.   Free of Charge: Let's Add a Twist

Okay, so now we have some connected chain segments that seem to be heading toward making a fabric of space. That's really pretty boring.

Look at a real untwisted chain that has its end links with their axes aligned. Now, remove one link and again align the axes of the end links. The resulting chain will have a twist in it.

A twist produces what we call charge. More specifically, a twist in a normal spatial chain segment from the removal of a link produces an electron.

Motion of the electron through space occurs in a peculiar manner. If untwisted chain segments have n links and the electron has n-1 links then to move the electron the distance of one chain segment requires only moving the connection that divides the segments by one link. In order to do that the previously charged chain segment must untwist and the new charged chain segment must twist.

In order for the electron to move there must be a string of connected chain segments that both twists and flows the opposite direction as the moving electron. The motion is more complex than just being a connected group of chain segments moving. As the chain segments move to adjust for the motion of the electron they also affect other chain segments that are connected to them.

12.   Size Matters: Chain Segment Length

Okay, let's say the reader buys into the developing image of the fabric of space. Just how would one propose determining the length of the chain segments? This is not as implausible as it might seem. It turns out that the motion of the electron provides the essential clue.

Recall that connecting donuts cancel motions that are not parallel when their donut particles connect. This creates a situation. A twist in the chain from an electron has no perfectly parallel solution without using an irrational number. An irrational number does not develop from a synchronized chain. Hence, we need to get pretty close using rational numbers.

People who have worked with irrational numbers know that you can never get an exact representation. They also know that you can get as close as you wish by simply using larger and larger integers in the numerator and denominator.

Clearly, we need some logical guiding principles to make sense out of this. Without appropriate guiding principles we are simply manipulating numbers to our heart's content. Even with guiding principles, we must be careful to distinguish real answers from a happenstance of numbers. This is of paramount importance. Anybody who has played with numbers knows how easy it is to follow a rabbit-trail.

Let's jump ahead and look at the answer. We will show how it was calculated in a later section.

The untwisted chain segment length of space is 138 donuts long including one of the connecting end links. The twisted chain segment length of the electron is 137 donuts long.

Those of you who are physicists may be saying: "Whoa, you just used a rough integer representation of the inverse of the fine structure constant." Not so. The inverse fine structure constant's closeness to the chain length was of more than casual interest, but it was not in any way used to bias the calculation of the actual length.

13.   Guiding Principles: This Sure Isn't Politics!

The guiding principles for determining chain length fall into two categories: configuration and stability. We are searching for candidates for the electron. This means continuous connected chain segments must move through space in the opposite direction as described under Configuration.

Configuration - Factors that affect the calculation or limit eligible solutions

Synchronization of contacts throughout a chain segment works best if the number of outer ring nodes is some multiple of the number of chain links in the segment. This will synchronize the end links if the successive intermediate links connect on any one of the nodes.

When a chain moves through space, the end connections must move to accomplish that motion. A pair of chain segments of lengths n become chain segments of lengths n-1 and n+1 after moving an end connection one link.

Stability - Criteria for choosing the most stable calculated solution

The stability of a chain segment ultimately determines what configuration will evolve. There may not be a clear choice of the appropriate stability measurements. Fortunately, over a considerable range of choices the same results are achieved. The author's choice of stability measurements are:

Chain Length - Longer chains (more links) are more likely to have some external disrupting event. Stability is inversely proportional to the Chain Length.

Elapsed Time - The more relative time it takes for contact to occur results in fewer reinforcing contacts and more opportunity to have some external disrupting event. Stability is inversely proportional to the square of the Elapsed Time.

14.   Counting the Links: This Sure Isn't Golf!

How does one even pretend to count the links on something as seemingly speculative as space fabric chains? It turns out the process is actually quite simple and provides a clear answer. We try each plausible solution to find the best one.

For each untwisted chain containing n links, we impose the node configuration from the preceding section while focusing on the electron. There are (n-1)·(n+1) nodes in order to support both of those chain lengths as the electron travels. The rest follows quite straight-forwardly.

Before doing the analysis, we don't know if an electron adds a link or subtracts a link from the normal chain size. So, in addition to looking at all plausible chain lengths we will also try both adding and subtracting a link to hopefully find a good answer.

The actual calculation is quite simple. Normally, chain links have a connecting path that is pi/4 radians (45 degrees) from the axes of both of the connected donuts. Now, we have an extra pi/2 radians (90 degrees) twist that needs to be spread uniformly over all of the connections in the electron chain segment.

The reader may visualize doing this calculation over a huge number of possible chain lengths. Stability criteria limit this range to around a few hundred links at the upper end.

Recall from Section 9. that r / q = R, where q is an integer equal to the number of nodes. In our case that integer is (n-1)·(n+1) nodes.

Now, we have an angle we are trying to achieve for each of the connecting links in the electron leg. The calculation is done by adding one trip at a time around the outer ring. Recall that the contact occurs on the inner side of the outer ring. So, we must always make a complete outer revolution to arrive at that position.

If we assume that we have achieved the desired angle, then each outer revolution will move us around the main donut circle by an amount equal to the tangent of the angle divided by the number of nodes in a complete circle. We also know that the main donut circle needs to by an integer amount to return back to the same point.

Combining the considerations of the previous paragraph, we keep adding the amount by which we move around the main circle for each time we make one revolution of the outer ring circle. We repeat this until we get a suitably stable integer number of main revolutions. We determine what is suitably stable by applying the Stability measures of Section 13.

The amount by which we move around the main circle depends on whether we are subtracting a link or adding a link to the underlying untwisted chain segment. It's a little tricky to visualize, but if we remove a link it elongates (i.e. increases) the amount by which we move around the main circle. The opposite applies if we add a link toe the underlying untwisted chain segment.

Thus, if we subtract a link then the angular amount by which we move around the main circle for each trip around the outer ring circle is equal to Tan(Pi/4 + Pi/2/(n-1))/((n-1)(n+1)), where (n-1) is the new link count and ((n-1)(n+1)) is the number of untwisted nodes.

Similarly, if we add a link then the angular amount by which we move around the main circle for each trip around the outer ring circle is equal to Tan(Pi/4 - Pi/2/(n+1))/(n-1)/(n+1), where (n+1) is the new link count and ((n-1)(n+1)) is the number of untwisted nodes.

15.   Chain Length Calculation: Please, Just Give Me the Answer!

The calculation produces an untwisted chain length of 138 donut links. For the electron a link is subtracted producing a twisted chain length of 137 donut links Nature's choice of this length plays an important role in all of physics.

This 137 / 138 chain length solution is well over an order of magnitude more stable than the next closest candidate. After 74445 outer ring revolutions (74445=3·5·7·709) nearly exactly 4 main ring revolutions have occurred. The deviation of the angle from perfectly parallel is 8.08727982582465E-11 radians.

The 74445 revolutions resulted from the original calculation.

In formula notation:

      atan(274*278/74445) - pi / 4 - pi / 274 = 8.08727982582465E-11

If the reader attempts to do this calculation, be careful to use a calculator with enough precision. There is a good calculator at If you register with your email, they don't abuse it.

The calculation of the chain lengths, angles and node patterns was first done in October of 1996. The results are of pivotal importance to the model. They remain today as they were first calculated and are the most vital ingredient in calculating, 20 years later, the ratio of gravitational to electromagnetic force between two electrons.

There is a download link for a zip file that includes a Windows executable (x86) program which allows one to experiment with the broad stability ranges over which 138 links prevails. If the reader examines the included source code, first browse through locating the appropriate section indicated by the embedded comments. The actual calculation is a very small part of the coding. Important comments are embedded in the source code and included in the attached help (.chm) file.

Click Here - --> download details and source code for 138 link calculation

16.   We're at the Right Place:   Now, Let's Check the Schedule

We are at the correct Main Angular Position in the direction of the adjacent donut. We are also at the correct Outer Ring Angular Position on the inside of the donut outer ring. So, where's the train?

Timing may not be everything, but it is important. It is not sufficient for the adjacent donut particles to be at the same place. They must also be there at the same time.

Do you remember the triangle from the previous section? The relative 'distance' around the Outer Ring Circle was 74445 (or, 3·5·7·709). The relative 'distance' around the Main Circle was 76172 (or, 4·137·139). The hypotenuse of this triangle needs to be an integer value in order for the timing of the two adjacent donut particles to match up with the other cycles. The reader is cautioned that this view doesn't match the physics to which they may be accustomed. This motion is happening in a local region devoid of the fabric of space.

The hypotenuse of a right triangle with legs of 74445 and 76172 is 106509.302922327. This is close to the minimum amount of precision we need in the hypotenuse to get what turned out to be the solution.

Even though this information was known in 1996, a solution was never attempted until March of 2016. The reader may ask why. I was convinced that trying to find 'the' correct rational expression used by nature would ultimately end up to be simply playing with numbers. It seemed likely that no clearly correct solution would present itself. If one looked far enough one could always find some combination that would produce the desired result. The trick would be to find one that gave the desired result by itself with no effort to force a result. So, why look now?

The results of the LIGO gravity wave experiment were released on February 11, 2016. They left no real doubt that gravity traveled at the speed of light; or, at least something very close to the speed of light. I had erroneously asserted that the speed of gravity had to be many orders of magnitude greater than the speed of light. The speed of gravity did not affect the nature of the calculations being done. However, it did affect the credibility of the model.

After the LIGO results were released, I decided to at least look at the possibility of getting the correct rational representation of the irrational hypotenuse. The process was so simple that it only took a couple of hours, most of which was just housekeeping, to get what almost had to be the answer: 4304360459 / 40413 (or, (7^2·347·253153) / (3·19·709).

So, why did this have to be the answer? The process used presented this solution as the eighth possibility. There was no attempt to force an answer. This solution was only about 22 times more nearly an integer than the next most accurate solution. However, three of its factors already appear in the triangle sides (3, 7, and 709). The combination of 22, 3, 7 and 709 made the solution about 320,000 times more accurate. The 709 that canceled (i.e. was already synchronized) in one of the triangle side legs was the clincher.

My excitement ran high as I plugged the values into the calculation that seemed most appropriate to produce the ratio of the gravitational force to the electromagnetic force between two electrons. Alas, the ratio was off by a factor of 2.13005 E-04 from the known experimental value.   The factor turned out to be very close to 4 times the square of the fine structure constant. This seemed to be something much more than a coincidence. This will be discussed more in a later section.

Why is it important to know how the number was developed? Because we must not be using a contrived number that forces a result.

How will we ever know if this is really the correct answer? Like other theoretical results, the experimental physicists will eventually do what they do so well and tell whether this answer is truly predictive. It is also possible that we will understand the precise nature of the fine structure constant at some point and be able to directly determine the correctness of the answer.

17.   Putting It Together:  Where and When do We Meet?

This summarizes previous sections for the combined effect of the Main Angular Position (Main Position), the Outer Ring Angular Position (Outer Position), and the Timing on synchronization.

Let's say the Main Position points in the direction of the adjacent donut once every third round. Let's also say that the Outer Position is on the inside of the donut every third round. In this case, it takes three rounds for each to be in the correct position for contact.

Now, let's say the Main Position still points in the direction of the adjacent donut once every third round. However, let's also say that the Outer Position is on the inside of the donut every fifth round. In this case, it takes fifteen rounds of each in order for both to be in the correct position simultaneously.

What did we just say? Common prime factors in each motion only need to be included one time. The same logic also applies to timing.

The Main Position and Outer Position have perpendicular motions and can be viewed as two perpendicular legs of a right triangle. The hypotenuse represents time and must very nearly be an integer value. It doesn't need to be exact because of the donut particle itself having size.

The Main Position triangle leg has the following prime factors: 2^2, 137, and 139.

The Outer Position triangle leg has the following prime factors: 3, 5, 7, and 709.

The Timing triangle leg (hypotenuse) has the following prime factors: 3, 7^2, 19, 347, 709, and 253153.

The following factors from the Timing triangle leg are not used for calculations: 3, one of the 7's, and 709. These factors duplicate factors already included in the legs.

The remaining unmatched factors in the Timing triangle leg (hypotenuse) needed before contact occurs are: 7, 19, 347, and 253153. The product of these factors is 11683264103.

Whoa! Just a minute here! How does the author expect one to believe that a 16 digit number is not contrived to produce a desired result?   This should be a paramount consideration regarding the validity of such a number. Ultimately, more precise physical constants may help us decide, but how about now?

It is important that the reader appreciate that these indeed are dots in a row and not contrived numbers.

Two of the triangle legs have steadfastly remained unchanged since their discovery over twenty years ago. Those legs are determined by the process of determining the chain length and minimizing the collision angle in Section 15.

The question remaining about the hypotenuse is legitimate. It was over a concern that a clear answer could not be found that the author never even looked for one for over 20 years until 2016. To his surprise upon searching, an obvious solution for the hypotenuse emerged immediately and clearly seemed to be the correct answer if one was ever to be found. This was before testing the results. Testing against experimental values confirmed that this was indeed the correct result!

18.   Calculated Ratio of g-force to e-force between Two Electrons

The standard value for gg/ee from 2014 Codata values is calculated below (the parenthetical value is the standard deviation in least significant digits).

2014 Codata ratio of g-force to e-force for two electrons

    Item   Value (std err) Units
a.   G   6.67408E-11(31) m3kg-1s-2
b.  x me2   9.10938356E-31(11)2 kg2
c.  / alpha   7.2973525664E-03(17) 1
d.  / hbar   1.054571800E-34(13) kg1m2s-1
e.  / c   2.99792458E+08 m1s-1
f.  = gg/ee   2.40053E-43(11) 1


There are at least three equivalent ways that different constants from the 2014 CoData values can be combined to produce the ratio of the gravitational to electromagnetic force between two electrons (ggee).

The three equivalent expressions for ggee are:

19.   Donut Model Ratio of g-force to e-force

There is a fundamental difference between GR and the Donut Model that allows us to determine the relationship between gravitational and electromagnetic forces. GR is continuous. The Donut Model (DDTC) is discrete.

In the following analysis, we look at gravity as if the electromagnetic field were not present. This caveat is included for those who might be delving into the precise nature of the underlying physical relationship.

Also, recall that DDTC explains the electron as a missing link in a chain that creates a twist. The twist that results causes the collisions of adjacent donut particles to be slightly misaligned resulting in a drag on local motion (i.e. a drag on time). Under DDTC, mass causes time drag which propagates as time dilation. Time dilation in turn causes gravity.

The sum of the source time drag in the e-leg multiplied by its frequency must equal the sum of the transferred time drag in each successive spherical shell. Otherwise, an equilibrium would not be established.

Time drag can be viewed as a ratio, d, of motion lost to complete motion. Since the time drag is extremely small, we will be considering d to be simply the sum of all time drags in the electron chain multiplied by their frequency. The motion remaining, r, is equal to 1 minus d.

Let's look at the motion (time) of two imaginary electrons superimposed on each other (note: r and d variable names are arbitrarily chosen).

r0 = 1 space motion with no electrons.

r1 = ( 1 - d ) space motion from one electron.

r2 = ( 1 - d )2 space motion from two electrons with the slower time caused by the individual electron affects on each other.

r2 = 1 - 2·d + d2

The quantity d2 is the mass loss that corresponds to the gravitational force.

The quantity 2·d - d2 is the entire mass loss that corresponds to the electromagnetic force. The mass of the electron is entirely due to its charge. Since d is on the order of 10-43, the d2 term is ignored for e-force calculations.

Thus, d2 / 2·d, or d / 2, is the ratio of the g-force to the e-force for two electrons.

If it is this simple, then why haven't we seen it before? The difficulty with GR there is that no discrete source exists that produces the gravitational source time drag. Reality is discrete and does not suffer from the infinities of the GR continuous functions. It is well established that GR does not work on the quantum level.

The DDTC donut model develops the ratio of gf to ef as being equal to the sum of the contact drag ratio per unit of donut time.  For the main electron chain segment this is equal to (all values are dimensionless).

DDTC model ratio of g-force to e-force for two electrons

    Value Description
a.   137 donut links in e-leg (138 - 1)
b. x (8.087280E-11)^2 / 2 one minus cosine of collision angle
c. / 76172 triangle leg one (main angular position) factors
(274 x 278)
d. / 74445 triangle leg two (outer ring angular position) factors
(3 x 5 x 7 x 709)
e. / 11683264103 triangle hypotenuse (Timing leg un-canceled factors)
(7 x 19 x 347 x 253153)
f. / 3 main angular position factor to achieve 3 way end split
g. / 2 d / 2 for g-force to e-force ratio
h. * 2.13005418 E-04 4 x alpha squared (2014 Codata)
i. = 2.40071078 E-43 vs 2.40053(11) E-43 experimental
(2014 Codata derived)
j. "G" 6.67458168 E-11 calculated value vs 6.67408(31) E-11 m3kg-1s-2
(2014 Codata)


The 8.0872.... collision angle results from looking for stable collision angle modes that match the twist in the donut chain.  Using 1 minus half of the collision angle gives the ratio of the motion lost due to a collision (i.e. the drag). See Section 15. for the calculation of the collision angle. This was first calculated in January of 1996.

The 76172 and the 74445 depend on matching the main revolution position and outer ring position between adjacent donut particles when they collide. These numbers come from the collision angle calculation just referenced.

The 11683264103 (7 x 19 x 347 x 253153) results from synchronizing the timing between adjacent donuts (see section 16). This was first calculated in March of 2016.

The factor 3 needs an explanation. There is already a factor 3 in leg two, but this doesn't help for the main angular position. The ends of the chain need a factor of 3 in order to achieve contact with the connecting space chain segments. This is a position issue, not a timing issue.

This ratio of the gravitational force to the electromagnetic force between two electrons is well within experimental error for the current 2014 Codata values. It is about 1.6 standard deviations larger than the 2014 Codata accepted value.

There is one aspect of this calculation that naturally raises questions. The use of α2 seems quite arbitrary, as it well should. The quantity, α2, was the balancing number needed after calculating the sum of the drag in the e-leg. In this sense, it was arbitrary. However, there is some number needed to complete the calculation. The number needed is the frequency with which the space fabric interacts with the e-leg. Whether the quantity, α2, is the same as this frequency is not known. However, the accuracy with which this reproduces experimental results make it a compelling choice.

It is worth noting that if the α2 factor stands, then this relationship allows calculation of the gravitational constant, G, with the same precision as the fine structure constant, α. This would extend precision by over 5 significant digits! The gravitational constant, G, is one of the least precise major constants in physics.

20.   Explanations of the g-force to e-force Ratio Calculation

An understanding of the g-force and the e-force requires an understanding of time. It is human nature to consider time as having some universal master clock against which one can measure events. Gravitational time dilation provides a strong clue that there is indeed no master clock.

Local time is simply a quantification of the underlying motion of the space fabric. One can't see the fabric, but one can see the results by how fast physical events happen. The fabric of space is connected in our universe so local time has some relationship to the rest of the universe by the nature of being part of a connected fabric.

Let us look at a simplified view of the process for an electron.

The mass of the electron is entirely due to the 'drag' created by the motion of adjacent donut particles not being perfectly aligned when contact occurs. It is the sum of these 'drags' that constitute the mass of the electron. The mass is equal to sum of the motion lost on contact multiplied by the frequency of contact. Let's call this the 'source drag'.

If the more distant space fabric did not continue to feed the motion of the immediately local space fabric, it would eventually grind to a near halt. This relationship is fairly easy to quantify using successive unit thickness spheres surrounding the central mass. For now we will ignore what is meant by 'unit thickness'. Let's call the sum of the drag transferred across the shell the 'shell drag'. This shell drag slows down space outside the shell, but speeds up space by a corresponding aggregate amount inside the shell.

In order for the local time to not deteriorate, it is necessary that the 'source drag' equals the 'shell drag'. The 'shell drag' must remain identical across each successive shell. Remember the 'shell drag' is the sum of the drag for each shell.

At a distance sufficient to treat it as continuous, the formula for time dilation using the drag approach outlined is identical to the time dilation from General Relativity (GR) for gravity from a spherical shell source. However, there are two major differences. In GR, gravity is continuous and considered to cause time dilation. In DDTC, time dilation is discrete and causes gravity. The discrete nature of DDTC permits calculation of the source drag as a specific quantity.

We know that calculation of the source drag for the e-leg does not include a needed factor for the frequency of interaction with the space fabric. The use of a factor equal to α2 produces a value for gg/ee that is within a reasonable range of experimental error to the accepted experimental value. This does not make it automatically correct. However, it does make it compelling.

21.  Fundamental Relationships with the DDTC Model

It is important to realize that DDTC is not a reworking of existing relationships. DDTC provides the fundamental relationships and calculations that allow gravitational constants to be calculated with the same precision as electromagnetic constants.

DDTC model summary equations for g-force to e-force ratio:

1.  Source drag = (1 / Τ2) · ΣΤ0ΔΤ · f1 , where f1 is the frequency of interaction of the e-leg with the adjacent space fabric.

2.  ggee = Source drag / 2 from Section 18.

Substitute equation 1. into equation 2.

3.  ggee = 1/2 · (1 / Τ2) · ΣΤ0ΔΤ · f1

Compelling value of f1 that produces ggee value in experimental range.

4.  f1 = 4 α2

Substitute equation 4. into equation 3.

5.  ggee = (1 / Τ2) · ΣΤ0ΔΤ · 2 α2

ΣΤ0ΔΤ from Section 19., Items a.-f.

6.  (1 / Τ2) · ΣΤ0ΔΤ = 2.25413119 E-39

2 α2 from calculation using 2014 Codata value.

7.  2 α2 = 1.06502709 E-04

Substitute equations 6. and 7. into equation 5. and multiply.

8.  ggee = 2.40071078 E-43

From standard physics relationships.

9.  ggee = G·me/c2/re

Solving for the gravitational constant, G.

10.  G = ggee·c2·re / me

Substituting ggee from equation 8. And c, re & me from 2014 Codata values.

11.  G = 6.67458168 E-11 m3kg-1s-2

Comparing to G directly from the 2014 Codata value.

12.  G = 6.67408 (31) E-11 m3kg-1s-2

G from equation 11. is 1.6 standard deviations larger than G from the 2014 Codata value.

22.   GR Relationships

Note 0:

Ignore this section. I just included it as a note to myself to look at again some day. Yes, this is an odd place to keep one's notes to self.

Note 1:

This Section is extraneous to the ggee and G value development from the DDTC model. It is only included for readers who might be curious about it.

Note 2:

GR relationships are determined by the distribution of matter and energy in spacetime. With DDTC in addition to matter and energy there is a negative contribution due to the presence of space itself. The negative contribution is proportional to the rate of time at the location of the space. This is a very, very small part of the equation. However, it does manifest itself significantly in regions of space that are large distances (tens of thousands of light years or larger) from large central masses.

There is a relationship that may seem unusual to those familiar with GR. The slowing of the universe increases the effect of gravity that was initially propagated at an earlier time. This could help explain MOND which appears to have been generally discredited using conventional GR.

It is interesting to look at standard constants restated based on GR gravitational time dilation relationships to the DDTC model. One could speculate that f2 is equal to f1. Alternatively, one could speculate that f2 is equal to unity. It isn't obvious that f2 is either of these values.

General Relativity time dilation for a mass me distributed in a shell centered at unit radius:

1.  t = t · (1 - (2·G·me) / (r·c2))1/2

2.  d t = t · (1 - (2·G·me) / (r·c2))-1/2 · 1/2 · ((2·G·me) / (r2·c2)) d r

Multiply equation 2. by equation 1. and divide by t2

3.  t d t / t2 = (G·me) / (r2·c2) d r

Sum equation 3. over a spherical shell with unit dimension assumed to be L.

4.  Sum = 4π · r2/L2 · (G·me) / (r2·c2) · L

5.  Sum = 4π · (G·me) / (c2·L)

Using a standard expression for g-force to e-force ratio (ggee),

6.  ggee = (G·me) / (c2·re)

Substitute equation 6. into equation 5. and introduce, f2 the frequency of interaction with the space fabric.

7.  Sum = 4π · ggee · re/L · f2

Substitute ggee from equation 3. of Section 21. into equation 7.

8.  Sum = 4π · (1/2 · (1 / Τ2) · ΣΤ0ΔΤ · f1) · re/L · f2

9.  Sum = 2π · (1 / Τ2) · ΣΤ0ΔΤ · re/L · f1·f2

Set equation 1., Section 21. (source drag) equal to equation 9. (sum of shell drag).

10.  (1 / Τ2) · ΣΤ0ΔΤ · f1 = 2π · (1 / Τ2) · ΣΤ0ΔΤ · re/L · f1·f2

Solve for L.

11.  L = 2π · re · f2

Solve for re.

12.  re = L / (2π · f2)

A standard expression for me.

13.  me = α·h / (2π·re·c)

Substitute re from equation 12. into equation 13.

14.  me = α·h / (L·c) · f2

Rydberg Constant restated in terms of L:

15.  R = α2·me·c / 2h

Substituting me from equation 14.

16.  R = α2·(α·h / (L·c) · f2)·c / 2h


17.  R = α3 · f2 / 2L

23.   Discussion

The dots are in a row. This model development has been presented in detail so that someone sufficiently motivated would be able to follow the logic from beginning to end. This is no small task, but the pivotal nature of the model calls for the logical path to be reproducible.

Note that the downloadable program used to determine the e-leg angles contains significant explanation and documentation both in the calculation source code and in the .chm help file. Anyone serious about studying this model should not overlook these resources. Click Here to download details and source code for e-leg calculations

In order for the G developed here to have the same precision as α, the effective interaction frequency f1 with space must equal 4 α2. If f1 is only close to this value, then linkage needed for the same accuracy does not yet exist.

After additional thought, it seems quite possible that instead of 4 α2 on the source electron that there is 2 α on both electrons. The 2 is likely due to a missing factor such as the need for the donut to communicate with both adjacent donuts, not just one. The α factor then would be the sole remaining factor needed to connect the electron with the fabric of space.

The factor, α, likely develops from the 137 link electron chain segment length. The contact between adjacent links in the electron chain segment is hugely less frequent than in space itself. Space fabric contact with the end link would transfer only 1/137th of the electron drag to the space fabric based on a uniform distribution among the 137 links. The drag is not uniformly distributed due to the nature of its propagation, it builds up more near the center links.

The frequency of contact with the space fabric on the leading and trailing links does not have as large of an effect as one might expect. It has some effect on the distribution and build-up of drag. If space fabric contact occurs less frequently, it transfers larger drag amounts on each contact.

The drag that matter creates on the space fabric may at first seem contrary to the well-established conservation of matter and energy. It is not. The drag on time in the universe caused by the matter it contains is in equilibrium. Time slows down at the same rate everywhere. This maintains the energy and matter relationships. The slowing of time never goes to zero since it is relative.

The full DDTC model has representations for Quantum Numbers, structure of the quark, and representations for the primary baryon octet, including decay routes. These omitted aspects of the model have no bearing on the topic presented here and have been omitted. They also are speculative in nature.

What Next?

The remaining sections contain further steps to be considered. These steps are indicated by the model, but not yet affirmed.

If someone chooses to pursue the steps indicated, an appropriate reference back to the model would be appreciated.

24.   One Step Dance

What? One step could hardly be considered a dance.

The model examines the relationship between two donuts in the electron chain segment as though they are the sole consideration. There is more to be considered than simply utilizing the fine structure constant as a convenient way to get to the result based on the dynamics of a single donut.

The following aspects of the electron chain segment need attention:

other interior donuts, dynamics of the relationship between more than one other donut in the interior portion of the electron chain segment;

trailing and leading chain segment end donuts, dynamics of the relationship between the interior donuts and the end donuts of the electron chain segment;

dynamics of a stationary electron, the possibility that the stationary electron could continuously be traveling in a closed loop;

dynamics of an electron in motion, the moving electron still needs to synchronize with the space fabric suggesting a quantum change in motion as dictated by synchronization needs;

distribution of twist, between the new and the old electron chain segment as the electron changes its position in the space fabric;

It's apparent that a great deal remains to be understood. This journey is far from over.

25.   New Tool

Physicists should not consider this model as diminishing the value of other models. It may best be viewed as a tool to gain insight into other models. While this model presents a physical understanding, other models contain relationships that go far, far beyond the scope of this model. Hopefully, this model would enhance other models.

26.   New Model of the Universe

DDTC indicates a likely candidate to consider for a new model of the universe where the fabric of space grows from the boundary of the universe. This growth forms new matter at the same time.

Boundary growth of the fabric of space is indicated by the following:

Logical growth option choices seem limited to either continuously growing the existing kernel of the universe; or to growing separate portions that later combine with one another;

Nearly equal rates of motion would likely be needed for separate portions to combine. This is almost impossible to achieve given different starting motions and variation in the speed of slowing;

Precise alignment of the fabrics would likely be needed for separate portions to combine. Again, this is almost impossible to achieve given random fabric orientations;

Donut interaction between the fabrics would be unlikely at any meaningful level given the infrequent interaction experienced by already synchronized donuts;

Parameter choices for the Boundary Growth Model could start with testing the following:

Uniform growth rate of the space fabric outward from the boundary. Growth occurs by random donut particles attaching themselves to the fabric. It seems reasonable to assume this growth would occur at a fairly uniform rate;

Uniform density of matter of the newly formed space fabric. Matter forms as part of the process of space malformations that occur in forming the new space fabric. It seems reasonable to assume this matter would form at a fairly uniform density;

Interesting characteristics of the Boundary Growth Model include:

Matter near the boundary would have double the effective gravity since no gravity can be directed outward due to the absence of space fabric in that direction;

Cosmic Background Microwaves would occur uniformly over the entire boundary surface during the formation of the highly energetic matter at the boundary;

Inner matter is accelerated outward and boundary matter is accelerated inward;

Tracking some specific matter from its creation at the boundary produces:

First, the matter is accelerated inward creating a tendency toward an increasing blue-shift;

Second, additional boundary matter creation slows and eventually reverses the initial inward acceleration. This leads to a lessening of the blue-shift and eventually to an increasing red-shift;

Third, the matter density of the inner portions of the universe would be less than the outer portions due to the accelerations. This results in an increasingly faster clock rate in the inner portions;

Forth, for inward traveling light waves, the faster clock rate on the leading edge compared to the trailing edge would increase the wavelength. This would be an extremely subtle difference that would require enormous distances to be significant. This red-shift may be greater than the blue-shift caused by initial inward acceleration;

Until some aspiring astrophysicist creates this model and compares the results to reality it is merely conjecture. A reference back to the DDTC Model would be appreciated.

27.   MOND Gravity

Note: numerical values for this model are under development. The mechanism being modeled is summarized below. The crucial features are that a large mass must be surrounded by a large low mass density region. This should be regarded as conjecture until further developed.

If the Universe were infinite and contained only a source point of matter, then the inverse-square relationship for gravity would be correct. Until seeing MOND, I had never looked beyond this. My initial reaction to MOND was that MOND was a contrivance. Now, I consider it is one of the most amazing relationships for someone to have developed. It has not been given the full credit it so richly deserves.

In order for MOND to occur, it seems that a spiral galaxy must contain a large central mass surrounded by a lot of sparsely populated space. We can view the mass in galaxy as determining the boundary of a region of space such that it will contain the average mass-density of the Universe. This region must be large enough that slowing down of the Universe will have a material effect on the gravity.

This local region of space is the "flywheel" that balances the "braking" action of the central mass. There is zero gravity at the boundary of this region.

As we travel outward from the central mass, two things happen. First, the portion of the "flywheel" traversed cancels out part of the effect of gravity. Second, the gravity created at an earlier time is operating on a more distant part of the region. This means the gravity created is stronger relative to the more distant regions because the Universe has slowed down during the travel time.

The "First" thing actually becomes more pronounced near the boundary of the region. Hence, it actually becomes pronounced later than the "Second" thing.

An interesting thing happens when the traversed volume becomes large enough with weak enough gravity, the weakening of gravity due to inverse square relationships can be offset by the slow-down rate of the Universe. One happens with space the other with time, but they both have an effect.

This article may be copied and distributed only in its entirety to others for non-commercial purposes.  Mass printed publication of this document or portions of this document alone or as a part of any other document, is prohibited without the express written permission of the author.  This copyright notice must be kept intact in any copy or partial copy distributed.

The author appreciates input or questions from others.  Please do not hesitate to contact him (click below).

Thanks for your interest, Richard Marker

Footnotes :

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1. William Berkson. 1974. Fields of Force. London. Routledge and Kegan Paul. Chapter 1, Faraday's Problem Situation; p. 23-24.

2. Pablo Acuña. 2013. On the Empirical Equivalence Between Special Relativity and Lorentz's Ether Theory. [Preprint].

3. Nigel Calder. 1977. The Key to the Universe. London. British Broadcasting Corporation. Chapter 2, Starbreaker; p. 54-55.