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

by Richard L Marker

New webpage with complete details of cause of gravity calculations:


ThursdayThoughts webpage with sequential thoughts leading to the donut chain:


This is an updated version of the original webpage that contains speculative sections not in the new webpage. Click on the link above to go to the newer version.

Richard in earlier years

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 the updated original webpage. It contains more general, easier to understand, and sometimes speculative descriptions of many aspects of DDTC.

Main index :

a. Introduction
5. Time, the Mystical Ruler
b. Copyright notice
6. Expansion of the universe
c. Preface
7. Ten dimensions
d. What is ddtc?
8. Left or right universe?
0. Overview
9. Ddtc details and calculations
1. Is ddtc physically real?
10. The Quark, dissected!
2. What is time?
11. Speculation from ddtc
3. What is gravitational mass?
12. Feedback
4. What causes charge?
13. Fun images not related to ddtc

Go directly to 138 chain length calculation

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.

c.  Preface

The Discrete Donut Twisted Chain theory(ddtc) is dedicated to my grandmother I never met. Margaretha Elisabeth "Grace" Fox Marker was born 3 Feb 1907, and died of tuberculosis on 4 Jun 1925. Grace brought my father into the world on 11 Dec 1924.

My grandfather worked for the railroad. Grace, already weak, endured pregnancy living in a railroad car in the bitter winter of Casper, Wyoming. My father's original birth certificate contains the number of the railroad car in which he was born.

Without Grace's sacrifice this model would never have been developed. I like to think of it as her legacy.

I initially developed ddtc over a twenty year period beginning in 1978. The fundamental form of the theory developed during the first year, but it was not until 1996 that useful calculations became possible. The theory developed separately from mainstream theories, although many parallels exist.

This is new and fertile ground... dig in!

d.  What is the "ddtc" space/matter model?

Most models of space and matter develop from the "top-down" as an explanation of observed phenomena. "Ddtc" develops from the "bottom-up" as a logical explanation of the nature of space, matter and time. The ddtc model conforms to observed phenomena.

Existing models do not provide a logical explanation of the nature of space, matter and time. They rely on many ingredients to define particles and force-carriers. The existing models fail to resolve the ingredients into a common and understandable construct. Indeed, a force simply results from the exchange of force-carrier particles. Existing models provide no detailed explanation of what drives a force or of what creates a particle.

All models originate from assumptions about the nature of reality. The ddtc model originates from extremely simple assumptions:

Can a lay person understand ddtc? Many lay people find the basic operation of ddtc understandable. Most would choose to avoid any of the physics or mathematics involved in developing or extrapolating the model.

Can a physicist understand ddtc? A physicist can understand ddtc if they allow themselves to explore the process involved. The physicist possesses much better tools for exploration, but also possesses many preconceived notions. An open and inquiring mind works best.

Ddtc contains some new concepts that must be learned. You must learn the ddtc view before you attempt to extend it to your non-ddtc views. The views do extend, but you miss the point by leaping too early. Standard physics does not possess all of the concepts needed to understand ddtc.

Ddtc provides concrete answers based purely on the geometry of the model without any inputted constants. It goes far beyond vague concepts.

Amazing beauty and structure develop from the simple beginnings of ddtc!

0.  Overview

Ddtc describes the fabric of space and matter as connected donut chains. Each link of the chain develops from the path taken by a "donut particle" (the only particle) traveling a helical path around the donut surface. A chain segment with a pi/2 twist (quarter turn) creates charge.

Including time, the donut possesses 10 dimensions, three standard spatial dimensions, one time dimension, and 6 donut dimensions that loop back on themselves. The discreteness and geometry of the donut particle path allow calculation of the gravity to electromagnetic force ratio from basic concepts involving no inputted constants or units.

Knowing the exact position of the donut particle along the donut path at any instant is crucial for calculating gravity. Theories that are continuous or lack the exact discrete form of the donut seem incapable of deciphering gravity. Treating the entire donut chain link as a discrete entity of continuous composition falls short. Each individual link (donut) must be viewed as a path that a discrete particle travels; rather than being viewed as a continuous link.

Do not confuse this donut model with other helical models that operate on a larger scale. The electron, for example, travels around a closed group of donut chains. This entire connected chain twists in a helical manner and resembles the donut of some other models.

1.  Is the ddtc model physically real?

The physical descriptions in this model are believed by the author to be the real processes, not simply analogies to the processes that occur. The ddtc model differs from most models in its development. Rather than matching mathematical models or functions to observed physical data, this model develops from a logical metaphysical approach (in a thought experiment sense, not a religious sense). Thus, the foundations of the model answer as much "why it is so", as they do "what it is". Indeed, important aspects of the discrete nature of space seem unreachable unless approached from this "bottom up" method.

The reader will find that this model appears to depart from conventional wisdom about general and special relativity and about stable paths of motion. The author recognizes the seeming absurdity at first glance of some of the conclusions or logic. These departures do not stray as far as appearances suggest. Often ddtc simply provides an alternative view of standard phenomena.

Do not make your understanding of this material more complicated than it is. The concepts, regardless of how foreign, are simple and few. So simple that you can easily scoot right past the heart of the issues.

2.  What is time?

Time is fundamental with ddtc.  Each major revolution of the donut particle about the major axis is a unit of local time. Adjacent donuts are very closely synchronized with each other, forming a giant interconnecting "gearing system" that can be considered to be a space clock made up of each donut particle in space connected to adjacent space clocks. Differences between adjacent clock speeds form time dilation and gravity. Before the existence of the donut chain fabric the term "time" would seem to have a vague meaning.

3.  What is gravitational mass?

All mass results from charge or the movement of charge. At least for the electron, the twist in the donut chain is compensated for by the donut adopting a different ratio of main angular revolving speed (around the donut center) to secondary rotation speed (around the donut edge) in an attempt to realign the angle of contact with adjacent donuts to zero. The stable pattern adopted does not quite close the angle of contact to zero. The resulting small angle of contact yields slight opposing motions that simply average each other and slow the donut particle speed ever so slightly due to this contact drag.

The sum of all the contact drags for the connections that constitute a particular real particle produce that particle's aggregate particle time drag (i.e. time dilation summed over a surface integral that encloses the various contact drags for the particle). Note that the particle motion (spin and translational) as well as simple angular donut chain twist adds to the contact drag. A donut time unit is treated as time it takes a donut particle to make one revolution around the donut.

The change over donut distance (i.e. donut unit distance) in the time dilation caused by a distant particle's contract drag produces the acceleration "warp" of space-time resulting from that particle.

The calculation of contact drag requires matching the elapsed number of donut revolutions until contact between adjacent donuts, together with the primary revolution speed and secondary rotation speed. All three of these numbers must be in phase before contact is achieved.

Note that the donut path traveled is not a continuous string but simply the path of a donut particle. The calculation of this number for the main electron donut chain segment is included in a later section, notice that prime numbers play an important role.

4.  What causes charge?

Charge (charge source, not charge field) results from twisting a donut chain segment. A donut removed from an untwisted chain segment requires a twist in order to reconnect with the adjacent donut chain segments.

Space fabric consists of donut chain segments that are 138 donuts long outside the range of the strong force.

Assuming you twist the donut chain segment the shortest distance needed to reconnect for every addition or removal you do, then the removal of a donut link will create a negative charge or the addition a positive charge. Thus, if the non-charged donut chain length is 138 and you remove one donut, the resulting length of 137 will carry the negative charge. If the non-charged length is 136 and you add one donut, the resulting length of 137 will carry the positive charge.

An electron moves through space by the leading end connection with space shifting by one donut. This shortens the next chain (the new electron) and lengthens the current chain (the old electron). The flow of the connections creates a distortion in the space fabric that is compensated by a string of donuts traveling through space to relieve the distortion. This flow is akin to the magnetic vector potential.

The "skin effect" where high frequency waves travel only in the skin of a conductor relates directly to the need for donuts to travel through space in order for electrons to flow through the conductor. At high frequencies the electron flow in the conductor and complementary donut flow in space only have time to penetrate a small depth into the conductor. The electron flow pressure has already reversed directions by the time the required donut flow can penetrate deeper into the conductor.

The fact that 139 and 137 constitute are prime adds to the electron stability since intervening in-phase modes are less likely.

5.  Time, the Mystical Ruler

Ddtc reduces time to the clock position of a gear (donut) in space, but elevates time to the master controlling all interactions. The gears of space synchronize so closely with one another that transmission of an enormous force through them creates hardly a whisper. When a donut frisks about sufficiently out of step with other donuts, we elevate it to the position of matter. Egocentricity is contrary to ddtc.

Before donut particles learned to travel in the donut path, time did not exist. The first part of a second after the big bang has limited meaning with ddtc, since the big bang gradually created the beginning of the sequence of current time.

Donuts are the engine driving the universe. Matter slows them down and gradually slows down the more distant donuts in space. These distant donuts revolve at higher rates than those close to matter to create the time warps that produce gravity. The donut is its own clock. You can slow it down all you want and it still behaves as though it were going the original speed since the measure of time slows in unison with it.

6.  Expansion of the universe

Does the slowdown of all of the clocks in the universe happen without other effects? The slowdown is extremely minute and occurring everywhere simultaneously, so effects from it would not be obvious in our local physical measurements; or, possibly not detectable at all.

Expansion of the universe is explored in greater detail with theoretical analyses in the ggee calculation link.

7.  Space has 10 dimensions

The casual usage of three dimensional space creates the appearance of naivete on the part of the author. What magic does three dimensions possess?

The author believes that our three dimensional world does not dictate that a three dimensional space could be the only answer. Rather, that the dimensions which our world displays are merely reflective of the primary dimensions of space.

Each donut possesses a major axis that can have three orientations and magnitudes. Additionally, each donut possesses a rotating axis for the donut particle to create the spiral aspect of the path. These rotating axes also can have three orientations and magnitudes. Including time this makes ten dimensions.

Several of the donut's dimensions are related to other dimensions. The primary donut orientations are limited by the chain relationships to one another. The secondary donut orientations of the rotating axes almost always would be parallel to the motion of the donut particle around its major axis. For most calculations it seems that this would be a likely occurrence.

8.  Space has handedness

Try to connect donuts of opposite handedness. You will soon discover that the same handedness is needed in order for adjacent donuts to have their donut particle motions be parallel at the point of collision (usually at 45 degrees to their major axis). This handedness of space violates one's sense of natural symmetry. The beta decay of the neutron produces a neutrino always with the same spin lending support to this aspect of discrete donut twisted chain theory.

9.0.  Ddtc Details and Calculations

9.1 Evolution of ddtc
9.2 Chain length of 138 links
9.3 Ratio of g-force to e-force
9.4 Fine structure constant
9.5 Proton to electron mass ratio
9.6 What is a quark?
9.7 Gravitational time dilation
9.8 Does ddtc work with relativity?
9.9 How big is the donut?

9.1.  Evolution of the ddtc structure

One assumption principally leads to the results of this model. No force can be communicated through empty space regardless of the smallness of the distance. Interaction occurs only between opposing contact motions of donut particles when they collide.

Ddtc assumes the universe consists of a single particle type called a donut particle residing in empty space (absence of everything, i.e. a void, not a vacuum). The donut particle is a single element of "something" with no special characteristics. Assume that it is a round uniform ball for simplicity. Also, note that the author uses the term "space" alone to refer to a vacuum as opposed to "empty space".

The discrete donut particle travels along the path of a spiral wrapped around on itself to form the overall shape of a donut. Adjacent donuts form the links of a donut chain. All donuts of a connected space must be either right or left-handed, but not both. This arises from the need for adjacent donuts to smoothly mesh on contact. Normal space forms from connected donut chains. Each link of the chain (donut) turns its major axis 90 degrees from the adjacent donut, just like a normal untwisted chain (i.e. it takes a chain of four donuts to return a donut with the same orientation).

The donut model bases many results on the fundamental torus shape (donut shape). A few mind games help us to understand how and why the donut particle achieves the dynamic path shape of a torus. Remember, the donut-like paths resemble chain links that connect to form a chain segment. These chain segments connect to form a space fabric constituting a preferred coordinate system.

Embark on a mental journey that begins before a space fabric, a chain segment, or even a donut path existed. The donut particles existed for they are the "something" residing in empty space. The donut particle at this early stage traveled a haphazard periodic path rather than its eventual donut path. Why did it travel this haphazard path?

Empty space possesses no preferred coordinate system whatsoever. All coordinate reference systems, accelerated or not, achieve an equal status in the total void of empty space. The idiot donut particles don't know their location. Each donut particle remained stationary relative to itself. Relative to another donut particle the motion could take on any periodic path of motion. The haphazard periodic motion could be represented by a series of circular motions superimposed on each other (similar to a Fourier series for you mathematicians). Note that this description of the donut particle motion occurs before achieving the donut path of motion.

The nonexistence of a preferred coordinate system in primordial empty space is why donut particle motion exists. Before the space fabric formed the donut particle didn't know how to stand still. Does this sound like total nonsense? You may want to reread these two paragraphs as they contain the crux move leading to ddtc.

This unwieldy view of space seems to only confuse matters. Indeed, until one donut particle contacts another donut particle nothing very interesting happens. Upon chance contact two donut particles average their opposing motions relative to one another. Eventually, more donut particles collide and average opposing motions to create a preferred coordinate system of the space fabric. The averaging of opposing donut particle motions lead to all changes in the universe. Note that the motions under discussion affect only donut particles. The donut particle doesn't correspond directly to a standard physics particle. Standard particles and forces develop from the dynamic configuration and interaction of the donut particles constituting the space fabric.

Recall that we described the early donut particle haphazard periodic motion as the sum of circular motions. It turns out in general that only two of these circles enjoy stability relative to each other and the space fabric. The major stable circle results from the donut particle motion about the major axis through the center of the donut hole. The second stable circle results from the helical path followed by the donut particle as it travels around the major circle. Stable non-canceling geometries usually require that the donuts be positioned with the line of contact at 45 degrees to the major axis of each donut (i.e. adjacent donuts have major axes at 90 degree angles). At the very instant of contact the motions of the colliding donut particles nearly parallel one another. Circular motions other than the two primary ones eventually cancel during the collision process. This leaves only the donut path that forms the basis of our model.

A circular stable path of motion belies our common sense. In ordinary life we view the physics of the space fabric rather than the physics of empty space. With no contact what-so-ever outside of itself, a particle has no way to "know" if it is traveling in a circle. Thus develops the first argument for the donut: a circular path of motion is stable in empty space.

The second argument for the donut comes from need. Recall our principal assumption that only pushing contact interactions occur between donut particles. We need a way for a donut particle to affect another donut particle without continually pushing it away. The circle (donut) provides a path in which pushing forces can exist and be stable. Descartes used interlocking circles in work he did centuries ago. The author did not have access to this work (still in French) to know how similar his circles may have been.

The third argument for the donut relates to consistency with views of the big bang. Originally, all donut particle were traveling randomly. This was before a big bang. Particles collided to form small separated regions of space of different sizes, different handedness and different "gauge". The 138 donut chain segment length of normal space described in the ddtc model resulted only after much transition. Initially each donut chain segment formed or combined in various lengths. After more activity (time passage) early on some of the stable space gauges could still have been different.

Probably the closest view resembling the big bang occurred some time after initial formation, when the different separated space regions ("mini-universes") collided with each other to form larger regions (same handedness) or destroy portions of each other (different handedness).

Mini-universes combined to decrease the total number, but likely not to a single universe. Indeed, a single universe may be impossible. If the handedness of universes sum to zero, then the final collision between the single remaining right-handed universe and the single remaining left-handed universe could destroy both universes. This may return conditions back to the near beginning of a big bang. (Note: In ddtc there really isn't a big bang as conventionally used.)

9.2.  138 Link Chain Length Calculation

9.2.a What causes 138 links to be stable?
9.2.b Table A --- Donut chain stability
9.2.c Table B --- Collision angle
9.2.d Features leading to 138 links
9.2.e Formulas leading to 138 links

9.2.a.  What causes 138 links to be stable?

Out of what magic hat did we pull the length of an untwisted chain in space? Why are space chain segments 138 links long and electron chain segments 137 links long? This is one of the most beautiful insights provided by ddtc.

The chain length of 138 donut links supports many of the numbers developed in ddtc. Nature's choice of this length plays an important role in all of physics. Due to this importance some discussion the development has been included. The following zip file includes a Windows executable (x86) program that allows one to experiment with the broad range over which 138 links remains stable.

download details and source code for 138 link calculation

The following two graphs demonstrate much greater stability for twisted donut chain segments 137 links long that develop from untwisted space donut chain segments 138 links long. The numbers shown have been limited to chains that are 180 links long for convenience of display. Numbers up to 400 have been tested. Chain instability contains an n^2 factor that lessens the chance of finding more stable longer chain segments.

9.2.b.  Graph A --- Donut chain relative stability

links image

click to view

Graph A. measures the stability of various chain lengths. The values shown are Log10(contact misalignment angle squared * time elapsed between contacts * number of major cycles). Values are normalized based on the 137/138 value that is the minimum.

9.2.c.  Graph B --- Donut chain collision angle

angles image

click to view

Graph B. measures the contact misalignment angle for the various chain lengths. Values are normalized based on the 137/138 value that is the minimum.

9.2.d.  Donut features leading to 138 links

The surface of the chain link (i.e. donut surface) is not real. The only real thing is the "donut particle". The "donut particle" travels a path of a circle on a circle. The major circle's axis is the major donut axis. The outside circle's axis is the center of the donut tube. This path looks like a slinky bent in a circle. It is important to remember that this is only a path the donut particle follows.

The "donut particle" is the only particle that exists in a real sense. The properties of the "donut particle" should not be confused with those of other particles in physics. Other particles develop from combinations of donut chain segments (except for the neutrinos, they are free donut particles possessing a particular motion). Key to the acceptance of donuts is an understanding that the donut path is stable as it exists. In fact any path of the fundamental "donut particle" is stable until it contacts another donut particle. Opposing motions then cancel.

A normal untwisted donut chain has links with the donut axes of adjacent donuts perpendicular to each other. At the moment of contact donut particle paths are at angles of pi/4 to the major axes. This angle is determined as the tangent of the main donut radius times its angular velocity divided by the minor donut radius times its angular velocity. Angular velocities are relative. The main donut radius is measured from the main axis to the inside surface of the tube. The minor donut radius is the radius of the outside tube. The number of donut nodes is the ratio of the minor angular velocity to the main angular velocity (i.e. the number of minor outside circles in a full revolution).

There are a number of conditions that apply to the donut particle paths. It is these conditions that permit us to determine the geometry and length of the donut chains.

Conditions affecting donut link interaction stability

1. The speed of adjacent donut particles at the moment of contact must be almost identical.
2. The path directions at the moment of contact must closely align.
3. The minor donut angular position at the moment of contact must align to inside of the donut tube.
4. The major donut angular position at the moment of contact must fall on a line between the two donut centers.
5. The time to traverse the donut path must bear some integer relationship with the time to traverse a standard untwisted donut (to synchronize).
6. The preferred number of donut nodes has a multiple of the chain length (to synchronize the ends).
7. The preferred number of donut nodes in a twisted chain segment (electron) has both an (n-1) and an (n+1) factor. These factors are necessary because as the electron moves, chain segments propagate the magnetic vector potential by connections in space moving such that two adjacent n-length chain segments temporarily become n-1 and n+1 chain segments during the propagation.
8. Higher contact frequencies within the chain (vs endpoints) are preferred to lessen outside destabilizing influences.

Long donut chains have more chance to be "crossed" by other donut chains during the time of space fabric creation. They are also more affected by a slight change in their motion.

Short donut chains have fewer nodes in the donut and less chance of "linking" up in the first place with another donut. They are also more limited in the angles of chain segment linkage permitted (i.e. where three chain segments come together in a common connection).

These criteria are by no means absolute or exclusive. They do help predict donut behavior. The matching condition (number 7., above) greatly limits the number of acceptable results.

9.2.e.  Formulas leading to 138 links

The calculation was correctly performed in the original version. The details can be found at the following link:
ggee calculation

This link includes the complete detail that is sufficient for someone to reconstruct the calculation. Be sure to download files on the linked site since they contain significant parts of the calculation.

9.3.  Gravitational to electrostatic force ratio between two electrons

The originally calculated value was clearly erroneous. However, it did include the 76172 leg and the 74445 leg used in the correct calculation.

The corrected calculation is detailed at the following link:
ggee calculation

This link includes the complete detail that is sufficient for someone to reconstruct the calculation. Be sure to download files on the linked site since they contain significant parts of the calculation.

9.4.  Calculation of fine structure constant

The following calculation is not intended to be a theoretical calculation of the fine structure constant. Rather, it is a very interesting calculation that uses some simple factors from ddtc. The chain length of the twisted e-leg is 137. The chain length of normal space is 138.

Calculation of fine structure constant

a.     1/137   fraction of e-leg contacted by space
b. /   (1382 – 22)   1st space e-m mode
c. x   (1382 - 32)   2nd space e-m mode
d. =   0.00729735325   Calculated
e. vs.   0.0072973525664(17)   2014 Codata

The calculated value does not fall very close to the experimentally established range for the fine structure constant. Regardless, it has considerable enough precision to make it quite interesting.

9.5.  Proton to electron mass ratio

A triangle with a chain segment for each side seems to be a likely candidate for a stable particle such as a proton.  The smallest such candidate that seems reasonable has a perimeter totaling seventeen donuts.  The particle spin requires that the dimensions of the sides change as the external connection to a triangle corner moves.

If mass is viewed as the sum of the inverse squares of the chain length times the charge squared, then the three seemingly most natural stages for the triangle produce:

Proton to electron mass ratio

  triangle sides mass units internal charge
 add external :
stage 1 (5+,5-,7+) 1/25 + 1/25 + 1/49 (0,0,+2)
stage 2 (5+,6,6) 1/25 (+1,0,+1)
stage 3 (6++,5-,6) 4/36 + 1/25 (+1,-1,+2)
average   0.062451 (-1,+2,+2)/3
x 1372   1823.96 vs 1836.15 exp.

Additional mass comes from charges on the attachments of the legs to the main space fabric.  An exact picture of the connections must be determined before completing this calculation.

There is a problem with this mass calculation. It seems to come from a different calculation than that of the electron g-force to e-force ratio.  This could mean it is wrong or simply not fully understood yet.

Initial calculations suggest that the proton, neutron, Sigma, Xi and Omega- particles may all have this basic triangle size (some with different charges).  It is too early to tell, but mass ratios of 2^1/3 and 2^1/2 appear to play a role if the basic core is the same.  The Lambda particle is an odd-ball with a (5+,5-,6; 5+,6,5-; 6++,5-,6) core triangle shape.

These calculations are all quite speculative, but intriguing.  The fractional powers of two seem possible with ddtc, but do not obviously provide an answer.  An understanding of what the mesons and muons are may provide clues that help to complete this area.

9.6.  Quarks

In discrete donut twisted chain theory (ddtc) the quark is simply the connection between three donut chain segments, the minimum number of segments needed in order to have separate segments.  As such it is difficult to discuss quarks as a separate entity.

The 1/3 fractional charge on the quark results from the proton, for example, having three donut chain sides with spin.  This results in the charge appearing to be the average charge from three (or more) different states.  Additionally, the attachment chain segments may have charges that add to the total apparent charge for each quark.

Deep inelastic scattering from the electron has been accurately calculated. The standard calculation varies to the square of the charge. This indicates that an average of 1,0,0 would not work the same as a continuous one-third charge since 1 + 0 + 0 is not equal to 1/9 + 1/9 + 1/9.

One explanation of this might lie with the structure of the electron. With its spin it has an orientation as well as a charge. Possibly electron-electron interactions have these spins aligned and interact fully. The electron-proton interaction may have random alignment of this orientation affecting the degree of interaction. This is quite speculative and may not pass muster under scrutiny.

Possibly a better explanation could come from the distribution of the effective charge. Rather than being 1,0,0; it may gradually transition from one stage charge level to the next stage charge level.

9.7.  Gravitational time dilation

The averaging assumption for donut collisions produces gravitational time dilation that equals that produced by General Relativity for point gravitational sources in an assumed infinite universe of space fabric.  However, there are important distinctions.  Donut collisions are a discrete process.  At distances much larger than the diameter of the electron, continuous approximations are reasonable.

For universes of a non-infinite size the donut model produces different gravity at larger distances from a point source. It also produces different gravity for non-point sources. Interestingly, gravity near the boundary of the universe; or, at zero gravity points within the universe approaches an inverse linear relationship instead of an inverse square relationship. This possibly could provide a theoretical basis for MOND

Gravity does not cause time dilation.  Matter is a cross-connection that slows the local "space-clock".  It is this time dilation that causes gravity.

Pure black holes with a "Schwarzschild radius" below which nothing can escape do not accord with donut theory.  The donut space is discrete long before reaching this small dimension making the calculation of such a radius moot.

Planck's constant results from the energy in the pi/2 twist in an imaginary donut chain one link long multiplied by the time to travel around the donut chain at the speed of light.

9.8.  Consistency with special relativity

Many readers will likely question how the space fabric can be consistent with special relativity.  On first (and second) glance it appears to be nonsensical that the two could co-exist.  Details of this co-existence are omitted from this write-up.  However, Special Relativity transformations can be developed directly from the assumption of an Euclidian space with no special characteristics except that a body in motion will seek the same electromagnetic equilibrium between its component particles as the equilibrium that existed at rest.

This view of relativity gives some insight into the paradoxes of Special Relativity.  The resulting transformations consist of a "real" part and "apparent" part.  If the observer is at rest relative to the space fabric then the entire transformation is real.  Otherwise, it is not.  If an observed body instantly changes speed, its apparent distance from an observer also changes.  It is this discontinuity of apparent measurement that needs to be considered in order for a paradox not to exist.  If s is the absolute speed on an observer and u the absolute speed of the observed, then the standard relativistic transformation (Beta) is equal to the product of an observational error and a real transformation part.  The observational error introduced by the motion of the observer is :
(c2 – s2) / (c2 – s·u).

The real portion of the transformation is :
(c2 – u2)1/2/(c2 – s2)1/2.

See the document referenced below for more details.

You may download a Word document that contains an unpublished development of Special Relativity transformations assuming an Euclidean Space with normal characteristics.
Click Here ---> download paper on special relativity -- pdf document

9.9.  Donut size

Determining the scale of the donut has been one of the most perplexing problems.  The Planck Length seemed to indicate a size many orders of magnitude smaller than the size eventually developed.  While it is possible that the structure may be that small, it seems more likely that the donut path size is closer to the traditional radius of the electron divided by 137.

The size of the donut particle traveling around the donut path likely could be estimated based on the recent(2016) calculations of ggee.

10.  The Quark, dissected!

Follow the pdf file link below for details of the configuration and decay routes of particles in the Spin 1/2 Baryon Octet.  A description of the contents of the pdf file follows.

The baryon core consists of three donut chain segments that form a triangle.  The three apexes of the triangle each connect with a short chain segment to portions of the space fabric (donut chains) outside of the baryon structure.  Each baryon core changes states three times as the apex connections to the core triangle change connections from one donut of the core to an adjacent donut.  This effectively subtracts one chain link from one side and adds it to another.

Take care to note that a side of the triangle with five links can either be negative or positive depending on the orientation of the end donuts.  If the end link orientations are the same as for four links with no charge (i.e. not twisted),  then five links has a positive charge.  If the end link orientations are the same as for six links with no charge,  then five links has a negative charge.

Also take care to note that the dynamic nature of the core triangle produces three states. For this reason, the average charge exhibited by a quark is a multiple of one-third.

The sum of the number of donut links in all three sides of the core triangle do not change for a particle.  Seventeen links produce strangeness 0.  Sixteen links produce strangeness -1.  Fifteen links produce strangeness -2.  Fourteen links produce strangeness -3.  Each link removed decreases the core charge by one which in turn decreases the visible charge by two since each charge is connected to two apexes.

The charge on the sides of the core triangle counts twice, once for each apex to which it is connects.  The charge on the outside connection to each apex counts only once in determining total charge.  The outside connection charge is shown in the linked file at the top of the small box for each quark.

For a given Baryon number, the Strangeness and Charge permit one to determine the directional Isotopic spin quantum number.
Click Here ---> Donut Chain Configurations and Decay Routes of the Light Spin 1/2 Baryon Octet

11.0  Ddtc Speculations

11.1 Antigravity, hot! ... Time travel, not
11.2 Antigravity, are your hopes up?
11.3 Time travel slowed down ...
11.4 Kooky free energy

11.1.  Antigravity, hot!  . . .  Time travel, not

How many of us have shared a seat with Walter Mitty* in wondrous travels through time?  Maybe we visited a race track of past  . . .  or lingered another moment with that special person.  These passions beget a disappointing chapter in the story unfolding.  Time travel, not  . . .

Disappointment may consume us, but fear not!  Mr. Mitty never dies if you truly hold rank as a daydreamer.

Antigravity and free energy entice the uninitiated.  Donut theory offers us a glimmer of hope that these animals exist.  But, can we harness them?  Most well bred scientists and all well behaved scientists likely consider free energy (I define as the creation of significant energy without the destruction of matter) to be elusive.

The dogma of science builds on all of itself, including its mistakes.  Those who most vigorously defend or deny something usually act from strong prejudice.  Prejudice that keeps eyes closed.  The messengers of free energy may sound like crackpots.  Indeed, they may even be crackpots.  Let them not shut our eyes anymore than one immersed in the dogma of science.  Antigravity and free energy, hot!

* James Thurber created Walter Mitty, daydreamer extraordinaire, as the lead character in one of his stories.

11.2.  Antigravity, are your hopes up?

Antimatter might at first glance seem to provide a path to antigravity.  Alas!  Both antimatter and matter produce positive gravity and attract one another.  They each slow the clock of space.

If free energy can be extracted from the fabric of space, it likely would slow down the space from which it was extracted.  Is it possible that slowing down the space (time) above an object could change the time dilation gradient and accelerate the object upward?

If an antigravity engine could developed, it could be used for propulsion in outer space to speeds near the speed of light assuming other difficulties of such high speed travel could be overcome.  Does this mean we could visit other solar systems?  And they visit us?  This is all very speculative.

11.3.  Time travel slowed down ...

The donut dances many exciting paths.  Contrariwise, the boredom of time is limitless.  All that adjacent donuts do when their donut particles collide is average opposing motions.

The averaging process is extremely inefficient as it slows down the local passage of time (time dilation).  In general, this doesn't matter since the passage of all time in the universe slows down at the same rate.

Time is not a dimension that can be traveled in two directions.  Or, even in one direction other than at its natural speed.  The averaging process has no corresponding "un-averaging" process.

11.4.  Free energy, does it belong to the Kooks?

At times there has been talk about extracting energy from the fabric of space.  Main streamers mainly discounted that this could be possible due to conservation requirements of energy and matter.

The donut fabric "wastes" limitless energy in its averaging process.  We don't notice this waste since we observe the resulting positions of the donuts themselves.  The fact that all clocks have slowed relative to an earlier clock does not appear to be detectable.

Can we extract the energy? Is it "free"?  This question has no clear answer.  The limitless energy available in the donut combined with so many reports of free energy arrangements of moving magnets stir one's imagination to hope that it could be true.

Reports of free energy machines becoming colder at times helps support their credibility.  If the donuts could collide in a pattern that inputs energy to the system, it seems likely that they could also do the opposite.

Free energy input/extraction could easily escape detection.  Consider a surfer on the ocean.  The surfer might go up and down all day long without gaining speed.  With skill the surfer learns how to ride the wave to extract the energy.  Can mankind learn how to "ride the wave"?

Ddtc doesn't actually suggest that the energy in the universe might increase from a mechanism that generates "free energy".  The mere presence of any generated "free energy" would slow the local speed of space.  This slowdown would in turn cause other neighboring energies to decrease until the original time equilibrium was restored.  The net effect of all of this would be to increase the amount of available energy rather than the total amount of energy in the universe.  In other words, "free energy" amounts to decreasing the entropy of the universe, and not actually violating conservation of energy and matter.

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The author appreciates input or questions from others.
Thanks for your interest,   Richard Marker