**The Mandelbrot Set
and Cosmology**

by Jonathan J. Dickau ©'97,'06 - all rights reserved

**Chapter Two**

**Early Forces - Cosmological Isomorphism at the beginning**

People have wanted to know the origin of all, for as long as the historical record shows, and probably much longer. We have not always had the means to study, nor the tools to examine the first moments of time, however. To study the very beginning, in an attempt to discern the origin of the universe, we must leave behind some of our conventional notions about time, space, and distance, and develop a more flexible framework of dimensionality. This is something that many physicists, including advocates of both String Theory (or M-Theory) and Loop Quantum Gravity, all seem to agree upon. Theoretical physicists of all persuasions know that, in the realm of exceedingly small size, the ordinary laws do not apply and the natural forces seem to all converge into one force. So; scientists have been working diligently, for almost 100 years now, to elucidate the nature of that realm. The study of the Mandelbrot Set, with the perspective of this theory, may allow us to understand more fully what we cannot hope to directly observe. We can examine the conditions at the very origin of the universe, using the Mandelbrot Set.

At the start of my
explorations into a Cosmology based upon the Mandelbrot Set, I was
trying to make it fit the concepts and tone of Big Bang theory, or
what has become the Standard Model of Cosmology. Along the
way I learned about Inflationary Universe theories, and how they
avoid some of the complications created when one tries to apply
the Big Bang model in combination with the Standard Model of Particle
Physics. More recently, I have come to grasp *why*
Inflationary theories can avoid some of those complications, and I
have also learned why a Cosmology derived from the Mandelbrot Set
must be framed as, or considered an Inflationary Universe theory.
It is my current working hypothesis that the Mandelbrot
Cosmology describes how 2-dimensional space curves toward the 3rd
dimension, creating broken symmetry and a universe of form along the
way. This insight is causing me to re-write this essay,
little by little, to incorporate and reflect my new hypothesis.
My present understanding is that the early universe arose from a
noncommutative space, and that the Mandelbrot Set was a natural
outgrowth of the process by which the property of commutativity
arose in the pre-bang era.

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The Mandelbrot Set

turned inside out,

or unfurled about

the origin at (0, 0i)

Modern Cosmology can
give us insight into events that happened a long, long time ago, in the
first few moments after time began. The detailed analysis of
new observational data from astronomy, radio-astronomy, and measurements
of the Cosmic Microwave Background radiation, can tell us a lot about
the early universe Combining this with data from high-energy
particle physics, we can surmise what must have happened from a small
fraction of a second onward, to create the universe we observe.
What about the beginning itself? Did the universe actually begin
with the beginning of time? Is that where the story *truly*
begins, or was there some pre-existing potentiality which shaped and
molded the forms things could assume? What were the forces
which actually gave birth to the Universe, and how may we best
understand them? Some of the answers may lie with the Mandelbrot
Set, or in the Mandelbrot Fractal Butterfly figure, and its family.
Modern Science has fostered much speculation, and continues to
push the clock back so we can see further into the past, but even the most
powerful theories deal only with events which happened a fraction of a
second after the Big Bang, and little is known for certain of how the
Bang began (or what happened before it). There may be an easy way
to observe this, however. It's my belief that the key to
understanding the universe and its evolution can be found in the
Mandelbrot Set, and by studying its intermediate stages of convergence.

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Mandelbrot Fractal Butterfly

is obtained by highlighting

areas where magnitude of the

dynamical variable is decreasing

Understanding this fully involves grappling with the question of how dimensionality arises, and how the properties of space become established. Space was likely very different in character, when the universe had its origin, from the way space appears today. It is my working assumption that the geometry of space at its origin (near the Planck time) was noncommutative (explained below), and it's my observation that the steps taken to evolve conventional spaces (with a commutative geometry and algebra) provide all of the tools needed to create the Mandelbrot Set. Specifically, the evolution of topological (having surfaces) objects and spaces, and the property of measurability, give rise to a vocabulary of concepts and processes sufficient to realize (or compute) the Mandelbrot Set. The form of the Mandelbrot Set itself allows us to chart the evolution of space beyond this point. The key concept of this theory is that the M-Set portrays the bounding surfaces which create symmetry-breaking, through the folding of space, and thus engender the fundamental forces with their familiar faces. A number of physicsists, including Paola Zizzi, have shown that the conditions of the early universe were sufficient to constitute a Quantum Computer. Knowing this can also be helpful to understanding this theory, in that it can explain how a mathematical object like the Mandelbrot Set could influence the universe's evolution before there were people with computers to explore it.

If one examines the
Mandelbrot function's behavior at the origin, or zero point, of the
complex plane (0, 0i), one finds that this is an instantaneously, and
infinitely repeating point. As one goes outward slightly, there are
rings where the function repeats the same value after one or two cycles
have occurred. The closest edge of the set is only .25 units away from
the center, in the direction of positive, real numbers. This is at the
cusp, or point, of the cardioid (heart-like) shape near the top of the
figure. This cusp corresponds to the Planck Time, or 10^{-43}
seconds into the Big Bang, according to this theory. The universe was
expanding, at this point, but was still very, very small indeed. By
conventional standards, the universe was quite tiny, almost infinitesmal.
For all practical purposes, the universe could be said to occupy a
single point. The understanding that is emerging, at this time, seems
to be that the realm of the exceedingly small appears to have a geometry
that is noncommutative, however. This means that the ordinary laws
of size and distance do not apply. Simply put, distances don't add
up, or can't be multiplied, when we examine space near the Planck scale.
And they seem to vary on their own. It would seem, therefore,
that reality began in a noncommutative space, but we now know that such
spaces tend to evolve over time.

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Magnification of

The Mandelbrot Set

centered at (0, 0i)

showing the cusp

at (0.25, 0i)

It's almost as though distance
and size don't matter, until space has assumed a particular dimension.
There is seemingly some critical value, however, where size *does* matter.
Once there is enough space to measure, it would appear, the conditions
are ripe for the universe to expand and evolve. At the Planck time,
the universe occupied a size of only about 10^{-33} cm, much smaller
than the nucleus of an atom. Once the universe arrived at this size,
and expanded beyond this point, the unfoldment of an entire universe of form
was inevitable. The laws of Physics take over, once the threshold is
crossed, and the familiar forms of those laws apply in increasingly greater
measure, as the Big Bang proceeds and very soon after the universe begins to
expand toward the familiar level of scale. A certain extent in space
is apparently necessary, or a certain level of dimensionality, before
the universe as we know it has the dimensions to expand into, or before
physical law can take shape. Beyond that point, it seems that
form appears automatically, or that the universe actively invites form
to inhabit it. Likewise; the fundamental forces take on their
familiar faces as things acquire measurability, taking on the property
of size or dimension, and as the emerging universe of form takes shape.

Similarly, the Mandelbrot Set exhibits complex behavior and develops detailed patterns of form only when a critical size, or distance, has been exceeded. All points on the complex plane within a radius of 1/4 from the origin (0, 0i) will converge under the Mandelbrot equation, or will not go to infinity no matter how many times the calculation is repeated. However, any positive real number greater than 1/4 will eventually escape all bounds, and fly to infinity, after a certain number of repeated iterations. This happens at the cusp, a point which has no imaginary component, and a real number value of 0.25. Once one exceeds this critical value of (0.25, 0i), or simply a real value of one fourth, the Mandelbrot function gets larger with each successive iteration. The dark body of the Mandelbrot Set seems to be drawn in toward the center (0,0i), at that point in the figure (-0.25,0i). It is at this point where the figure is perfectly symmetrical on the imaginary axis that we see symmetry broken for real numbers. We see that the left and right side of the image below are mirror images of each other. However, we see that the top and bottom are very different, in their character of form. Negative real numbers can be as large as -2, and still not grow out of bounds upon repeated iteration.

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The Mandelbrot Set

But some points in the
Mandelbrot Set neither fly to infinity, nor shrink away upon repeated
iteration. Instead they arrive at some constant value, after a
while (within any level of precision), and this value repeats endlessly.
All of these points are part of the Mandelbrot Set, and all of
the early repeaters fall within the main body of the Set. This
can be clearly seen on the figure below with 1st and 2nd order Repeating
Points highlighted (repetition with a period of one or two).
What is shown highlighted in this image are those areas where the
function repeats the last value, or the one immediately previous.
Near (0,0i) and (-1,0i) all points repeat after only a few
iterations, but as one moves out toward the edges of the
Mandelbrot Set, more and more iterations are required before the
* size* value repeats. This figure seems to depict
the luminous aspect of the universe, in terms of its vibrancy and
expanding energy. The image was created by highlighting
those areas displaying cyclical variation, which is characteristic
of waves and wave-like phenomena. What the image reveals,
in my view, is the purely wavelike nature of reality, rendered as
a halo within the Mandelbrot Set.

If one examines the cardioid, or heart shaped, region in the upper half of the image above, it is easy to imagine that the repeating points are a ball of energy expanding outward from the origin, toward the edges of the set. However, as it expands, its symmetry is broken. What we see is that this expansion starts in roughly concentric circles, but as the expansion proceeds toward the cusp these cirlces become more and more dimpled, and morph gradually into a cardioid-like shape. It is notable, however, that there appear to be holes in that shape, which do not resolve even at very-high iterations. These spaces appear, in the halo, in the region near each bulb (except for the largest circular region), with the magnitude of the corresponding missing spaces depending on its size. If one were to create waves in a liquid, within a Mandelbrot shaped vessel and starting at the point of origin, these waves would break against the cusp and be forced to go one way or the other. This is another kind of broken symmetry, as the energy from waves is swept into clockwise motion on the right, and counter-clockwise motion on the left. The different-sized bulbs (or voids) around the periphery also serve as a kind of filter for various wavelengths of vibration.

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Magnification of Mandelbrot Set

with Repeating Points highlighted

centered at the origin (0, 0i)

In the enlarged image above, one can see all of the detail described in the preceeding paragraph, but there is also another feature to describe. There are radiating spokes extending out from the center, that seem to reach toward some bulbs and avoid others. In the central region, these spokes appear fairly evenly spaced, but successive color bands seem to have more, and more of them. As these spokes approach the periphery of the object they appear to bend toward the edge, as though they are trying to approach it from a perfect right angle, or perhaps to exactly match the angle of the bulb or void it is approaching. If the expanding energy ball of the early universe is represented by the repeating points, then the radiating spokes may correspond to the angular variations we observe in the Cosmic Microwave Background radation, and the density variations in the expanding fireball. I have not attempted to quantify specific correspondences, however. But having only examined the repetitive and cyclical nature, or wave-like aspect, we must now look to its complement, the substantial part of the equation and the particle-like aspect. Does the Mandelbrot Set offer us insight into how matter arises, as well? It seems that it does indeed. It give us a lot of information about this aspect of the universe's birth too.

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Mandelbrot Fractal Butterfly

The particle-like
aspect, which constitutes the protons, neutrons, and electrons that comprise
our bodies and planet, can be seen in the disks around the periphery of the
Mandelbrot Fractal Butterfly figure above. This figure highlights areas
with diminishing size (of the dynamical variable). If we magnify the
same area of the M-Set as in the previous paragraph, one observes that the
ands at the uppermost tips of the butterfly wings curve toward the cusp
from either side to approach the surface from an angle of about forty-five
degrees. However the points on these wing-like figures are blunt
and in this area the upper and lower faces come together at an angle
slightly in excess of 90 degrees. These bands require many fewer
iterations, or computational cycles, to resolve than either the
repeating size function, the Mandelbrot points, or the surrounding
repeller sets. The image below required *only a few
minutes* to calculate, where the repeating points image of the same area
took over an hour. In the part of the set near the cusp, the
Butterfly wings appear to overlie the detail around the edge of the set,
obscuring it entirely. All of the bulbs around the slope of the
cusp have been completely covered over. What does this imply?
If those wings depict energy exchanges, this would imply that
the early universe was too hot for objective form to congeal.
The primal energy itself would push out the boundaries of space,
and energy bands would sweep away any form before it had a chance to
emerge. This is reasonably consistent with accepted cosmology
theory.

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Magnification of the

Mandelbrot Fractal Butterfly

centered at origin

The fact that these wing-like structures require less time or energy to resolve, or calculate, as compared to the alternative forms which they overlie or displace, is significant in terms of how the Mandelbrot Set might encode quantum probability, for early events. Form in our universe will normally assume the shape, or state, which allows it to use, or occupy, the least energy. In other words, a wood block on its edge, rather than on its side or its end, is more stable. Water would rather flow downhill than sit on top of the hill, but it will collect in a lake or pond. Crystals form because they allow dissolved minerals to give up some of their energy, and relax into a stable configuration. The ability of form in the primeval universe to soak up Energy into shapes of Matter requires this form to be favorable to, or somehow easier to attain, than alternatives. Easier, that is, than simply remaining energy. So; when considering what makes particles emerge from a sea of energy, we can reasonably assume that the shapes they assume are easier to maintain than it is for them to revert back to pure energy.

The favorability of one form over another may depend upon the shape of bounding objects, as when water fills a pond, or upon properties contained within that which is assuming form, as when water becomes a snowflake. In most cases, the form assumed will be more orderly, or at a lower energy than the amorphous state, or whatever form it is replacing, and this is a key point to consider. What causes matter to congeal from energy in the first place? It could be that the very possibility for energy to assume form is enough to make it happen. But it would be just as likely for the shape assumed to transform back into energy, unless there is an advantage to a particular formation, though it wouldn't stay that way for long, so long as it was hot enough. The image below is a magnification of the Mandelbrot Fractal Butterfly figure, around the right-hand arc from the cusp, with discs emerging from behind the butterfly wings. Could this depict the first sub-atomic particles tentatively emerging from the energy 'soup' of the universe's earliest moments?

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Discs begin to emerge

from behind wing-like bands

on Mandelbrot Fractal Butterfly

The early universe was very much like this; it was full of energy with exotic forms so fleeting as to be almost impossible to re-create in the laboratory. Even our most powerful particle accelerators are far shy of the energies needed to create the earliest and heaviest particles. When stable sub-atomic particles finally did emerge from the energy soup, this allowed some of the energy of the Big-Bang to be encapsulated in form. We imagine that the earliest moments of the universe contained free strings and quarks that were not bound into particles, but the tendency exists for energy to flow in circuits. If a specific form allows loops of energy to relax into a lower energy state, this can be a more stable configuration. In the earliest moments of the universe, however, the opportunity for this to happen did not exist, or rather didn't last for long as energy was so greatly predominant over matter, and the universe had to expand and cool a bit, in order to allow form as we know it to completely coalesce. At some critical moment, the Color Force, or Strong Nuclear Force, went into action - binding energy into sub-atomic particles. First were predominantly heavy particles from a group known as Hadrons, which includes Protons, Antiprotons, and Neutrons among others.

As one goes outward, along the edge of the Mandelbrot Butterfly, toward either (0, i) or (0, -i), there is an emergence of independent forms, which look more like a circle or disk as you go around the curve. In addition to the disks, there are other forms which are more like a segment or a fragment (e.g. - a crescent moon) thereof. If one examines the Mandelbrot Set in the area (see image above) near either of these points (+i or -i), one will see shapes that are very reminiscent of the tracks in a bubble chamber, or some other particle detector, connected to a particle accelerator. Looking at the Fractal Butterfly in the same region (as in the image below), one can easily imagine that the circular forms we observe are the particles which are participants (materials) or reaction products in sub-atomic interactions, and the threads connecting mini replicas are the reaction (or energy) paths of those interactions. At these points on the figure (near i or -i), it appears to be the Strong Nuclear Force (or the Color force) which is mainly depicted, and the creation of the first stable sub-atomic particles involving the primal energy, by binding it into independent or free-standing forms.

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Same magnified area of Mandelbrot

Fractal Butterfly, centered at (-0.1, i),

shows particle interactions

If we zoom out slightly, we can see that there is what appears to be a particle nursery of sorts, in this part of the Set. Note that the discs are not competely circular, but rather they are squashed, and they have a piece missing on the side facing away from the body of the Mandelbrot Set. If we compare this to the mode of action of the strong nuclear force, there is a parallel. When the strong force binds sub-atomic particles to form atomic nuclei, the individual particles give up some of their mass to form a single entity (the nucleus) that binds to itself. The color force, which is the precursor of the strong force acting to bind the sub-atomic particles themselves, is presumed to act in a similar manner. The correspondence here is that regions that are almost circular in shape are areas of enhanced probability, according to Mandelbrot Cosmology. The implication here is that the ease of computing the disc-shaped regions of the Fractal Butterfly figure seems to represent the tendency for energy to congeal into particles, because such forms can be manifested with greater ease than the alternatives.

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Lesser magnification of Mandelbrot

Fractal Butterfly near (0, i)

shows particle nursery

The initial binding of energy into stable particles of form (Hadrons) took place during the first millisecond of the universe, according to Big Bang Cosmology (and most of its variations). When a certain time had passed, the relatively heavy particles were once again swept away by bands of energy in exchanges which resulted in the annihilation of most of these particles. To a large degree, lighter particles (Leptons) began to predominate, at that point in the evolution of the universe, and other interesting things began to happen. If you examine the behavior of the form along the edge of the Butterfly figure, as it swings around the corner of the cardioid (past +i and -i), the ending of the Hadronic Era (by more wing-like figures) is clearly depicted, where the effects of the Strong Nuclear Force give way to other forces, and the next phase or cosmological era begins. I take this era up in the next chapter of this essay, The Mandelbrot Set and Cosmology, entitled Approaching Decoupling - the Matter/Energy Cloud Expands.

©'97,'06 Jonathan J. Dickau - all rights reserved

Single copies of this document, for reference or personal use,
are allowed,

but reproduction for commercial purposes is not permitted.

This work constitutes part of a theory first published in November
1987,

in a publication called Amygdala. These are the most detailed records

of my findings to date. I hope you enjoy reading them. - 7/31/06

**What happens next?**

**the next chapter
**

the next chapter

Approaching Decoupling -

the Matter/Energy Cloud Expands

annotated image of fractal butterfly

Shows Graphically Overview of Theory

1st close-up of fractal butterfly

Shows Spawning Ground of Particles

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Last revised on

August 4, 2006

- jd