Close to the Edge, Event Horizons, Black Holes and the Mandelbrot Set (or everything I know about the Universe I learned from the Mandelbrot Set)

by Lori Gardi

Abstract

Using the complex plane as my space-time manifold and the iterative function (system)

z=z^2+c to model the dynamics and evolution of this manifold, I am able to reproduce to

a great extent the complex dynamics that we observe in our Universe. In this manner, the

Universe could be thought of as an iterated function system that has emergent properties

that manifest as fractal patterns and complex dynamics. This greatly differs from the

current model of the universe where only differentiable manifolds are allowed.

Introduction

The concept of fractals is relatively new, developed and popularized by (the late)

Benoit Mandelbrot in the 1980's. Mandelbrot actually coined the term "fractal" to

describe geometric shapes that have fractional dimensions. Before that, there was no

concept of "fractal" and therefore no language with which to describe them. Even

Stephen Hawking admits that he does not know to fit fractals into the standard model of

cosmology. In the introduction to his paper “Virtual Black Holes”, he makes the

following statement:

“One might expect this space-time foam to have very complicated structure, with an involved topology. Indeed, whether space-time has a manifold structure on these scales is open to question. It might be a fractal. But manifolds are what we know how to deal with, whereas we have no idea how to formulate physical laws on a fractal.(1)”

Exploring the fractal nature of the universe requires a paradigm shift in the way we

think about the structure of our universe. A fractal „manifold‟ is very difficult to deal with

using standard techniques because there are no clear boundaries with which to make

measurements. In other words, calculus does not work on fractal manifolds and

therefore, the main tool with which all of cosmology is based, cannot be used. New tools

need to be developed in order to study and quantify the fractal nature of our universe.

(See: “Mathematical Model for Fractal Manifold”, Fayca (2))

In this paper, I present an alternative way of looking at the universe through the eyes of

one of the simplest iterative function systems that we know; The Mandelbrot Set (M-Set).

Using the M-Set as my “microscope/telescope/particle-accelerator”, I was able to explore

every aspect of this mathematical anomaly in great detail. What I discovered was a

universe not unlike our own. I found black holes and event horizons; galaxies and galaxy

clusters. I was also able to simulate particle dynamics much like what you would see in a

bubble chamber experiment from a particle accelerator. Last but not least, I was able to

create a universe, or something that looks very much like a universe from scratch, using

only the Mandelbrot Set iterative function system.

Mandelbrot Set

In my model, the Mandelbrot Set refers to the set of all points in the complex plane as

iterated through the function z=z^2+c. (See: Appendix A for details on the Mandelbrot

Set and Julia Set algorithms.) Technically, the set called Mandelbrot refers to the set of

points in the complex plane that never reach escape velocity during the iteration process.

These are the points that are traditionally painted black as seen below. These are the

points that are analogous to black holes in my model.

Figure 1. The Mandelbrot Set. Black = Inside : Blue = Outside.

In the above image, the blue points represent points that are able to reach escape

velocity (given a finite number of iterations), and the black points represent points that

can never escape the boundary (given an infinite number of iterations).

The Mandelbrot Set is a Universe in its own right. It is a perfect example of bounded

infinity. The M-Set curve is an infinitely complex one dimensional curve that is bounded

in two dimensions. It is technically a bounded boundless boundary. This is the boundary

that separates OUTSIDE points (blue) from the INSIDE points (black). It could be

thought of as a horizon, and in my model, is analogous to the event horizon of a black

hole.

Space-Time Manifold

In my model, the whole of the complex plane is my space-time manifold. Each point

from the complex plane (r,i) represents a particular point in space (i) and time (r) which

makes it a space-time manifold by definition. This is a fractal manifold when iterated

through the function z=z^2+c, therefore, it cannot be differentiated. Instead of

integration, we will be employing various iterative techniques or algorithms in order to

analyze and understand this particular manifold, and its behaviours.

Figure 2. M-Set, Space-Time Manifold

Notice that I assign the dimension of space to the imaginary component and the

dimension of time to the real component of the complex plane. When you think about it,

it makes sense to assign asymmetric dimension to time, since time is asymmetric (as in

the arrow of time). As well, since space is seen to have symmetry, the symmetric

dimension (the imaginary dimension) should be assigned to space.

Defining time as real and space as imaginary is contrary to the standard model of

cosmology. For example, in “A Brief History of Time” (1), Hawking defines time as both

real and imaginary which effectively clears up the singularity of the Big Bang. In this

case, the complex plane would be map of time-time rather than space-time. It is also

possible that each dimensions of the universe has both a real and imaginary components

(2). This may be the case, however, for the sake of this analogy; I will stick with the

notion that the complex plane is a space-time manifold.

We can then extend this manifold to four dimensions by using quaternions (r,i,j,k) in

place of complex numbers. In this case, there is one real dimension of time and three

imaginary dimensions of space.

Imaginary/Space

Real/Time

Black Holes

Black holes are cosmic objects with massive gravitational fields that curve space-time

so drastically that nothing, including light can escape the boundary or event horizon of

the black hole, not even light. A big part of my hypothesis hinges on the idea that the M-

Set is analogous to a black and/or is a black hole generator. As it turns out, the M-Set

does generate similar dynamics to that of black holes which I will be demonstrating in the

next few pages.

Iteration = Energy

Figure 3. Black region = point of no return.

In my model, iteration is analogous to energy and therefore an increase in iterations

represents an increase in energy and vice versa. It is well known that the closer one gets

to the event horizon of a black hole, the more energy it takes to escape its gravitational

field. In a similar manner the closer you get to the Mandelbrot Set boundary, the more

iterations it takes to for the algorithm to reach escape velocity.

Conversely any object that finds itself on the inside of a black hole will find that it

takes an infinite amount of energy to escape, and therefore it will be trapped forever on

the inside of the black hole. In a similar manner, I find that the points from the INSIDE

of the M-Set (as iterated through the function z=z^2+c) never escape the boundary of the

M-Set and are forever trapped on the inside of the Mandelbrot Set even after an infinite

number of iterations (or energy). In this manner, the black hole analogy holds true.

Event Horizons

An event horizon is the boundary that separates the outside of a black hole from the

inside. Once inside, nothing can escape the event horizon of a black hole. It is often

referred to as the boundary of no return.

Figure 4. M-Set Event Horizon

The boundary of the Mandelbrot Set is an event horizon in the truest sense. It is the

boundary that separates the outside of the M-Set from the inside; it is the boundary of no

return, just like in the black hole model. This horizon, unlike the event horizons in

theoretical physics, is a fractal, and consequently, this black hole DOES have hair. This

fractal horizon has the property of self-similarity and scaling so you will find many self

similar copies of the original “black hole” at different scales each with its own event

horizon. Contrary to the standard model of theoretical physics, where it is said that black

holes have no hair, I will demonstrate in this paper that the black hole in my model (the

M-Set black hole) behaves very much like the real black holes found in nature.

In my model, the whole of the galaxy represents the event horizon of the black hole

(black whole?). See the “Galaxy” section for more details on this. In other words, the

event horizon of a black hole is a fractal.

Gravity

In order to explain the black hole nature of the M-Set, I need to have an analogy for

gravity in my model. It is well known that large gravitational bodies such as the earth

and the sun are not homogeneous; rather they appear to have an increase in density

toward the center and a decrease of density toward the surface. This change in density is

referred to as a density gradient. In Einstein‟s theory of relativity, massive celestial

bodies curve space-time creating giant vacuum density gradients. The largest

gravitational fields, especially around black holes, will produce the largest gradients.

Figure 5: M-Set as a Density Gradient

In the above image, each region (alternating black and white) represents a region of

similar mass/energy/iteration. The points farther away from the M-Set black hole have

lower “mass/energy/iteration” and therefore can reach escape velocity easier. The points

closer to the black hole have higher mass/energy/iteration and therefore take longer (more

iterations) to reach escape velocity.

In this image, you will notice that space-time curvature increases as you look closer

to the black hole region, much as you would expect from a gravitational field around a

black hole. Also, since iteration is equivalent to energy/mass in my model, then the

gradient generated by this algorithm can be interpreted as a density gradient. Notice how

the slope of the gradient appears to increase as you look toward the black hole region?

This reinforces the notion of gravity in my model. In this manner, gravity can be seen as

an emergent property of the space-time manifold.

In my model, all the forces are merely the emergent properties of the complex non-

linear dynamical system(s) that generates the universe.

Singularities

To further my analogy, I must be able to demonstrate how the Mandelbrot Set

generates singularity. As I mentioned earlier, each point on the complex plane generates a

unique dynamic when iterated through the function z= z^2+c. The points from the outside

of the M-Set eventually reach escape velocity fly away to infinity. The points from the

inside of the Mandelbrot however, appear fall toward a single point in the complex plane.

This is point represents singularity in my model.

Figure 6: M-Set Fractal Dynamic Field

Above is an example of what happens to the points from the inside of the M-Set as

they are iterated through the function z=z^2+c. The red dot represents the starting point

or initial conditions of this system. I refer to this as a “space-time fluctuation”. The

yellow dots represent all the points generated by the iteration process. I refer this as a

“fractal dynamic field”. This dynamic appears to be a collapsing field where the points

are spiralling in toward a centre region or what I refer to as singularity. Given this

analogy, I could say that a “space-time fluctuation” generates a “fractal dynamic field”

that collapses to “singularity”.

I believe I am justified in calling this a singularity, at least in my model where iteration

is equivalent to energy. In the above example, it will take an infinite number of iterations

to generate or resolve this field. It will never escape and it will never stop collapsing. In

my model, this is analogous to the infinite mass/energy of a black hole singularity.

Singularities continued.

Each point in the black part of the Mandelbrot Set generates a different singularity,

some more complex than other depending on how far the initial point is from the event

horizon. Below is an example of a beautiful singularity generated from a point from the

inside of the Mandelbrot Set. The initial point for this singularity was close to the event

horizon of the M-Set and therefore it is much more complicated than the one on the

previous page.

Figure 7. M-Set Singularity: Close to the Edge of the Event Horizon

What is interesting about these “singularities” is that they are not centred with respect

to the rest of the dynamic. If the singularity were perfectly centred, then this figure would

have circular symmetry and therefore, it would be completely predictable. Since this is a

chaotic system, the output of this system is not predictable. In fact, after following these

singularities down to 500 decimal places or more (I had to write my own math library to

extend the digits of precision of my 32-bit computer), I discovered that these fractal

singularities are actually falling toward and/or generating irrational numbers. Each time

you iterate, the dynamic collapses further so you need more digits of precision. As you

add more digits of precision and continue to iterate the dynamic collapses further. This

can only continue indefinitely if the resulting singularity is an irrational complex number.

Turns out, the points from the inside of the Mandelbrot Set are complex irrational number

generators when iterated through the function z=z^2+c

Singularities continued.

Here is another example of a Mandelbrot singularity.

Figure 8. M-Set Singularity

Figure 9. Plant Growth

Interesting how these singularities seem to replicate they dynamics of plant growth.

Galaxies

Below is an example of a real galaxy, NGC 1232 (top image), and a fractal galaxy

(bottom image) generated using the traditional M-Set algorithm. Clearly you can see the

similarity in structure and texture; bright spots vs the voids; the mini galaxy extending to

the left; the distinctive paths of the spiral arms.

Figure 10. Galaxy NGC 1232

Figure 11. M-Set Set Galaxy

Although the fractal galaxy is a bit “contrived” compared to the real galaxy, the

similarities are striking and suggestive that a similar dynamical process was involved in

the creation of these patterns. This demonstrates that at least this galaxy has fractal-like

morphology which suggests that some non-linear dynamical feedback process was

involved in its creation and evolution.

Galaxies continued

Here is another example showing the fractal morphology of galaxies. Notice the

finger-like configurations coming off both the fractal galaxy and the real one.

Figure 12: M-Set Galaxy

Figure 13: Colliding Galaxies? Arp 292

This unlikely configuration could not have evolved by accident. More likely, it

evolved through some non-linear feedback process that promotes fractal morphologies.

Since black holes are found at the centre of all galaxies, I believe that black holes might

actually participate in the feedback process that generates these patterns. In this sense,

black holes could be considered as creators rather than destroyers. I also believe that

there is a possibility that the two supposedly “colliding galaxies” (above) are not actually

colliding at all. This image could represent the morphology of a dynamical system that is

creating this fractal pattern. In other words, this is how they formed and will continue to

form in the future. Is there any proof that these galaxies are colliding?

Galaxy Clusters

Using the same method used to generate the singularities, I am able to generate a

pattern that looks very similar to a real galaxy cluster. The image at the bottom (left) was

generated by plotting the trajectory of a point selected from inside one of the mandel

buds near the boundary of the M-Set.

Figure 14. M-Set, point selected inside “MandelBud”

Figure 15. M-Set Galaxy Cluster (left) – Abel-370 Galaxy Cluster (right)

Notice the similarity between the clustering of the fractal pattern (left) and that

of the real galaxy cluster (right). As can be seen from the above image, this algorithm is

able to reproduce the morphologies of galaxy clusters very accurately.

Generating a Universe From Scratch

In the image below, I randomly selected points from the complex plane and iterated

them through the function z=z^2+c. I plotted the points accordingly. Using this

algorithm, I was able to generate a universe from scratch, at least something that looks

very much like a universe as promised.

Figure 16. Random space-time fluctuations.

Figure 17. Cascading random space-time fluctuations.

In the bottom image, I selected one point from the complex plane. I iterated that point

(100 iterations), and plotted its trajectory. Then, I repeated that process for all the points

in the original trajectory. The results of that experiment are shown in above image. This

looks very much like a star field to me. Many of the galaxy shapes are represented here.

Orbits

Now I'm going to show you what happens to a point or particle as it “falls” toward the

event horizon of the black hole of the Mandelbrot Set.

Figure 18: Falling point/particle near the M-Set black hole.

Figure 19: Black Hole orbits left, Mandelbrot orbits right.

The image on the left represents 16 years of data consisting of the most detailed

observations yet of the stars orbiting the centre of our galaxy, bolstering the case

that a monstrous black hole lurks there. The image on the right represents the orbits of

a test particle “falling” into the “event horizon” of the Mandelbrot Set. Needless to

say, the similarity is striking; and the implication are that the Mandelbrot Set is either

a black hole, a meta black hole or a black hole generator.

Test particle falling into

the event horizon...

Particle Dynamics

In general relativity, the equivalence principle states that local effects of a gravitational

field are indistinguishable from those arising from acceleration, and vice versa. From

this, it could be argued that extreme gravity (ie. Gravity near a black hole) should be

indistinguishable from extreme acceleration like what you would see in particle

accelerators. In other words, particle dynamics near a black hole should exhibit similar

dynamics to that seen in particle accelerators.

Now, I will argue that the only way we can "see" a particle is via the bubble chamber

pictures generated by smashing particles together in particle accelerators. We have never

actually seen these accelerating particles directly, only via these bubble chamber pictures.

In fact, what we are looking at when we look at these "pictures" are the "dynamics"

generated by accelerating (smashing) these particles in the particle accelerators.

Everything we know about these particles is inferred from the "pictures" generated by the

bubble chamber experiments. (indirect measurements)

Given that, the equivalence principle states that accelerating particles in a particle

accelerator is equivalent to “falling” or accelerating particles near a black hole, then we

should agree that falling particles near a black hole should generate similar dynamics (or

pictures) as the particle accelerator experiments. In other words, a bubble chamber device

near a black hole should generate similar "pictures" as the particle accelerator bubble

chamber experiments. In fact, this turns out to be the case as seen in this article from

CERN. See link below:

Black holes could act like cosmic particle accelerators.

http://cerncourier.com/cws/article/cern/29320

In order for the Mandelbrot Set analogous to a black hole, it should be able to

reproduce the dynamics of a bubble chamber pictures according to the equivalence

principle.

Figure 20. CERN bubble chamber image (left); M-Set experiment (right)

The Big Bang

The image below is created by generating and keeping track of all the trajectories of all

the points from the INSIDE of the M-Set. A histogram is generated where the brighter

regions in the image contain points that have a higher probability of intersecting a

trajectory and the darker regions contain points that have a low probability of intersecting

a trajectory. Put simply, this image is a “probability distribution map” of the all the

trajectories of all the “singularities” from the complex plane.

Figure 21. M-Set Probability Distribution Map

In my model, each point in the complex plane generates a different “dynamic” when

iterated through the function z=z^2+c. Each of these points can also be used to generate a

unique Julia Set curve. The M-Set can be thought of as the mother ship of all the Julia

Sets. Each Julia curve is a universe in its own right, especially the ones generated using

points from the INSIDE of the M-Set. These Julia Sets are referred to as connected sets

and have a similar bounded boundless boundary to that of the Mandelbrot Set.

Figure 22. Connected Julia Set

In the context of my analogy, I could argue that each point in the complex plane

generates a unique “universe”, each with its own emergent properties (or laws of

physics). The Mandelbrot Set then can be thought of as a a meta-universe or multiverse

containing all the universes each with its own set of unique properties.

Chaotic Inflation

Physicist Andrei Linde, and others, have found fractals helpful in modeling the

behavior of the early universe. Their work suggests that the fractal nature of space may

actually be the cause of gravity. More recently; the work of Alain Connes has shown that

non-commutative spaces evolve 'fractality' naturally. Andrei Linde's work titled “The

Self-Reproducing Inflationary Universe”, describes a self-replicating fractal universe

that sprouts other inflationary universes each with its own very specific law of physics.

In Linde's theory, quantum fluctuations correspond to regions of rapid inflation,

creating “bubble universes”, making the structure of space fractal at even the largest of

scales, certainly larger than the observable universe. Here‟s what Linde has to say about

his theory:

“When I invented chaotic inflation theory, I found that the only thing you needed

to get a universe like ours started is a hundred-thousandth of a gram of matter.

That‟s enough to create a small chunk of vacuum that blows up into the billions

and billions of galaxies we see around us. It looks like cheating, but that‟s how

inflation theory works – all the matter in the universe gets created from the

negative energy of the gravitational field.”

His description of “creating a universe from a gram of matter” reminded me of

simulation I described in the section “generating a universe from scratch “ where I

generated a whole “universe” of stars and galaxies from only one point in the complex

plane (space-time fluctuation) . In Linde's theory, quantum fluctuations generate gravity

(contraction) which in turn causes inflation (expansion) “which blows up into the billions

and billions of galaxies we see”.

Figure 23. Cosmic Inflationary

http://www.wired.com/wired/archive/3.07/rucker_pr.html

Expanding Universe

In my model, the inside points of the M-Set represent contraction and the outside

points represent expansion. The inside points generate a collapsing dynamic and the

outside points generate an expanding dynamic. Another way of looking at is is the

outside points are attracted to the infinitely large (fly off to infinity or expand) and the

inside points are attracted to the infinitely small (collapse to singularity or contract). Put

simply, the emergent properties of the universe exist or 'come into being; at the interface

between the infinitely large (or expanding universe) and the infinitely small (or

contracting universe). In the case of the Mandelbrot Set fractal construct, this is exactly

where all the 'good stuff' seems to be going on.

Figure 24: M-Set Event Horizon

This suggests an intimate relationship between the expanding part of the universe and

the contracting part. I think the idea of feedback loops is the key to understanding why

this might be so. First, we let the expanding part represent a positive feedback loop; and

we let the contracting part represent a negative feedback loop. It is well known that the

integration of both positive and negative feedback loops generates homoeostasis. The

universe is an example of a bistable regulator using both positive and negative feedback

loops to stabilize the system. In other words, a universe that is both expanding and

contracting, will generate universal homeostasis.

Life

In the paper, “Toward the Definition of Life”, Peter MacKlem defines life as

follows:

“Life is a self-contained, self-regulating, self-organizing, self-reproducing, interconnected, open thermodynamic network of component parts which performs work, existing in a complex regime which combines stability and adaptability in the phase transition between order and chaos...”

If you replace the word “life” with the word “fractal” and put it in the context of

my model, then you can see how life might be and emergent property of some non-linear

dynamical system that generates fractal patterns. In other words, life is a fractal. We

know that trees are fractals and most other plants. Animals are made of fractals in that the

vascular systems are fractal and our lung structures are fractal; the brain is fractal as well.

The dynamics of our heart also has a fractal nature. In fact, health of the heart can be

directly determined by measuring the fractal dimension of the phase space diagram

generated by analyzing the temporal differences in our heart rate.

Utilizing fractal geometry and chaos theory in the medical field is fairly new but

things are starting to happen in that area. Here are a few headlines:

Physicists use fractals to help Parkinson's sufferers.

http://www.innovations-report.com/html/reports/medicine_health/report-25345.html

Fractal Analysis of Nuclear Medicine Images for the Diagnosis of Pulmonary

Emphysema.

http://www.ajronline.org/cgi/reprint/174/4/1055.pdf

Fractal Mechanisms in the Electrophysiology lf the Heart (IEEE)

Complex-Dynamical Extension of the Fractal Paradigm and Its Applications in the Life

Sciences.

http://cogprints.org/4140/

“Life arises from self-contained, self-organizing, self-replicating interconnected

dynamical systems that generate complex fractal patterns or dynamics.”

Entropy

In my model, I conclude that the event horizon of a black hole is a fractal. In in other

words, the event horizon of a black hole is an infinite boundless bounded boundary. As I

will demonstrate, the Bekenstein-Hawking entropy equation can easily be interpreted in

such a manner that allows the event horizon of a black hole to be a fractal surface.

.

This equation basically states that the entropy of a black hole is directly proportional to

the surface area of the event horizon of a black hole. In order for entropy to increase (as it

always does in a closed system according to the second law of thermodynamics) the area

of the event horizon of a black hole must always increase. There are two ways for the

surface area of a black hole to increase:

1) Increase the radius of the black hole.

2) Reduce the “measuring stick” of a (fractal) event horizon.

In “How Long is the coast line of Britain? Statistical Self-Similarity and Fractal

Dimension”, Benoit Mandelbrot examines the surprising property that the measured

length of a stretch of coastline depends on the scale of measurement. Empirical evidence

suggests that the smaller the increment of measurement, the longer the measured length

becomes. In other words, reducing the “measuring stick” increases the measurement.

What do “measuring sticks” have to do with black holes?

Einstein (in his theory of relativity) showed that the length of a measuring stick

decreases, as one gets closer to the event horizon of a black hole (or as one accelerates

toward the speed of light). In fractal theory, reducing the measuring stick increases the

overall length/area/surface measurement. Now we can interpret the Bekenstein-Hawking

equation in the following manner:

.

From the above equation we see that, as entropy (of a black hole) increases, the surface

area of the event horizon must also increase. Using Mandelbrot's fractal coastline theory,

the measured length/area/volume can now be increased by reducing the measuring stick

of the system.

Evolution

Einstein‟s theory of relativity clearly states that measuring sticks decrease, as one

gets closer to the event horizon of a black hole. What does it mean to decrease a

measuring stick? In order to decrease a measuring stick, one must increase the resolution

of the system. In order to increase the resolution of the system, one must increase the

digits of precision of the system. This is equivalent to adding another bit to a computer

processor or CPU. By adding more bits to the computer, not only are you increasing the

resolution, you are increasing the complexity or “complexity potential” of the system.

What does all this mean?

First, we agree that black hole entropy is directly proportional to the surface area

of a black hole. Then, we agree that universal entropy is always increasing over time. If

we then allow the surface area of the black hole (event horizon) to increase by reducing

the measuring stick, which can only be done by increasing the resolution of the system,

(which is equivalent to adding another bit to the computer processor) then what we

should observe is a universe that increases in “complexity potential” over time.

Therefore, an increase in BH entropy will brings about an increase in BH

complexity. This goes contrary to the standard interpretation of entropy which is

traditionally thought of as as a measure of disorder of a system. This may be true for

closed systems, or systems far away from black holes. Like gravity, entropy has “special”

meaning in the vicinity of a black hole. This new interpretation of entropy could explain

why life and the universe seems to move toward an increase in complexity (evolution)

rather than decrease in complexity (devolution).

Figure 25. M-Set Entropy Gradients

In his paper, “On the Origin of Gravity and the Laws of Newton”, Verlinde presents a

holographic scenario for the emergence of space and argues that gravity is an emergent

property of entropy gradients. Although out of the scope of this discourse, I just wanted

to point out that there is work going on out there describing holographic principles that

closely resemble the fractal principles that I present in this paper.

Buddhabrot

The term “Buddhabrot” was coined by myself (Wikipedia) to refer to the algorithm

and rendering technique of the M-Set as seen below. This image is generated using the

same technique as in the “Big Bang” image, only here, we use the points from the

OUTSIDE of the M-Set instead of the INSIDE points.

Figure 26. M-Set Buddhabrot

Buddhabrot is one of the most unexpected “emergent properties” of the Mandelbrot

Set iterative function system. This image represents a “probability distribution map” of

(the trajectories of) the points from the OUTSIDE of the Mandelbrot Set. The Julia Sets

that are associated with these points are referred to as the “disconnected sets”. This image

represents the sum of all the disconnected sets. It a “Unified Field” and may be analogous

to consciousness and/or represent the “Buddha Nature” of our universe. For what ever

reason, this image is an emergent property of the Mandelbrot Set and is one of the most

complex fractal patterns ever generated..

Conclusion

Using the complex plane as my space-time manifold, and the Mandelbrot Set

iterated function system my space-time generator, I am able to produce fractal patterns

that are not unlike the fractal patterns that we observe in our universe. I am able to

replicate the morphologies of galaxies and galaxy clusters. I am also able to reproduce

the non-linear orbits around black hole and the very peculiar trajectories generated by

particle accelerators. The M-Set as described in the paper does seem to have a lot of the

properties of black holes including singularities and event horizons.

It is my contention that the Mandelbrot Set represents a black hole and/or the set

of all black holes. If it does turn out to be a black hole or black hole like, then we can use

it to study the INSIDE of a black hole for the very first time; something that the current

model of black holes strictly forbids.

Biography

Lori Gardi is a computer scientist working in the field of medical biophysics. She

develops software for mechatronic and robot devices that are use in biopsy and surgical

procedures. She is co inventor of several of these devices and co author on dozens of

papers related to this field.

Lori Gardi graduated from University of Western Ontario (UWO) in 1985 with a

bachelor‟s degree in computer science. She worked for the Astronomy Department at

UWO from 1983 – 1993, first as a summer student, then full time employee. There, she

used her skills in software and electronics to develop instrumentation for research

purposes.

In 1993, Lori began working for the Robarts Research Institute where she began

her career developing 3D ultrasound guided medical procedures. She develops software

for mechatronic and robotic aided biopsy and surgical procedures. She is co inventor of

several medical devices and is co author on many papers in this field. She is currently

involved in commercialization projects taking these medical devices to market.

Lori's interest in fractals and chaos theory started in the mid 1980's when the

beautiful Mandelbrot images were made popular due to advancements in computer

processing and graphics. Her love for the Mandelbrot Set imagery continues to this day.

She has written hundreds of programs and simulations relating this particular iterated

function system. There is no other fractal algorithm that is more interesting than the

Mandelbrot Set according to Lori. She loves fractals so much she refers to herself as

FractalWoman on the internet and in the blogospheres.

When Lori is not developing software for medical devices, spend time studying

cosmology and developing her own Fractal Cosmology based on the Mandelbrot Set

fractal algorithm.

APPENDIX

The Mandelbrot

The Mandelbrot Set is generated by iterating the function:

Z = Z2 + C where Z and C are complex numbers.

Since Z and C are both complex numbers, we can replace Z and C with points from the

2D complex plane, where one component represents a real number and the other

component represents an imaginary number:

And, since i2 = -1, you get:

Which factor out to these two equations:

These two equations are used to generate all the fractal images in this paper.

Z = Z2 + C

(R1 , Ii) = (R1 + Ii)2 + (Rc + Ici)

= R2 + Ii

2 + 2*R*Ii + Rc + Ici

= R2 - I

2 + 2*R*Ii + Rc + Ici

R new = R2 – I

2 + Rc

I new = 2*R*Ii + Iic

The Mandelbrot Set Rendering

To render the traditional Mandelbrot Set image (below), take a set of points from

the complex plane (between –2.0 and +2.0) and iterate each point through the function Z

= Z2 + C. For each point in the complex plane, initialize C to the complex point and Z

to zero (or C since the first iteration will essentially make Z=C) . You then iterate that

point until either the real component or the imaginary component is greater than a certain

value such that subsequent iterations will cause it to fly off to infinity. This is referred to

as escape velocity. If and when the point reaches escape velocity, we stop iterating and

count the number of iterations it took to escape. We then map that point to a particular

color from a color lookup table.

For each point in the complex plane:

Initialize: Z=current point, C=current point

Iterate: Z = Z2 + C

Stop: when Z2 > escape radius?

Count: the number of iterations it took to reach escape velocity.

Colour: the point based on number of iterations.

There are some points in the Mandelbrot Set that never reach escape velocity no

matter how many times we iterate them. These are the points that we traditionally paint

black and these points are said to be on the INSIDE of the Mandelbrot Set. The points on

the OUTSIDE can always reach escape velocity given a finite number of iteration. The

points INSIDE the M-Set never reach escape velocity. It turns out that the reason the

inside points do not ever reach escape velocity is because they are collapsing into the M-

Set rather than flying away from it. Note: there are also points in the M-Set that neither

fly away or collapse. These are referred to the 'zeros' of the M-Set

The Julia Set Rendering

The Julia Set is named after the French mathematician Gaston Julia who

investigated its properties in the early 1900‟s. Basically, every point on the Mandelbrot

Set generates a Julia Set curve. To generate a M-Set image, you vary both R and C for

each point on the complex plane. In the case of the Julia Set, you vary R but you keep C

fixed for each point on the complex plane. So the above algorithm changes slightly:

Select a fixed point C in the complex plane.

For each point in the complex plane:

Start: Z=current point, C=fixed point

Iterate: Z = Z2 + C

Stop: when Z2 > escape radius?

Count: the number of iterations it took to reach escape velocity.

Colour: the point based on number of iterations.

One of the most beautiful results in complex dynamics was proved independently

by both Gaston Julia and Pierre Fatou. It states that either a Julia Set is connected, or it

consists of infinitely many pieces, each of which is a single point. It‟s interesting to note

that the points from the INSIDE of the Mandelbrot Set generates CONNECTED Julia Set

curves and the points from the OUTSIDE of the M-Set generates DISCONNECTED

Julia Set curves.

The disconnected sets are often referred to as “dust” or more appropriately, a

cantor dust. The connected sets generated by the INSIDE points of the M-Set consist of

points that seem to act as one piece or unit; that is, they seem to be bound by some

unknown force holding it together instead of tearing it apart. Keep this in mind as we

discuss the true nature of the Mandelbrot Set and the meaning behind its magic.

Lori Gardi

1459 Oakdale St.

London, Ontario, Canada

N5X 1J5

(519) 432-9159

www.ButterflyEffect.ca

lgardi22@hotmail.com lgardi22@gmail.com

Words are flying out like endless rain into a paper cup They slither while they pass

They slip away across the universe

Pools of sorrow waves of joy are drifting thorough my open mind

Possessing and caressing me

Jai guru deva OM Nothing's gonna change my world...

© The Beatles

TO DO

References

Index