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Vancouver Canada, BC 2017 Proceedings of the CNPS 1

The Mandelbrot Set as a Quasi-Black Hole

Lori Gardi

lori.anne.gardi@gmail.com

Black holes, ﬁrst identiﬁed by general relativity, are of the most mysterious objects in the universe. Scientiﬁc

consensus is that black holes do in fact exist in nature. Not only that, but they are considered as an important

feature of our universe. The equally mysterious concept called fractal geometry, popularized by Benoit

Mandelbrot, is also considered a very important feature of nature. Since fractals appear just about everywhere,

it seemed reasonable to wonder if the geometry of the Mandelbrot set (M-Set) might also appear somewhere

in nature. The main property that distinguishes fractal geometry from other geometries is the property of self-

similarity. That said, it is well known that black holes come in many sizes. Stellar-mass black holes are typically

in the range of 10 to 100 solar masses, while the super-massive black holes can be millions or billions of solar

masses. The extreme scalability of black holes was the ﬁrst clue that black holes may in fact have the property of

self-similarity. This ultimately led to the quasi-black hole analogy presented in this essay. Here, the anatomy of

the Schwarzschild black hole is used as a starting point for the analogy. All of the main features of black holes,

including the singularity, the event horizon, the photon sphere and the black hole itself, are mapped to features

of M-Set. The concepts of time, space-time curvature and black hole entropy are also addressed. The purpose of

this research is to see how far this analogy can be taken. Consensus is that both black holes and fractals exist in

nature. Could there be a mathematical fractal that describes black holes, and if so, do they also exist in nature?

Can this approach make a prediction and if so, is it testable? It turns out that M-Set as a quasi-black hole does

lead to some interesting predictions that differ from standard thinking. Given the evidence presented herein,

further investigation is suggested.

Keywords: fractal geometry, Mandelbrot set, self-similarity, black hole, singularity, event horizon, photon

sphere, iteration, time, entropy, relativity, space-time curvature, chiral symmetry, morphology

Figure 1. Mandelbrot set traditional rendering.

1. Prelude

Due to the controversial nature of this research, a few

things need to be clariﬁed. First, the author admits to the

circumstantial and highly speculative nature of the "evi-

dence" presented in this essay. This idea was developed

independently by the author over many years of inves-

tigation. It began as a "thought experiment" and a few

simple questions. What if relativity was never invented?

Figure 2. Schwarzschild black hole.

What if the concept of fractal geometry predated relativ-

ity? Would we still have the concept of the black hole?

Would we still be talking about event horizons and space-

time curvature? This "thought experiment" is in no way,

meant to replace any of the currently accepted theories of

black holes. This is a philosophically different approach

to cosmology that looks nothing like the the way it was

done before.

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2 L. Gardi: The Mandelbrot Set as a Quasi-Black Hole Vol. 10

2. Introduction

"There is always another way to say the same thing

that doesn’t look at all like the way you said it before."

Richard Feynman

The "Mandelbrot set" is one of the most recogniz-

able mathematical fractals. The formula, z:=z2+c,

gives no clues as to the vast complexity and unending

beauty hidden within this simple iterative system. At ﬁrst

glance, a connection between M-Set and black holes may

seem improbable. For example, M-Set, as depicted in

Figure 1, doesn’t look anything like Figure 2, i.e., the

Schwarzschild black hole. What does fractal geometry

have to do with black holes? Recent research suggests

that a relationship between black holes and fractal ge-

ometry does in fact exist. Using a mathematical duality

between Einstein’s relativity and ﬂuid dynamics, simula-

tions show that fractal patterns can form on the horizons

of feeding black holes [1]. This important point shows

that relativity and fractal geometry may be intimately

linked.

One of the main objections to this body of work is

that it does not reference the many successes of relativ-

ity. Aside from comparing M-Set to the Schwarzschild

black hole, this research doesn’t look anything like rel-

ativity. This is a completely different approach to cos-

mology that is founded on different principles. Whether

relativity is successful (or not) does not affect this line of

thinking. What if relativity was never invented? What if

the discovery of fractal geometry (and M-Set) predated

relativity? Would we still have intuited the existence of

black holes? Can the geometry of M-Set tell us anything

new about black holes that we didn’t know before? Can

it make any predictions and if so, are they testable? This

essay is an attempt to address all these questions. If this

"thought experiment" can give us insights into the inner

workings of nature, then is it not philosophically worthy

of further investigation?

This essay starts with the generalized methods used to

generate the M-Set related images presented in this essay

(Section 3), followed by a brief discussion (Section 4).

The anatomy of the Schwarzschild black hole, Figure 2,

is then compared to the anatomy of the M-Set as depicted

in Figure 1 (Sections 5 - 9). Next, some discussions about

dimensionality, black hole entropy and space-time curva-

ture (Sections 10-12) are presented. This is followed by

a controversial discourse on "the atom as a quasi-black

hole" (Section 13) and a brief discussion about symmetry

as it relates to the fractal geometry found within M-Set

(Section 14). Finally, in Section 15, a prediction is made

and some evidence presented that, if found to be true,

would answer the question "do quasi-black holes exist in

nature?".

3. Methods

The term M-Set refers to the complex plane as iterated

through the following function:

z:=z2+c(1)

where, zand care complex numbers. Because "z" is on

both sides of the equation, the ":=" notation is used. Us-

ing this notation, equation (1) better reads, "z transforms

into z squared plus c". Below is pseudo code for the gen-

eralized algorithm for M-Set. This algorithm applies to

all the computer generated images presented in this es-

say:

1: c = (a , bi)

2: ClearPointList()

3: z = c

4: while (!done)

5: z:=z*z+c

6: AddToPointList(z)

7: if (StoppingCriteria == true) done = true

8: AnalyzePointList()

9: GoTo 1:

Explanation: 1) Select a test point cfrom the set of

complex numbers. This corresponds to one pixel in Fig-

ure 1. 2) Clear the list. This list will be used to store the

sequence of points generated by the iteration process for

test point, c. 3) Initialize zto the test point c. 4) Begin the

iteration loop. 5) Iterate the function (1). This generates

a new complex point, z. 6) Add the new zto the list of

complex points. 7) If the iteration process reaches some

stopping criteria, then end the iteration loop. 8) Analyze

the data from the list. Update the image accordingly. 9)

If there are more test points, repeat steps 1 through 9.

It is found that each test point, c, from the complex

plane generates a different trajectory and each trajectory

has a different behaviour and/or structure. Put simply,

no two points from the complex plane make the same

picture. Note that in this model, we are only concerned

with the complex points inside the 2.0 radius circle of

the complex plane (outer circle in Figure 1). All points

outside this boundary are outside of the scope of the M-

Set model.

The trajectories are divided into three regions or

domains. The ﬁrst is referred to as the domain of con-

vergence. This corresponds to the central black region of

Figure 1. The second domain is referred to as the domain

of divergence and corresponds to the outer grey-scale

region of Figure 1. The third region is the domain of

uncertainty. These are the points whose trajectories

do not diverge nor converge, even after the maximum

number of iterations. The uncertain region is depicted in

Figure 1 as the bright halo surrounding the black region.

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Vancouver Canada, BC 2017 Proceedings of the CNPS 3

Below is a description of the three tests used to stop

the iteration loop.

Divergence test: If the trajectory extends outside the

2.0 radius circle of the complex plane, then the iteration

loop is stopped and the initial point is assigned to the

diverging domain (grey-scale region of Figure 1).

Convergence test: If the trajectory contracts beyond the

digits of precision of the computer, then the iteration

loop is stopped and the initial point is assigned to the

converging domain (black region of Figure 1).

Undecided test: If the maximum number of iterations

is reached before convergence or divergence, then the

iteration loop is stopped and the initial point is assigned

to the domain of uncertainty (bright halo surrounding

black region of Figure 1). Note that, given more itera-

tions and more digits of precision, most of these points

will eventually be resolved. However, since computers

have limits, there will always be some uncertain points

on this list.

4. Methods Discussion

The points from the diverging domain of M-Set are

well studied since these points have a very clear stopping

criteria. These are the points whose trajectories escape

the 2.0 radius circle after a ﬁnite number of iterations.

The non-escaping points (the black region) are a bit

more illusive. It is commonly thought that the trajecto-

ries from the black region of M-Set fall into periodic

orbits or cycles. This, however, is only partly true. It

was found through experimentation that a majority of

these cycles were merely an artifact of the limit to the

digits of precision of the computer. When more digits

of precision are added to the computer program, then

the collapse can continue past the previous limit. In

theory, given an inﬁnite number of digits of precision

(and an inﬁnite number of iterations), the collapse could

continue indeﬁnitely.

In short, aside from some special complex points such

as (0,0) and (-1,0), there are no stable periodic orbits in

M-Set. This realization is the key to the quasi-black hole

analogy presented in this essay.

5. Anatomy of the Schwarzschild Black hole

The simplest black hole described by general relativity

is the Schwarzschild black hole (SBH). The anatomy

of an SBH is depicted in Figure 2. It is characterized

by a region of space referred to as a black hole (black

region in Figure 2). At the center of the black hole is

an inﬁnitely dense, inﬁnitely small volume of space-time

called a singularity (white dot at the center of the black

region in Figure 2).

The lesser known region just outside the black hole

is referred to as the photon sphere (outer grey region in

Figure 2). The photon sphere is a region of space just

outside the black hole where photons are forced to travel

in complex orbits due to the extreme curvature of space

within this region [2]. According to relativity, there are

no stable orbits within the photon sphere of an SBH.

Finally, the boundary that exactly separates the black

hole from the photon sphere is referred to as the event

horizon (white circle surrounding the black region in

Figure 2). The distance from the central singularity to the

event horizon is known as the Schwarzschild radius. Ac-

cording to relativity, nothing, including light, can escape

the event horizon of a black hole. Put simply, "things"

can fall into a relativistic black hole, but "things" can

never come out.

6. M-Set as a Quasi-Black Hole

Figure 1 depicts three regions of M-Set that are analo-

gous to the three regions of the SBH as described in the

previous section. The black region of M-Set is analogous

to the black hole of the SBH. These are the points whose

trajectories collapse past the digits of precision of the

computer after a ﬁnite number of iterations. The outer

grey-scale region of Figure 1 is analogous to the photon

sphere of the SBH. These are the points whose trajecto-

ries travel in (complex) orbits until they escape the 2.0

radius circle of the complex plane after a ﬁnite number

of iterations. Finally, the region exactly separating the

black region from the grey-scale region in Figure 1 is

analogous to the event horizon of the SBH (the white

fuzzy boundary just outside the black region). The white

dot within the black region of Figure 1 is the (0,0) point

of the complex plane. This is analogous to the zero point

or singularity of the SBH model. Each of these concepts

will be discussed in greater detain in the next sections.

7. Quasi-Singularities

The trajectories generated by the points inside the

black hole region of M-Set (Figure 1) are referred to as

quasi-singularities. These are the points whose trajecto-

ries collapse or converge toward inﬁnitely small regions

within the complex plane. Like the singularities of rela-

tivity, quasi-singularities never escape the boundary con-

dition of the quasi-black hole. Unlike the singularities

of relativity, quasi-singularities can collapse/converge to-

ward more than one region in the complex plane. For ex-

ample, the middle image in Figure 3 is a quasi-singularity

that is converging toward 9 regions within the complex

plane.

It was found, through experimentation, that the points

closer to the event horizon of M-Set take more iterations

to collapse past the digits of precision of the computer

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4 L. Gardi: The Mandelbrot Set as a Quasi-Black Hole Vol. 10

than the points closer to the origin (0,0). It was also

found that the points nearest the event horizon generate

quasi-singularities with extremely complex behaviours

as depicted in Figure 3. As stated earlier, each point

from the complex plane generates a different trajectory

and each trajectory generates a different picture. The

quasi-singularities depicted in Figure 3 are only a small

sampling of the inﬁnite number of unique and interesting

"singularities" generated by M-Set.

8. Quasi-Photon Sphere

The grey-scale region just outside of the quasi-black

hole of Figure 1 is referred to as a quasi-photon sphere.

These are the points whose trajectories can and do reach

the escape condition of M-Set after a ﬁnite number of

iterations. These trajectories appear to travel in ever ex-

panding "orbits" up until the point where they reach the

escape condition, i.e., the 2.0 radius circle of the complex

plane. The behaviours of these trajectories are quite inter-

esting. Like the quasi-singularities from the previous sec-

tion, it was found that the points close to the edge of the

event horizon take the more iterations to reach the escape

condition. The dynamics of these trajectories also appear

to be more chaotic and exhibit complex morphologies

when plotted directly. A hand full of these trajectories

are depicted in Figure 4. The bottom row shows the com-

puter generated trajectories and above each of these is a

morphologically similar (self-similar) cosmological ob-

ject. Figure 5 is an example of a complex trajectory that

looks a lot like a galaxy cluster. On the left is the com-

puter generated trajectory and on the right is the Virgo

Cluster. Keep in mind that the ﬁgure on the left was gen-

erated using only one seed point from the complex plane,

as with the all the other trajectories.

Also, it should be noted that all the cosmological

objects depicted in Figures 4 and 5, which include plane-

tary nebula, galaxies and galaxy clusters, are all thought

to be associated with black holes. The most surprising

quasi-photon trajectory in Figure 10 is the one that looks

similar to Einstein’s cross. In standard cosmology, the

appearance Einstein’s cross is thought to be caused by

gravitational lensing, but what if gravitational lensing is

not the only way to explain this object? If the discovery

fractal geometry and M-Set predated relativity (and in-

directly the concept of gravitational lensing) then M-Set

as a quasi-black hole would have been able to predict

the appearance of objects such as Einstein’s cross along

with the other objects depicted in Figures 4 and 5.

9. Quasi-Event Horizon

The event horizon from the standard model is de-

scribed as a theoretical boundary surrounding a black

hole beyond which nothing, including light, can escape.

The photon sphere is rarely mentioned in the black hole

discourse. Technically, the event horizon exactly sepa-

rates the black hole from the photon sphere as depicted

in Figure 2. The photon sphere will be described in more

detail in the next section.

In a similar manner, the boundary that exactly sep-

arates the quasi-black hole of M-Set from the quasi-

photon sphere is referred to as a quasi-event horizon.

This is an event horizon in the truest sense as it exactly

separates the escaping "events" from the non-escaping

"events". But what is an event?

In a previous paper by the author [3], iteration is

considered the mathematical analogy for time. Iteration

generates change and change gives us the sensation of

time. Here, one iteration generates one unit of change

and one unit of change is one event. Thus, the complex

plane as iterated through the function z:=z2+ccan

be considered as an event generator. When the sequence

of events is plotted, it produces images like the ones in

Figures 3, 4 and 5.

The M-Set event horizon (the fuzzy boundary in Fig-

ure 1), separates the collapsing events from the expand-

ing events (the converging domain from the diverging

domain). It was found, through experimentation that tra-

jectories, whose seed points are close to the quasi-event

horizon, require more iterations to reach the stopping cri-

teria than the seed points farther away. This is true on

both sides of the horizon. In other words, quasi-black

holes exhibit asymptotic behaviour on both sides of the

event horizon. This is a huge departure from standard

black hole theory.

The implication here is that it is just as hard to get into

a "black hole" as it is to get out of one. If this were true,

then there should be some evidence of this in nature.

Recent Chandra images of the black hole at the center of

our galaxy, Sagittarius A*, indicate that only a fraction

of a percent of the gas (if any) actually fall into the black

hole. It appears that most of the material that approaches

the black hole gets ejected before long before it reaches

the event horizon [4] or "point of no return". Asymptotic

behaviour on both sides of the event horizon may help

explain this observation. This behaviour is further dis-

cussed in Section 13.

10. On Time and Dimensionality

The most common concern about M-Set as a quasi-

black hole has to do with dimensionality. Before we

can continue, the concept of dimensionality needs to be

addressed. Since M-Set (as depicted in Figure 1) is a

2-dimensional static structure, how can it possibly tell

us anything important about the 3-dimensional dynamic

universe we observe? This is a good and important ques-

tion.

In the standard model of cosmology, space and time

are combined together into a 4-dimensional "space-time"

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Vancouver Canada, BC 2017 Proceedings of the CNPS 5

Figure 3. This ﬁgure depicts three distinct singularities from the quasi-black hole of M-Set. Each singular-

ity, including the cluster, was generated from a single seed point (real, imaginary) from the complex plane.

Left: (0.11421610355396709, 0.59570907391827810); Middle: (0.33751130482196073, 0.41014244936187033); Right:

(0.25161670227079957, 0.49833246356572231).

Figure 4. This ﬁgure depicts a handful of the trajectories from the quasi-photon sphere of M-Set together with some familiar

cosmological objects (credit NASA). The top row from left to right: Cartwheel Galaxy, Stingray Nebula, Catâ ˘

A´

Zs Eye Nebula,

NGC 5315, Einsteinâ ˘

A´

Zs cross. Below each of these is a computer generated (M-Set) trajectory with a similar morphology. Each

trajectory depicted in this ﬁgure was generated using a single seed point from the complex plane as reported in Appendix A.

manifold where time is treated as another spatial dimen-

sion. In order to understand M-Set as a quasi-black hole,

we need to decouple the concept of time from the spatial

manifold. In the cosmology outlined in this essay, time is

an emergent property of change brought about by an iter-

ative feedback process. In this manner, time is analogous

to iteration, and vice verse. The other important point

to make here is that the change associated with iterative

time cannot be undone. As one cannot un-break a glass

or unborn a baby, one cannot undo iteration. This is yet

another way of looking at entropy. Quite simply, entropy

means that change cannot be undone. "It is the unending,

unknowable uniqueness of each moment that gives us the

sensation of time and the arrow of time"[3]. Without un-

ending, unknowable, irreversible change, there would we

no sensation of time and no arrow of time. That said, the

only math that can mimic this kind of unending, unknow-

able, unrepeatable, entropic change over time is iteration,

especially as it relates to chaos theory and fractal geom-

etry. This, of course, includes M-Set.

With time out of the way, we need to rectify the

spatial dimension problem. M-Set clearly resides in a

2D complex plane. How can a 3D dynamical system be

explained by a 2-dimensional (mathematical) structure?

It is well known that 2D complex numbers (r,i) can

be extended to 4D complex numbers using quaternions

(r,i,j,k) [5]. Unfortunately, 4D complex structures are

difﬁcult to depict in a 2-dimensional format such as an

5

6 L. Gardi: The Mandelbrot Set as a Quasi-Black Hole Vol. 10

Figure 5. This ﬁgure depicts an M-Set cluster from the photon sphere of M-Set on the left and the Virgo Cluster on the right. Credit

NASA. This cluster was generated using a single seed point from the complex plane: (real, imaginary) = (0.05103771361715907 ,

0.64098319549560490)

article or paper. That said, it is found that projecting

a 4D M-Set onto a 2D plane gives us back M-Set. In

other words, the "physics" of the 4D M-Set set is no

different from the "physics" of the 2D M-Set. Thus, the

2D complex plane is used throughout this essay with the

assumption that it can be extended to the higher spatial

dimensions using quaternions, not unlike what is done in

standard physics.

Logically, if space is 3-dimensional, as we perceive

it to be, then the manifold housing 3D-space must be

4-dimensional. As it takes the 2-dimensional complex

numbers to represent 1-dimensional angles and curves, it

takes 4-dimensional complex numbers (quaternions) to

represent 3-dimensional angles and curves. The complex

curve depicted in Figure 1 (i.e. the quasi-event horizon)

is technically a 1D fractal curve housed within a 2D

complex space. Analogously, the curvature of space

(previously curved space-time) could be modelled as a

3D fractal curve housed within a 4D complex space.

This new concept of "curved" space is discussed further

in Section 12.

11. On Black Hole Entropy and Evolution

The purpose of this research is to see how far we can

take the black hole analogy. What can M-Set tell us about

black hole entropy? Bekenstein was the ﬁrst to make

a connection between the area of a black hole horizon

and entropy[6]. He concluded that the area of the event

horizon of a black hole must continuously increase over

time as expressed by the following equation:

S=A

4(2)

Using Euclidean geometry, the only way that the sur-

face area of a spherical object (such as the Schwarzschild

black hole) can continuously increase over time is by in-

creasing the radius of the object. Thus, to satisfy black

hole entropy, black holes must get bigger over time.

Black holes cannot shrink. Hawking later proposed that

black holes might evaporate, but let’s assume that the

area of the horizon of a black hole must continuously in-

crease over time as originally proposed. How does fractal

geometry rectify this problem?

In 1967, B. Mandelbrot wrote a paper called "How

long is the coastline of Britain?’[7] where he proposed

that the measurement of a rough geometric shape such

as a coastline would change depending on the size of the

measuring stick used for the measurement. In short, the

smaller the measuring stick, the longer the coastline. This

turns out to be true of all fractal structures. Einstein also

had something to say about measuring sticks by arguing

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Vancouver Canada, BC 2017 Proceedings of the CNPS 7

that measuring sticks "shrink" as one get closer to the

event horizon of a black hole. This is commonly referred

to as length contraction.

What does this mean for the quasi-black hole? The

event horizon of M-Set (the bright halo in Figure 1)

is a rough geometric shape with the property of self-

similarity, i.e., it is a fractal. Fractal geometry gives

us another mechanism for continuous increase in the

event horizon area over time. If the "measuring sticks"

of the universe are allowed to "shrink" over time, then

the surface area of quasi-event horizons (if they exist)

would also increase over time. In this manner, black hole

entropy would be valid for all black holes with fractal

event horizons (i.e., quasi-black holes).

This of course is a huge departure from standard

thinking about how the universe evolves. If the mea-

suring sticks (i.e., pixels) are shrinking over time, then

the "pixels" of the early universe must have been much

"bigger". One way this could manifest is if the atoms

from the past were "scaled bigger" than the atoms of

today. This leads to the idea of "scale relativity" where

the earlier universe, although scaled differently, could

still experience the same laws of physics. Atoms of

the earlier universe would emit light of a much longer

wavelength, thus, this line of thinking gives us an alter-

nate mechanism for observed cosmological red-shift. In

this case, accelerated expansion is an optical illusion in

that universal expansion only appears to be accelerating

because the universal measuring stick is shrinking. This

idea is difﬁcult to visualize using the current paradigm

but easy to visualize within the fractal paradigm. A

recent paper by independent researcher Blair Mac-

Donald (Fractal Geometry a Possible Explanation to

the Accelerating Expansion of the Universe and Other

Standard ΛCDM Model Anomalies) offers great support

to this alternate line of thinking [8]. An ever shrinking

measuring stick would allow ever increasing frequencies

(of light) to appear over time. Ever increasing frequen-

cies would guarantee the irreversibility of time (i.e., the

arrow of time). This line of thinking may also explain

why evolution always increases in complexity over time

since, like the computer generated fractal, an increase

in resolution would allow more details and thus, more

complexity to be represented.

12. On Space-Time Curvature

According to Einstein’s general relativity, the phe-

nomenon of gravity is caused by the curvature of

space-time. But what if relativity was never invented?

What if fractal geometry proceeded general relativity?

Would we still have the concept of space-time curvature?

Let’s begin by remembering that we previously decou-

pled the concept of time from the concept of space. In a

fractal universe, time is merely an emergent property of

change brought about by an iterative feedback process.

Time is not a spatial dimension in that it does not include

any coordinates or locations that we can physically

return to. With time out of the picture, we need only

be concerned with the curvature of space. What does it

mean to curve space?

Figure 6. This ﬁgure is a zoomed in regions of the M-Set

quasi-event horizon. An edge ﬁlter was applied to this image

to highlight the topological gradient. Notice that the regions

of complex curvature are also the regions with the steepest

gradients.

If we simplify all forces, including gravity, to the con-

cept of the gradient, then it is easy to visualize the curva-

ture of a manifold of space. Think of a topographic map.

When the lines are closer together, it means the slope of

the gradient (and indirectly, the force) is greater. Tech-

nically, M-Set can be thought of as a gradient generator.

Figure 6 depicts a zoomed in region of M-Set. An edge

ﬁlter was used to make it look more like a topographic

map.

Notice how the lines of the gradient get closer to-

gether as you look closer to the black region (the M-Set

quasi-black hole). When one zooms into the M-Set frac-

tal, what is happening is the slope of the gradient is ever

increasing. In order to accommodate this, the computer

must continuously decrease the measuring stick of the

fractal generator. As the measuring stick (pixel dimen-

sion) decreases, the slope of the gradient increases.

Figure 7 depicts a sub-region of the complex plane

along side a sub-region of space known as the Grand

Spiral Galaxy, NGC 1232. Notice the morphological

similarity between the way that M-Set "curves space"

and the way nature "curves space". Hence, fractal ge-

ometry, in particular M-Set, gives us an analogy for the

curvature of space that is similar to (self-similar to) the

curvature of "space-time" that we observe in nature.

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8 L. Gardi: The Mandelbrot Set as a Quasi-Black Hole Vol. 10

Figure 7. On the left is a region of M-Set depicting a fractal gradient analogous to "space-time curvature" of relativity. On the

right is a region space showing the "space-time curvature" surrounding the Grand Spiral Galaxy, NGC 1232, (Credit: FORS1,

8.2-meter VLT Antu, ESO). In both images, the bright regions correspond to regions of high curvature and the dark region

correspond to regions of low curvature. The morphological similarities between these images is of particular interest. (Real

Center: 0.238291710520393 Imaginary Center: 0.38038103197243328 Real Extent: 0.00020176911157024794 Imaginary Extent:

0.00015132683367768596)

13. The Atom as a Quasi-Black Hole

A black hole with asymptotic behaviour on both sides

of the event horizon leads to an interesting line of think-

ing. Since quasi-black holes have the property of self-

similarity, and they exhibit asymptotic behaviour on both

sides of the event horizon, then there is nothing pre-

venting the atom from being a quasi-black hole. Self-

similarity implies scale invariance and so why should

the scale of the atom be an exception? In the atom as a

quasi-black hole, the nucleus plays the role of the "black

hole" and the electrons shells play the role of the "pho-

ton sphere". As quarks cannot and do not escape the nu-

cleus of the atom, electrons cannot and do not fall into

the nucleus of the atom. In other words, the atom exhibits

asymptotic behaviour on both sides of the event horizon.

But what is the event horizon of the atom? Here, the weak

force plays the role of the event horizon.

Event horizons of the standard model are generally

associated with strong gravity. As strong gravity can

bring things together, it can also tear things apart. In

a similar manner, the weak force is responsible for

bringing things together (fusion) and tearing things apart

(ﬁssion). In other words, the (poorly named) weak force

at the atomic scale is self-similar to (the poorly named)

strong gravity at the cosmic scale. The domain (range) of

the weak force is quite small ( 10−18 meters) compared

to the size of the atom ( 10−10 meters). The domain of

the M-Set event horizon is also quite small compared

to the escaping and non-escaping domains. Again, if

quasi-black holes exist in nature, then there is nothing

preventing the atom from being a "black hole". If these

are the black holes that nature makes, then black hole

theory will need to be revised to incorporate the prop-

erty of self-similarity and account for the asymptotic

behaviour on both sides of the event horizon.

14. On Chiral Symmetry

Another interesting feature of M-Set has to do with the

various kinds of symmetries found within the standard

M-Set rendering. First, you will notice the obvious left-

right symmetry of the quasi-black hole in Figure 1. This

can be considered an axial symmetry since a rotation or

ﬂip about the vertical axis leaves the image unchanged.

Another kind of symmetry can be found when you look

closely at some of the smaller scale quasi-black holes

buried deep within the event horizon of M-Set as seen in

Figure 8. Notice that the black hole still has the left-right

symmetry but the region far away from the black hole

exhibits a different kind of symmetry known as chiral

symmetry. In general, a chiral object is an object that is

not superimposable on its mirror image. Chiral symmetry

is commonly seen in the spiral arms of galaxies. In

general, it is found that late-type spiral galaxies have a

stronger chiral signal than the early-types as depicted in

the Hubble tuning fork diagram or sequence[9].

Chrial symmetry breaking is an important phe-

nomenon in theoretical physics; from quantum chromo-

dynamics and the study of mesons [10] [11] to the study

of nanoparticles in semiconductors [12]. According to

standard cosmology, quantum ﬂuctuations of the early

universe were greatly expanded during the inﬂationary

epoch. Thus, macroscopic chirality is thought to be

caused by some primordial process shortly after the

big bang. In the fractal cosmology (presented herein)

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Vancouver Canada, BC 2017 Proceedings of the CNPS 9

Figure 8. This ﬁgure depicts a quasi-black hole found deep

within the quasi-event horizon of M-Set. Here we see that the

region close to the black hole exhibits a left-right symmetry

while the region far away from the black hole exhibits chiral

symmetry. This is one of the signatures of a quasi-black hole.

any similarities between the quantum scale and the

cosmic scale are thought to be due to the property of

self-similarity.

What does this have to do with black holes? Spiral

galaxies (with supermassive black holes at their cen-

ters) exhibit a strong chiral signal in their spiral arm

structures. Lenticular galaxies also show the signature

of spiral arms with a weak chiral symmetry, as recently

detected by the IRAC instrument on the Spitzer Space

Telescope [13]. It is also found that galaxies in galaxy

clusters exhibit a strong chiral symmetry far away from

the central black hole and weak chiral symmetry closer

to the central black hole [14][15]. The M-Set model also

exhibits strong chiral symmetry far away from the cen-

tral black hole and a weak chiral symmetry close to the

central black hole. As a side note, it turns out, chirality

is also related to quaternions; the 4-dimensional exten-

sion of the 2-dimensional complex plane as discussed in

Section 10 [16]. In short, a transition from chiral symme-

try to axial symmetry is a signature of a quasi-black hole.

15. Do Quasi-Black Holes Exist in Nature?

It is generally accepted that black holes exist in nature.

But what kind of black holes does nature actually make?

There are currently two contenders: 1) The standard

black holes that Relativity describes, and 2) the quasi-

black holes as described in this essay. Do quasi-black

holes exist in nature? In order to test this, we need to

be able to make a prediction. What is the signature of

a quasi-black hole? In the previous section, we saw that

the transition from chiral symmetry to axial symmetry is

a signature of a quasi-black hole.

General relativity also makes very speciﬁc predictions

about the morphology of the region surrounding a black

hole. The movie, Interstellar, used computer simulations

to accurately depict the behaviour of a relativistic black

hole. These simulations predict a bright ring of light

surrounding a shadow cast by the black hole. It has been

a long standing goal of astrophysics to directly observe

the environment surrounding the event horizon of a black

hole. The Event Horizon Telescope, a large telescope

array consisting of a global network of radio telescopes,

is uniquely suited to observe the black hole at the center

the Milky Way (catalogued as Sagittarius A∗). The data

from the Event Horizon Telescope was collected in April

2017 and, as of the writing of this paper, the data has

not yet been analyzed. It could be the end of 2017 or the

beginning of 2018 before the results are released to the

public.

The prediction made by M-Set as quasi-black hole is

quite different. In this case, the prediction is based on

the shape of the halo of the photon sphere. Here, the

author speculates that the shape of the photon sphere

surrounding the black hole will appear either oblate or

slightly pear shaped depending on how far we can see

into the photon sphere of the black hole, if at all. If we

can see deep into the photon sphere, we should begin

to see the shape of the quasi-black hole as depicted in

Figure 1. If a pear-like shape is found, then a prediction

can be made as to the exact location of the singularity

in that region. In the case of the primordial quasi-black

hole, singularity is at (0,0) of the complex plane (white

dot in Figure 1).

Luckily, we don’t have to wait until the results come

in from the Event Horizon Telescope to test this predic-

tion. Messier 87 is a super-giant elliptical galaxy in the

constellation Virgo. It was recently discovered that the

super-massive black hole at the center of M87 does not

currently reside at the measured center the galaxy but in-

stead, is 22 light years away [17]. Figure 9 is an image

showing the region in question.

Notice that the shape of the bright region housing the

black hole is slightly pear-shaped. Figure 10 shows a

close up of the pear-shaped region (left) along side a

rendering of M-Set after only two iterations (right). The

image on the left of Figure 10 shows the pear-shaped

region of M87 aligned to and alpha blended with the

M-Set image on the right. When this was done, it was

noticed that the location of black hole (singularity) of

M87 overlays exactly with the (0,0) point of M-Set. In

other words, the black SMBH of M87 is exactly where it

is predicted to be by the M-Set model.

Quasar 3C-186 in an example of a quasar that is also

offset from the center of gravity [18] as seen in Figure

11. Notice that the bright region in the vicinity of the

9

10 L. Gardi: The Mandelbrot Set as a Quasi-Black Hole Vol. 10

Figure 9. Astronomers found that the location of the black

hole at the center of M87 is offset from the center of the galaxy

by 22 light years. The light dot indicates the center of the

galaxy’s light distribution and the dark dot is the location of

the black hole. Image credit Nasa.

Figure 10. The image on the left shows the region surround-

ing the M87 black hole registered to and blended with the

M-Set quasi-black hole on the right. The white dot in the M-

Set model (0,0 of the complex plane) aligns precisely with the

SMBH of M-87.

quasar is slightly pear-shaped (as outlined in white) and

the bright quasar itself is exactly where it is suppose to

be according to the M-Set model.

The author agrees that this is highly speculative,

however, if quasi-black holes do exist in nature, then

all quasi-black holes, at all scales, should exhibit this

signature. Interestingly, recent studies have shown that

the nuclei of some atoms, such as Radium-224 and

Barium-144, are found to be asymmetrical and pear-

shaped [19]. This supports the idea that the atom is a

quasi-black hole, as suggested in Section 13. Before

discounting the existence of quasi-black holes in nature,

let’s wait and see what the evidence shows us.

Figure 11. The bright source in this image, Quasar 3C 186,

is shown displaced from the center of its host galaxy, indicated

with a grey circle.The white arrow points to a blob like feature

of unknown origin). Image: Hubble

16. Conclusions and Future Work

It is well known that fractal geometry mimics nature

very accurately. Benoit Mandelbrot wrote a book on this

called "The Fractal Geometry of Nature"[20]. The evi-

dence presented in this essay shows how Mandelbrot’s

signature fractal is able to mimic the morphological fea-

tures of a multitude of cosmological objects which, coin-

cidentally, also happen to be associated with black holes.

Of course, this kind of mathematical model cannot tell

you the exact trajectory of an actual gravitational body

orbiting the black hole at the center of our galaxy. That

said, dynamics like this are strangely encoded in the M-

Set model as seen in Appendix A (Figure 12). It also

may not be possible to predict exactly how a set of parti-

cles will interact with each other in a particle accelerator,

however, dynamics like this are also strangely encoded

into M-Set as seen in Appendix A, Figure 13. Future

work will be to determine the correspondence (if any)

between images such as these and physical reality.

Fractal geometry is illusive. Using extremely simple

rules and a simple formula with no arbitrary constants,

one is able to generate an endless set of "ﬁgures" that

strikingly resemble familiar cosmological objects as de-

picted in Figures 3, 4, 5 and 7. At the very least, we have

to admit that the universe is very "Mandelbrot-like". If

fractal geometry did preceded relativity, the author sug-

gests that we would have been able to intuit the existence

of black holes using fractal geometry alone. Then the

question becomes, "Could the laws of physics be emer-

gent properties of fractal geometry?"

10

Vancouver Canada, BC 2017 Proceedings of the CNPS 11

17. Appendix A

Parameters from Figure 4

Below are the seed points for the trajectories found in

Figure 4. From left to right (a,b,c,d,e):

4a: (0.07023590960694610, 0.61486872535543668)

4b: (0.25520149659878483, 0.49473324077959746)

4c: (0.13811184749188554, 0.58354392530509225)

4d: (0.26464143869779005, 0.48473638288631671)

4e: (0.24725611046013074, 0.50292165702256331)

Orbital Dynamics

Figure 12. 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 [21]. On the right is

an M-Set experiment that depicts similar dynamics. These are

preliminary results that need to be investigated further.

Particle Dynamics

Figure 13. The left half of this image comes from an ac-

tual bubble chamber experiment depicting particle collisions.

Source: Fermilab. The right half of this image is an M-Set ex-

periment that shows similar dynamics. These are preliminary

results that need to be investigated further.

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