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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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