A 21st century scientific perspective on Buddhism

A 21 st century scientific perspective on Buddhism

By Minh-Khong

© Copyright 2011. All commercial rights reserved. Any reproduction in whole or part, in any form, for sale, profit or
material gain is strictly prohibited. The reprinting of this in its entirety, for free, unconditioned distribution is allowed after
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Abstract

2500 years ago, things were simple, life was simple, and science was in an infant stage.
Today, things are complicated, life is complicated, and science has advanced so much that reading
Buddha's teachings about emptiness nature of all phenomena is incomprehensible to most people.
Buddha's teachings were adapted to the understanding of the people at the time, using the language
and knowledge of people at the time. Those don't seem to suit people's scientific quest for knowledge
nowadays anymore. People don't believe unless things are explained in a scientific way, especially
those with higher education.

This paper is an attempt to use today's science methodology to convey, in 21 st century terms,
Buddha's teachings especially on emptiness nature of all phenomena. Scientific model using fractal
transformation for the law of universal causation commonly known as karma is introduced to explain
rebirth, Buddhist's six realms of existence, enlightenment concepts and the emptiness nature of all
phenomena.

Introduction:

Today's science has advanced so much that scientists take on the roles of authorities
regarding the objective, physical world [1]. However, when it comes to our minds, little is known.
Scientists are troubled on how to character or analyze the mind (consciousness) using similar
methodologies as they have done with the physical world. Cognitive and behavior sciences have
emerged but can not explain consciousness, experiences of growing numbers of people reporting
"experience after death" or the state of the minds of monks in deep meditation.

Growing numbers of conferences, symposiums and dialogs [2] between scientists and
Buddhist monks have happened to bridge the gap between scientific methodologies in materialistic
science and Buddhist methodologies in mind cultivating.

However, the differences between scientist's and Buddhist's approaches are deep rooted from
the very foundation of science and Buddhism. This paper examines the foundations of science and
Buddhism and proposes a scientific model to represent Buddhism theology.

"Buddhism and Science: Confrontation and collaboration", presented at International Conference on "Buddhism and Science"
Central Institute of Higher Tibetan Studies, Sarnath, Varanasi by Dr. B. Alan Wallace

http://www.newdualism.Org/papers/B. A. Wallace/Wallace-Buddhism%20and%20Science-2008.pdf
2 nd Buddhism & Science symposium, July 15, 2010 in Singapore.
The 3rd World Conference on Buddhism and Science, December 1-2, 2010 in Thailand

Buddhism and Science: Cross road of the future, December 4-5, 2010 at Maitripa College, Portland, Oregan, USA.
The dialogs between HH Dalai Lama and quantum physicists -

http://www.youtube.com/watch?v=Zid26JSaq64
The dialogs between university researchers and the HH Dalai Lama-

http://www.voutube.com/watch?v=OvfkU8ihdAA&feature=relmfu

http://www.voutube.com/watch?v=R AGhAP7pzc&feature=relmfu

Foundation for science:

In science, according the Oxford dictionaries a postulate is "a thing suggested or assumed as
true as the basis for reasoning" [3]. Therefore, a postulate is a proposition that is not proven or
demonstrated but considered to be self-evident. Once you've accepted the postulate, other rules or
laws can be deduced from these basic "foundation" and the whole science on this is formed.

For example, in Euclidian geometry, the first postulate [4] is that "you can only draw one
straight which is also the shortest line between two points". Another postulate is that "you can have
one and only one plane containing any 3 arbitrary points in space". These are fairly "obvious" to
accept. This is the "Euclidian geometry" that we know and it is used in every day's life to build
geometry shapes as our house, table, chairs.

On the other hand, in the theory of relativity, Einstein's postulate [5] is that "light is always
propagated in empty space with a definite velocity c which is independent of the state of motion of the
emitting body". This is a little bit harder to be "obvious" that is if you travel at the speed of light, you
still perceive light traveling at that same speed c which is approximately 300,000 kilometers per
second. This is as opposed to if you travel on a car at 60 miles per hour, you would see other cars
traveling along at same speed as if they were stationary. However, since the rules deduced from this
postulate are able to explain phenomena observed with stellar movements and the famous E=M*C 2
equation deduced from this rule agrees in phenomena observed in atomic phenomena, this is
accepted by scientists.

Next to postulates is a set of basic rules or laws that can not be proven but observed to be
true. For example, in Newton's dynamic mechanics, the 2 nd law [6] states that the acceleration of an
object asserted by an external force is proportional to the force and inversely proportional to the mass
of the object (a = f/m or f = m*a). This law is observed by experiments but can not be proven (via
mathematical means). Things around us - cars, machinery - are the results of applying this law.

All other laws can be proven, deduced from the postulates and those basic rules to become a
science. Any theory, phenomenon that doesn't conform to those "science laws" is deemed to be not
scientific.

People used to think science is absolute, is truth. In fact, science is only relative truth,
"perceived truth". The postulates are formulated via empirical experiments or observations. The
formula is then "generalized" to include all objects. Once a phenomenon is discovered that violates
the postulate, another postulate is formed to cover the case and a "new" science is formed.

In science history, the Newtonian physics can not explain the observed movement of stellar
bodies or atoms. Einstein's theory of relativity is able to explain that and becomes a super-set of
Newtonian physics. Einstein's relativity physics can not explain today's observation of sub-particle
behavior. Quantum physics is realized to explain those.

3. Oxford's definition of "postulate" - http://oxforddictionaries.com/definition/postulate

4. Euclidian geometry postulates -
http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Postulates-and-Theorems.topicArticleld-18851, articleld-18770.html

5. Einstein's theory of relativity postulates - http://home.att.net/--anti-relativitv/Einsteinl.htm

6. Newton's dynamic mechanics laws - http://www.qlenbrook.kl2.il.us/GBSSCI/PHYS/CLASS/newtlaws/u2l3a.html

2

Foundation for Buddhism:

With Buddhism, to be seen as a scientific model, it is proposed that the first postulate is that
"all sentient beings have the same nature and this nature is basically emptiness". This is much harder
to accept as "obvious", most would accept this via their belief instead.

Next is a set of rules on dependent-arising phenomena [7], namely

Wi
W

w

Wi
Wi

w
w

Wi
Wi
W
W

th Ignorance as condition, Mental Formations arise

th Mental Formations as condition, Consciousness arises

th Consciousness as condition, Name and Form arise

th Name & Form as condition, Sense Gates arise

th Sense Gates as condition, Contact arises

th Contact as condition, Feeling arises

th Feeling as condition, Craving arises

th Craving as condition, Clinging arises

th Clinging as condition, Becoming arises

th Becoming as a condition, Birth arises

th Birth as condition, Aging and Dying arise

These are so important in Buddha teachings that Buddha himself said "Deep is this dependent
co-arising, and deep its appearance. It's because of not understanding and not penetrating this
Dharma that this generation is like a tangled skein, a knotted ball of string, like matted rushes and
reeds, and does not go beyond transmigration, beyond the planes of deprivation, woe, and bad
destinations" [8].

Last is the laws of karma i.e. "cause and consequences" of good/bad actions/behavior as the
seed for subsequent actions or behavior (analogous to the "force" in Newton's 2 nd rule of mechanics
sense). These are disclosed in very detailed in the Patthana section of the Abhidhamma [9].

Only when you fully accept the postulate, dependent-arising rules and karma laws, then the
rest of Buddha teachings would make sense. Otherwise, there would be areas where doubts arise
since they don't have a solid foundation to base on.

On 12 links of dependent arising -

http://www.buddhanet.net/e-learninq/depend.htm
http://buddhism.about.com/od/basicbuddhistteachinqs/tp/twelvelinks.htm

Transcendental Dependent Arising, A Translation and Exposition of the Upanisa Sutta by Bhikkhu Bodhi
http://www.accesstoinsiqht.orq/lib/authors/bodhi/wheel277.html

Maha-nidana Sutta - http://www.accesstoinsiqht.Org/tipitaka/dn/dn.15.0.than.html

The conditionality of Life - http://www.archive.org/details/TheConditionalitvOfLife

Scientific model for Buddhism:

In science, there is a mathematical field called fractals or cellular automation [see appendix].

Since fractal patterns are observed in nature and material-like shapes can be composed using
fractal transformation, it is suggested that fractal transformation can be used to model the karma
causal effect in Buddhism. However, in mathematical fractal transformations, the formula for the
transformation is fixed from cycle to cycle. In this model, it is generalized so that transformation
changes from cycle to cycle using dependent-arising rules and karma laws as described earlier in the
foundation of Buddhism section. But since karma composes of common (collective) karma and
individual karma, the transformation formula is "similar" for the most part for the collective karma but
with minor "variants" for the individual karma. This is similar to where DNA of siblings is similar for the
most part but there are minor variations between them. In analogy to fractal decompression process
of images, one can achieve any form or formless material or beings after eons of the karma fractal
transformations from a uniform "emptiness" as a starting point.

This is our scientific model of the progress of this world, our lives, and science. Each stage of
mental or scientific development corresponds to a fractal detail level. But unfortunately karma so far
can only be described qualitatively. It is a challenge to quantify karma using scientific tools.

We went from the "4 basic materials" - earth, water, fire, wind - to describe things we know to
a molecule level than to atomic level, sub-atomic and now quantum level. It is not surprising that as
we progress, science will keep finding more details, the never ending fractal details. This model
explains the materialistic view of science, a perceived reality (toward more fractal details) and
Buddhism mind purifying process (toward less fractal details). This model explains our lives, our
thoughts, as time progresses, we'll get more "sophisticated", more "detailed", more "materialistic" but
farther away from our basic "true nature", the root of all things.

Science is moving into the "more detailed" direction whereas Buddha teachings are trying to
get us to go to in "coarser level" direction i.e. severing the dependent arising cycles and unwinding
the fractal cycles so that ultimately we would get closer to our "true nature" - the root, the
"emptiness".

Recently the Dalai Lama offers science community to probe/measure brain waves of highly
trained monks in meditation [10]. No doubt they will find explanations in science terms the meditation
process and why the stages of meditation or state of the minds progresses in such way. They would
be all true, at "that scale" in the "fractal structure". These details are more "materialistic", "perceived
details" and are farther away from the "real truth", the "true nature of things".

10. Scans of Monks' Brains Show Meditation Alters Structure, Functioning -

http://psyphz.psych.wisc.edu/web/News/Meditation Alters Brain WSJ ll-04.htm

Now, the Venn diagram [11] can be used to describe the relationship of Buddhism's view and
scientific views - Buddhism postulate, dependent arising rules and karma laws are denoted as the
heavy circumference, all dharmas as the result are indicated as the grey area inside. Smaller circles
indicate various science postulates and their internal grey areas are the resulted associated sciences
- math, geometry, calculus, physics ....

Other ^"^^^^^ Buddhism

Buddha f x^ y^ postulates

worlds

X \ / \

Science

postulates

Saha
world

Other

"global"

postulates

Sciences that don't see the same view as that of Buddhism are likely operating under a
different set of "global" postulates as indicated in the dotted line and its internal area.

From this, we can see that there are areas that are common and we all can agree and there is
area where there are differences. It all depends on the "original postulates" that we base on whether
they are the same or not. So, in arguing on a subject whether it is right or wrong, it is important to
see whether the implied assumptions are the same. Today's science is still searching for the "global"
postulates such as the big-bang theory to explain the existence of this world (note that the big-bang
theory is not absolute truth, it is as valid as its postulates assume). Various sciences today are in
agreement with Buddhism as in the diagram above especially quantum physics in the emptiness
nature of particles and interactions among objects. Those sciences that are in agreement with
Buddhism are put inside the "Buddhism circle". For example, various "science circles" are within the
"saha world circle" which is the world we perceive and live in today. The global postulate circle, the
big bang theory for example, encloses the saha world circle. Note that all sciences to date have 2
common postulates - space and time. The Buddhism postulates and dependent arising rules would
lead to space and time concept (as explained in page 7 below). That's why "science circles" are
enclosed by "Buddhism postulate circle". This diagram can also explain the existence of countless
other existing Buddha worlds, Amitabha for example, besides our saha world.

11. Venn diagram - http://en.wikipedia.org/wiki/Venn diagram

Traditional "emptiness" [12] in Buddha's teachings can be interpreted in this scientific model
context. In the past, there have been discussions as what this "emptiness" really means [13]. From
the Venn diagram above, you can see that since the "Buddhism circle" is outside the "our saha world
circle", there are phenomena that are beyond the grasp of our world and its science which are limited
in knowledge via the "established postulates" at the time. As the result, at the time of Buddha
teachings, there wasn't any other proper word other than the word "emptiness" to explain the true
nature of sentient beings. While this true nature itself is unchanged, but to better describe this true
nature in today's terms i.e. within the bounds of today's science postulates, "pure energy life form"
would be more appropriate to represent our true nature. While this still can't convey the true nature of
"emptiness" (as it would be comprehended by Buddha or any enlightened being), it would be "closer"
than just "emptiness" since the science at the time of Buddha teachings, people knowledge was much
more limited than what we have today.

This is similar to as using a circle in 2-dimensional geometry to describe a sphere in 3-
dimensional space. Expressing this using Diamond sutra style - "the circle we are referring to is
neither circle nor not circle, it is not really such, it is just called by that name" i.e. the circle is not really
circle since it's just a name to describe the sphere which is inconceivable in 2-dimensional terms.

This is an analogy to a similar verse in the Diamond sutra - "And Buddha answered: Subhuti,
those to whom you refer are neither living beings nor not-living beings. Wherefore? Because "living
beings", Subhuti, these "living beings" are not really such; they are just called by that name." [14]

Why is "emptiness" being energy ? First refer back to the Heart sutra:

"The characteristics of the voidness of all dharma are non-arising, non-ceasing, non-defiled,
non-pure, non-increasing, non-decreasing" [15]

In today's words, this refers to the "law of conservation of energy" - energy can not be created
nor destroyed (non-arising, non-ceasing). Second, this also refers to, according to the 2 nd law of
thermodynamics, an energy body in perfect equilibrium where its entropy [16] is at maximum i.e.
perfectly uniformed (non-defiled, non-pure, non-increasing, non-decreasing).

12. On Buddhism emptiness - http://www.thebiqview.com/buddhism/emptiness.html
Emptiness - Cula-suhhata Sutta http://www.accesstoinsiqht.org/tipitaka/mn/mn.121.than.html

13. The Essence of Other-Emptiness, by Taranatha, translated and annotated by Jeffrey Hopkins
Snow Lion Publications, 2007

14. Diamond Cutter Sutra (vajra prajna paramita sutra) Published by the DaeJang Gyong Research Institute of Haein-sa Monastery,
Korea, p-12.

15. Prajna Paramita Heart sutra by translated by Tripitaka Master Hsuan Tsang, commentary by Grand Master T'an Hsu ,
English translation by Ven. Master Lok. http://www.buddhanet.net/e-learninq/heartstr.htm

16. Entropy is an indication or measurement of how well a "system" is distributed uniformly. For example, a glass of water and ice
would have smaller entropy than that when the ice is melted (more uniformed). The 2 nd rule of themo-dynamics states that for an
isolated system, the entropy of the system can only be the same or increasing i.e. can not decrease. For example, a drop of ink in
a glass of water would tend to spread to get the ink dissolved evenly in the water i.e. more uniformed. Any machine, device such
as "perpetual motor" that would decrease the entropy of an isolated system is scientifically impossible.

If applying Einstein's famous equation E=M*C 2 that describes the transformation between
energy and matter, one can easily see why in the Heart sutra - "All things are marked by emptiness"
where "emptiness" is energy as the interpretation above. Note also that energy in its pure form has
never realized in today's science. We always perceive energy in forms like heat, light, electricity,
gravity, motions, chemical bonds ...

This "energy body" is not similar as "passive" energy we know in today's science but this
energy body is a "life form with ultimate wisdom". Keep in mind that this is the "perceived emptiness"
within the bounds of current science, the mapping of "true emptiness" into today's "scientific systems".

Now, starting from "emptiness", if for some reason, this "emptiness" is "shifted" i.e. not in
perfect distribution state any more, "ignorance" i.e. not knowing the truth would result in a "mental
formation" that there is some part of this "emptiness" is "different" than "other part" of this emptiness.
The consciousness of "self" and "non-self" is formed (and also the consciousness of space). Once the
consciousness of "self" and "non-self" is formed, there would be a consciousness of the "boundary" of
the "self" and the "non-self" i.e. form is realized. Once the "form" is realized, at this boundary, there
would develop a "sense" consciousness to be able to "feel" the other side and eventually a sense
organ is formed (consciousness of changes leads to consciousness of time). If the rules of
dependent-arising and karma keep applied, it would end up with the first simplest "living beings"
whether they are viruses or micro-organisms. Imagine the rules of karma and these fractal dependent
arising cycles keep going for eons, what we have is our world, our societies, our science as we know
today. Note also that this example doesn't imply the "first cause" i.e. how this universe if formed. The
explanation is still within the bound of today's science to clarify the 12 dependent-arising phenomena.

Fractal transformation has been used for image compression. In fractal image de-compression
[17], one starts from a uniform gray sheet and the fractal decompression formula is applied over and
over to restore original object. Similarly, one can start from pure uniformed energy to end up with a
real objects or beings after eons of fractal transformations of dependent arising cycles and karma
laws.

-> Fractal de-compression ->

On the other hand, if all materials in this universe transform into pure energy and the energy is
distributed perfectly evenly i.e. at maximum entropy state, it would become "one", "no-self" because
of the uniformness; no part could be differentiated from "other part". The time concept would cease
since the state of this uniform energy stays unchanged, there is nothing to "mark" the past, present or
future. Space concept would also cease since there is nothing to mark "here" and "there". Keep in
mind that this interpretation is still bound by the limit of today's science. As science progresses in the
future, the science's postulates are "enlarged" to be a superset of today's science, the "perceived
emptiness" and "fractal transformation of karma laws" would (or should) be then re-modeled to be
closer to the "real truth".

17. Fractal image compression, AK Peters, Ltd. by Michael F. Barnsley & Lyman P. Hurd
Fractal decompression sample -
http://www.cs.northwestern.edu/~aqupta/ projects/image processinq/web/FractallmaqeCompression

To further verify this fractal model against Buddha's teachings, from the Diamond sutra:

"Yet when vast, uncountable, immeasurable numbers of beings have thus been liberated,
verily no being has been liberated" [18]

That can be visualized symbolically as described by a fractal structure below. If each sentient
being is represented as a "detail" on a Von Kock fractal structure [19]. When a bodhisattva helps to
liberate many sentient beings as symbolically depicted in the heavy line below, none was liberated
(sentient beings that were liberated are all the detail levels from that heavy line if the fractal
transformation were applied from that level).

Through this, we can see that when all sentient beings are "liberated" (i.e. getting to the "root"),
"no sentient being was liberated" and the end result is a unified "one", "no-self", the "emptiness".

This example is fairly simplistic to convey the idea. The actual dependent-arising rules
according to the karma of each sentient being and the complex network of inter-karma of beings,
when applied, sentient beings would move toward or away from the "root" depends on their good/bad
karma in the past (lives). The world would move away or toward the root also depends on the
collective karma [20]. The quantum physics view of interaction among particles or objects is very
similar to this inter-karma among sentient beings.

18. Diamond Cutter Sutra, published by the DaeJang Gyong Research Institute of Haein-sa Monastery, Korea, p-1.

19. Fractal Von Kock's curve - http://www.iimloy.com/fractals/koch.htm

20. Cakkavattisihanada Sutta - http://www.basicbuddhism.org/index.cfm7GPI D=29

In Buddhism, sentient beings are born according to past karma into each of the 6 basic
realms- god, demi-god (asuras), human, animal, hungry ghosts or hell realms [21].

The picture below is to symbolically depict how the 6 realms of existence in Buddhism are
perceived from this scientific model.

At the center core is our "Buddha nature".
The 1st cylinder layer is the god realm where their karma is "very light".
The 2 nd cylinder layer is the demi-god realm where their karma is "thicker" then that of the god realm.

,rd

The 3 cylinder layer is the human realm

-th

The 4 cylinder is the animal realm where their karma is worse than that of human's.
The 5 th cylinder is the hungry ghost realm where their karma is worse than that of animal's.
The 6 th cylinder is the hell realm where their karma is worst.

Buddha nature

God realm

Demi-god realm
Human realm
Animal realm

Hungry ghost realm
Hell realm

access
channel

21. Six realms of existence in Buddhism -

http://buddhism.about.com/od/basicbuddhistteachinqs/tp/Six-Realms-of-Existence.htm

The karma can be visualized as a cloud covering and hindering our Buddha nature to see
things the way they really are. This "karma shield" causes us to perceive things (including our present
forms) according to the realms we are in whether we are a god, a human or an animal ... The "thicker
the shield", the "farther" we are from our true nature. We perceive life starting and ending as this
karma shield changes its state according to our past karma. That's how we "die" from one realm and
"reborn" to the other realm or reborn in the same realm. When one reaches enlightenment, all these
"karma shields" are shredded, one returns to his/her true nature.

This gives insight to the re-birth process that most westerners find it's hard to believe.

Buddha taught various dharmmas according to the past karma of each person. Therefore,
there are recorded teachings in various levels. Unfortunately, that makes it difficult for us to know
where to start and where to go next. The best is to try everything first, once you see a pattern on a
particular sutra or methods easy to understand or feel more "suitable" to you. Then try to go deeper in
to that line of sutras or method in more depth and details.

Various dharma practices such as zen, chanting, pure-land and tantras are methods to
effectively sever the dependent-arising cycle. This effectively opens a "channel" through these "karma
shields" to gain access to our Buddha nature. That is why when this channel is open, people often
feel a vast tranquility upon reaching Samadhi state (concentration or absorption state of mind). This
access allows us to "see" things "better", to understand the "Buddha teachings" better via our own
Buddha nature. Doing it frequently, it's like a "boot-strap" process with positive feedback. It makes it
easier "next time" we try to open this channel again since we are "wiser", we understand Buddha
teachings better which in turn helps the next time. People meditating, who frequently open this
"channel", can keep the duration of the channel longer and/or "widen" channel. This explains why
some sages have gained the "god/divine eye" ("eye" that can see through material objects, can see
sentient beings in other existence), the "wisdom eye" ("eye" that can see the true nature of things)
and/or the "dharma eye" ("eye" that understands all dharmas, can see karma in the past and lives in
the future as the result of past karma) as they are able to easily access their Buddha nature that has
the ultimate wisdom [In contrast, today's science can not predict exactly an outcome of throwing a
dice since all the factors affecting its movement are too numerous to be accounted for. Only a few
primary factors are included within computing power means to calculate the probability of the dice
landing position].

That's why (be able to access to the Buddha nature) Nagarjuna was able to comprehend the
"real emptiness" in his analysis of emptiness [22]. That's why Vimalakirti comprehends the
"inconceivable liberation" ability of Bodhisattvas [23]. Describing these using the words at the time are
mostly difficult to comprehend and are often being disregarded by today's science.

If the positive feedback is strong enough, it causes an avalanche effect and all the "karma
shields" are shredded, one achieves enlightenment.

When a person dies or has near-death experience, there is a brief moment he/she has access
to the Buddha nature when the "karma shield" is transformed. This explains how they could see their
past lives and karma. For experienced dharma practitioner, this gives him/her an opportunity to
escape the birth/death cycle [24].

22. Causality & emptiness, The Wisdom of Nagarjuna - http://www.buddhanet.net/pdf file/naqarjuna.pdf

23. Vimalakirti Nirdesa Sutra - http://www.fodian.net/world/0475 06.html

24. About Bardo - http://buddhism.about.eom/od/vairayanabuddhism/a/bardothodol.htm

10

Conclusion

The foundation of Buddhism has been analyzed using scientific approach of postulates and
basic laws. Venn diagram has been used to represent the Buddha dharma in relation with other
sciences. Fractal transformation has also been used coupled with those postulate and karma laws to
represent the form and formless formation of materials and sentient beings as seen in Buddhism
theology. It also gives insight to rebirth concept, the Buddhist's six realms of existence as well as
enlightenment concept. This scientific model is also verified against traditional Buddhist scriptures.

We hope that these would give those with scientific mind a good solid foundation to spark their
belief in Buddhism and those who are already on the path something to re-assure their belief in this
scientific world. In reality, this world and our perceptions seem very real to us, we have to deal with
them the best we can to reduce our bad karma. In practice, just like science, one doesn't dwell into
details of the basic formulas for quantum physics or classical Newtonian physics to believe in every
day's phenomenon as how a car, a boat or a space vehicle works. In Buddhism, one would use the 8-
fold path as the general compass to make sure the direction taken is, in general, correct. The
teachings, stories, advises in the Tripitaka as well as analysis, explanations by dharma masters
would provide more details and examples suited to each person's level, depending on where he/she
is on "the path".

To conclude, "If there is any religion that would cope with modern scientific needs it would be
Buddhism" which is believed to be a quote from Einstein.

11

Appendix - Fractals

In fractals, if the same rule, formula, or algorithm is applied over and over again, it results in an
interesting outcome. For example, if the "rule" is to make a "mountain" in a middle of the line segment
(Von Kock curve [19])

Now apply the same rule on the new line segments

If this rule is applied over and over again, the figure would tend to "converge" to some shape

th

called "conjecture" of this "rule". Here is the 5 order Von Kock curve

12

These fractals have been recognized in natural shapes [25], for example a "fractal fern".

Another example is Romanesco (a cross between broccoli and Cauliflower, which accentuates
the great fractal spiral patterns on the top)

25. Fractal in nature and synthetic fractal - http://www.miqel.com/fractals math patterns/visual-math-natural-fractals. html

13

Or it could be totally synthesized with pure mathematic equations as in this Iterative Desert
Landscape: this windswept landscape, looking a bit like sandstone is completely generated by
math/fractal equations

DNA structure and its passing from parents to children is another example of a fairly
complicated iterative structure in nature.

14

An amazing characteristic of fractals is that at any "coarser" level, one can always go to a
deeper "detailed" level and find more details which is often referred as "artificial details" as in the well
known Mandelbrot set [26].

Excerpts from http://en.wikipedia.org/wiki/Mandelbrot_set:

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually
called "zooming in". The following example of an image sequence zooming to a selected c value gives an
impression of the infinite richness of different geometrical structures, and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10,000,000,000 to 1. Relating to an
ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border
would show an inconceivable number of different fractal structures.

Start

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26. Mandelbrot fractal and artificial details - http://en.wikipedia.orq/wiki/Mandelbrot set

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