Trigonometric Fractal Differential Equations

Abstract

1)Solve
2)Graph
3)Take Fractal of Graph

Full text

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Alexander Ohnemus
Engineering
24 June 2024
Trigonometric Fractal Differential Equations
Table of Contents:
Prologue
1. Triggernometry
2. Fractals
3. Differential Equations
4. Engineering
5. Conclusive Formulas

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Prologue

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A difficult discipline necessitates critical rationalism(skepticism avoiding contradictions),
deductive reasoning, rigor and practicality. Triggernometry study consumes time. Thus, fractals
acknowledge reality’s unpredictability thus, deducing, and differential equations practically
segway towards engineering(lucrative discrepancy testing).
See discussions, stats, and author profiles for this publication at:
https://www.researchgate.net/publication/381469939
Critical Rationalist Physics
Presentation · June 2024 DOI: 10.13140/RG.2.2.21290.63689
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Alexander Ohnemus Physics
16 June 2024
Critical Rationalist Physics
Criterion of falsifiability is synonymous with critical rationalism. “criterion of

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falsifiability, in the philosophy of science, a standard of evaluation of putatively scientific
theories, according to which a theory is genuinely scientific only if it is possible in principle to
establish that it is false”(Britannica 2024).
Critical rationalism(falsifiability) matches the scientific while skeptical empiricism does
NOT. Thus, critical rationalism epistemologically maps physics more than skeptical empiricism
does. “He (Popper) rejects Hume’s psychological account of induction, specifically his theory of
belief formation by repetition”(Parusniková 2018). Hume excessively dismisses rationalism.
Specifically, Hume forgets the importance of identity, while excessively referencing repetition.
Popper, a self described critical rationalist, correctly moderates between the known and unknown
with his deductive reasoning. Fortunately, both Hume and Popper rejected inductive reasoning.
Objectivist epistemology correctly emphasizes the law of identity but continues inductive
reasoning. “Leonard Peikoff and David Harriman have denounced modern physics as
incompatible with Objectivist metaphysics and epistemology. Physics, they say, must return to a
Newtonian viewpoint; much of relativity theory must go, along with essentially all of quantum
mechanics, string theory, and modern cosmology. In their insistence on justifications in terms of
“physical nature,” they cling to a macroscopic worldview that doesn't work in the high-velocity
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arena of relativity or the subatomic level of quantum mechanics. It is suggested that the
concept of identity be widened to accommodate the probabilistic nature of quantum
phenomena”(Gibson 2013). Objectivism correctly notes reality exists. Rejecting scientific
advancements lead to both epistemological ignorance and stagnation. Science ONLY

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approximately derives from philosophy. Science observes literally. Science can overlap with
epistemology. But ethics are philosophical. Scientism is the excessive reliance on science, which
this essay will later cover more explicitly. The Newtonian viewpoint was abandoned, on some
level, out of practicality. Relativity matters. Quantum mechanics also. String theory is a valuable
attempt to merge relativity with quantum mechanics. Modern cosmology is probably necessary
for space exploration. Not everything is macroscopic. Some of the most vital information is
invisible to the naked eye. The law of identity applies at any speed. The law of identity goes for
any size. Only viewing the macroscopic view would greatly hinder science. Adjusting the
concept of identity does NOT mean keeping the premises after contradictions. Rather, the law of
identity applies to non macroscopic views as well. The law of identity applies to quantum
phenomena.
The law of identity deserves a formal definition. “a statement of an identity is the
expression of an abstract relation of identity symbolized by a term (as A in "A is A") that
apparently refers in its separate instances to the subject and predicate
respectively”(Meeriam-Webster). A is A in relativity. A is A in quantum mechanics. A is A in
string theory. A is A in modern cosmology. A is A across all of modern physics and all
contradictions, at any speed and or size, warrants adjusting both premises and concepts.
Critical rationalism is self critically rationalist. Nominalism could be critical rationalism
without the self criticism because the former engages with less rationalism. “nominalism, in
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philosophy, position taken in the dispute over universals—words that can be applied to
individual things having something in common—that flourished especially in late medieval
times”(Britannica 2018). Both critical rationalism and nominalism boil down to the most
fundamentals then the former slightly utilizes universals. Empiricism without any rationalism is
haphazard and mindless. While rationalism without any empiricism, abandons reality. Either
every entity is unique or too different for too perfectly to predict the future. One detail from the
past could have changed an outcome due to dimensionality. Thus, for rigorous physics, the law
of identity must apply at all levels to prevent contradictions and foster practicality.
Even if time is relative, each being across is unique. If everything is unique then all scales
apply.
All vibrations are also unique.
Every being across the cosmos is unique.
Even in Newtonian physics, contradictions ignore reality, thus preventing rigor. The law
of identity always applies. Misidentification is most obvious macroscopically.
Science cannot answer everything. “Scientism is Faith Theorem (group theory) Given
that the roots are humanism(faith that humans are ethical despite being the most unethical
species), induction(faith that because x is the case in particular situations then x is always the
case), and credentialism(OFTEN trusting in intellectuals whose end products are ideas thus
without direct enough consequences to encourage improvement)”(Ohnemus 2023).
Intimately ending contradictions may lead to completely understanding the afterlife.
“(RESPECTFULLY, Quantum Entanglement ≠ Reincarnation) Quantum Entanglement does not
offer enough reincarnation evidence. "Entanglement is at the heart of quantum physics and future

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quantum technologies. Like other aspects of quantum science, the phenomenon of
entanglement reveals itself at very tiny, subatomic scales. When two particles, such as a pair of
photons or electrons, become entangled, they remain connected even when separated by vast
distances. In the same way that a ballet or tango emerges from individual dancers, entanglement
arises from the connection between particles. It is what scientists call an emergent
property"(Caltech). Science cannot explain all, at least not yet. PERHAPS engineering can
demonstrate what science has attempted to explain. Scientific theories must be falsifiable. One
cannot prove science. One can prove a scientific theory is wrong. A scientific theory MUST be
debunkable or it is metaphysics NOT science. The law of identity also applies to subatomic
scales. The law of identity also applies to photons. The law of identity also applies to electrons.
Science can be highly counterintuitive BUT, contradictions prompt adjusting both concepts and
premises. Parts, by definition, are NOT the same. Two beings can connect in some ways yet
separate IN OTHERS. The law of identity PROHIBITS contradictions. Ignoring reality does
NOT avoid consequences. Quantum entanglement does NOT defeat these arguments against
reincarnation:...”(Ohnemus 2024).
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Works Cited

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Britannica, The Editors of Encyclopaedia. "criterion of falsifiability". Encyclopedia
Britannica, 29 Apr. 2024, https://www.britannica.com/topic/criterion-of-falsifiability. Accessed
16 June 2024.
Parusniková, Z. “Popper and Hume: Two Great Skeptics.” Link.Springer.Com, Cham
Palgrave Macmillan, 14 July 2018, link.springer.com/chapter/10.1007/978-3-319-90826-7_17.
Warren C. Gibson. “Modern Physics versus Objectivism.” The Journal of Ayn Rand
Studies, vol. 13, no. 2, 2013, pp. 140–59. JSTOR,
https://doi.org/10.5325/jaynrandstud.13.2.0140. Accessed 16 June 2024.
“Law of identity.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/law%20of%20identity. Accessed 16 Jun. 2024.
Britannica, The Editors of Encyclopaedia. "nominalism". Encyclopedia Britannica, 6 Apr.
2018, https://www.britannica.com/topic/nominalism. Accessed 16 June 2024.
Perkowitz, Sidney. "relativity". Encyclopedia Britannica, 7 Nov. 2023,
https://www.britannica.com/science/relativity. Accessed 16 June 2024.
Britannica, The Editors of Encyclopaedia. "special relativity". Encyclopedia Britannica,
12 Apr. 2024, https://www.britannica.com/science/special-relativity. Accessed 16 June 2024.
Britannica, The Editors of Encyclopaedia. "general relativity". Encyclopedia Britannica,
2 May. 2024, https://www.britannica.com/science/general-relativity. Accessed 16 June 2024.
Squires, Gordon Leslie. "quantum mechanics". Encyclopedia Britannica, 20 May. 2024,
https://www.britannica.com/science/quantum-mechanics-physics. Accessed 16 June 2024.
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Greene, Brian. "string theory". Encyclopedia Britannica, 16 Feb. 2024,
https://www.britannica.com/science/string-theory. Accessed 16 June 2024.
Shu, Frank H.. "cosmology". Encyclopedia Britannica, 5 Apr. 2024,
https://www.britannica.com/science/cosmology-astronomy. Accessed 16 June 2024.
Britannica, The Editors of Encyclopaedia. "Newton’s laws of motion". Encyclopedia
Britannica, 6 May. 2024, https://www.britannica.com/science/Newtons-laws-of-motion.
Accessed 16 June 2024.
Ohnemus, Alexander. “Scientism Is Faith Theorem.” ResearchGate.Net , Ohnemus
University , 7 Aug. 2023, dx.doi.org/10.13140/RG.2.2.18100.83843.
Ohnemus , Alexander. “RESPECTFULLY, Quantum Entanglement ≠ Reincarnation.”
ResearchGate.Net , Ohnemus University , 14 June 2024,
dx.doi.org/10.13140/RG.2.2.15798.72007.
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Triggernometry

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“Trigonometry
“The study of angles and of the angular relationships of planar and three-dimensional
figures is known as trigonometry. The trigonometric functions (also called the circular functions)
comprising trigonometry are the cosecant csc x, cosine cos x, cotangent cos x, secant sec x, sine
sin x, and tangent tan x”(Weisstein).
“Examples for
Trigonometry
Trigonometry is the study of the relationships between side lengths and angles of
triangles and the applications of these relationships. The field is fundamental to mathematics,
engineering and a wide variety of sciences. Wolfram|Alpha has comprehensive functionality in
the area and is able to compute values of trigonometric functions, solve equations involving
trigonometry and more”(Wolfram).

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Fractals

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“A fractal is an object or quantity that displays self-similarity, in a somewhat technical
sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the
same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph
versus scale then gives a straight line, whose slope is said to be the fractal dimension. The
prototypical example for a fractal is the length of a coastline measured with different length
rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline
paradox”(Weisstein).
“A fractal is an object or quantity that exhibits self-similarity on all scales. Use
Wolfram|Alpha to explore a vast collection of fractals and to visualize beautiful chaotic and
regular behaviors. Examine named fractals, visualize iteration rules, compute fractal dimension
and more”(Wolfram).

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“See discussions, stats, and author profiles for this publication at:
https://www.researchgate.net/publication/380729229 Fractal Vindication of CRT
Research Proposal · May 2024 DOI: 10.13140/RG.2.2.34117.67043
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Alexander Ohnemus Applied Mathematics 20 May 2024
Fractal Vindication of CRT
DNA is SO unpredictable that they are either fractals or something less predictable, thus
a gene is never known to manifest into a trait, debunking hereditarianism and vindicating
CRT.
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Works Cited
Britannica, The Editors of Encyclopaedia. "fractal". Encyclopedia Britannica, 5 Jan.
2024,

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https://www.britannica.com/science/fractal. Accessed 20 May 2024.
Ohnemus, Alexander. (2024). Dimensionality, dysgenic trends, somatic mutations,
genetic
engineering and information theory all support constructivism(with privilege heritability)
over hereditarianism. I welcome further elaborations.. 10.13140/RG.2.2.19415.04002.
Dimensionality, dysgenic trends, somatic mutations, genetic engineering and information theory
all support constructivism(with privilege heritability) over hereditarianism. I welcome further
elaborations.
Ohnemus, Alexander. (2024). My Current Condensed Political Inclination.
10.13140/RG.2.2.29875.40489. Conservatives may have more kids but, liberals are more likely
to use life extension and to have increased socioeconomic status. Removing almost all doubt, the
self-imposed demographic decline of the North Western Europeans is an attempt to filter SO,
only their GENERALLY smartest remain. Progressivism rightfully did NOT emerge from the
Jews but, from North Western Europeans recognizing that under their continued empire, proving
constructivism more so than hereditarianism, they would be subject to dysgenic forces and then
become petty nationalists, and they have mysterious impulses to check their own unchecked
power, as the most powerful group. As that was the case with the Mediterraneans. Plus,
GENERALLY the highest IQ North Western Europeans are liberals and they GENERALLY get
more progressive on race relations the more intelligent they become. ALL DISPARITIES are the
fault of the North
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Western Europeans, thus the MOST ENLIGHTENED North Western European people
should be cloned so they and their descendants can pay reparations. And the recessive privileges
of being an Enlightened North Western European should MAYBE be distributed through
genetically engineered somatic mutations.
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“(Ohnemus 2024).

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Differential Equations

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“A differential equation is an equation involving a function and its derivatives. It can be
referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE)
depending on whether or not partial derivatives are involved. Wolfram|Alpha can solve many
problems under this important branch of mathematics, including solving ODEs, finding an ODE
a function satisfies and solving an ODE using a slew of numerical methods”(Wolfram).

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Engineering

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See discussions, stats, and author profiles for this publication at:
https://www.researchgate.net/publication/372503312 Fundamentals and Applications of
Engineering Thesis - Book · September 2022
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READS
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Alexander Ohnemus No Professor Engineering/Education 15 September 2022
Fundamentals and Applications of Engineering Thesis
Engineering is a subject matter. Engineering has fundamentals and applications and
advice that could lead to more vocational success. The credibility of a source can be
determined by its reputation, incentives, and the logic of its information.
Engineering should be more formally defined for this essay. Engineering is "the
application of science and mathematics by which the properties of matter and the sources of
energy in nature are made useful to people"(Merriam-Webster). This dictionary definition of

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engineering is very useful. Engineering is complex in that it is the application of other subjects.
Yet it has fundamentals just as every subject does.
As stated previously in this essay engineering has fundamentals. In the fundamentals of
engineering curriculum of a certain university the teachings are "Calculus and differential
equations (with engineering applications)
Foundational knowledge areas needed by engineers engineering professionalism,
computational and programming skills, communication (graphical, written and oral),
problem solving, design analysis,
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teamwork and project management"(University of Louisville). Ironically engineering is
the application of other subjects yet it too has its own fundamentals. The fundamentals of
engineering are the unchanging elements of the subject.
Differential equations may be the most tangible fundamentals to engineering and
therefore are imperative to learn for vocational success. All other fundamentals may be too
dependent on change to explain in this essay. So the essay will focus on differential equations.
"In Mathematics, a differential equation is an equation that contains one or more functions with
its derivatives. The derivatives of the function define the rate of change of a function at a point. It
is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of
the differential equation is the study of solutions that satisfy the equations and the properties of
the solutions"(BYJU'S). The source is probably credible because it could face economic losses

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for stating falsehoods. The answer also is logical. In conclusion, a differential equation has been
defined for this essay.
Taking a derivative is important to engineering so the rules will be included in this essay.
"Frequently Asked Questions on Differentiation Formulas
What are the formulas of differentiation?
The formulas of differentiation that helps in solving various differential equations
include:
Derivatives of basic functions
Derivatives of Logarithmic and Exponential functions Derivatives of Trigonometric
functions
Derivatives of Inverse trigonometric functions
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Differentiation rules
What are the basic rules of differentiation?
The basic rule of differentiation are:
Power Rule: (d/dx) (xn ) = nx{n-1}
Sum Rule: (d/dx) (f ± g) = f’ ± g’
Product Rule: (d/dx) (fg)= fg’ + gf’
Quotient Rule: (d/dx) (f/g) = [(gf’ – fg’)/g2]
What are the derivatives of trigonometric functions?
The derivatives of six trigonometric functions are:

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(d/dx) sin x = cos x
(d/dx) cos x = -sin x
(d/dx) tan x = sec2 x
(d/dx) cosec x = -cosec x cot x
(d/dx) sec x = sec x tan x
(d/dx) cot x = -cosec2 x
What is d/dx?
The general representation of the derivative is d/dx. This denotes the differentiation with
respect to the variable x.
What is a UV formula?
(d/dx)(uv) = v(du/dx) + u(dv/dx)
This formula is used to find the derivative of the product of two functions"(BYJU'S). A
way to determine the veracity of the source is following the money. The source could face
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negative financial consequences for being incorrect. Therefore the source is probably
correct due to its financial risk.
Differential equations also have types and can manifest beyond only words. "We can
place all differential equations into two types: ordinary differential equations and partial
differential equations. A partial differential equation is a differential equation that involves
partial derivatives. An ordinary differential equation is a differential equation that does not

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involve partial derivatives"(Green 2021). Furthermore there are the manifestations of differential
equations. "((d^2)y)/(dx^2)+(dy)/(dx)=(3x)sin(y)(2.2.1)
is an ordinary differential equation since it does not contain partial derivatives. While
(∂y)/(∂t)+x(∂y)/(∂x)=(x+t)/(x−t)(2.2.2)
is a partial differential equation, since y is a function of the two variables x and t and
partial derivatives are present"(Green 2021).
As such solving differential equations can be done either manually or using software. "A
differential equation is an equation involving a function and its derivatives. It can be referred to
as an ordinary differential equation (ODE) or a partial differential equation (PDE) depending on
whether or not partial derivatives are involved. Wolfram|Alpha can solve many problems under
this important branch of mathematics, including solving ODEs, finding an ODE a function
satisfies and solving an ODE using a slew of numerical methods"(WolframAlpha).
WolframAlpha's software is most likely useful for solving differential equations because the
business is subject to economic consequences if it is defective. Noting those at market risk is a
feasible way to follow the money and discover the veracity of information. WolframAlpha is
most likely a useful source of differential equation solving software.
4
Vocabulary is important in mathematics as it determines what is what and the term
derivative should be defined for this essay. "The essence of calculus is the derivative. The
derivative is the instantaneous rate of change of a function with respect to one of its variables.

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This is equivalent to finding the slope of the tangent line to the function at a point"(MIT 1999).
That is what a derivative is.
The definition of a derivative covers the ordinary ones. A special definition is required
for the partial ones. "The reason for a new type of derivative is that when the input of a function
is made up of multiple variables, we want to see how the function changes as we let just one of
those variables change while holding all the others constant"(Khan Academy). Khan Academy is
describing a partial derivative. Partial derivatives are necessary information for the fundamentals
of engineering. Partial derivatives must be learned to do engineering.
Once the fundamentals have been explored the applications are next. "In broad terms,
engineering can be divided into four main categories – chemical, civil, electrical and mechanical
engineering. Each of these types requires different skills and engineering education"(Cote 2022).
The next information in this essay shall be the types of engineering.
Chemical engineering deserves a definition. Chemical engineering is "engineering
dealing with the industrial application of chemistry"(Merriam-Webster). As such the skills
necessary for chemical engineering differ from those of other types of the vocation. A knowledge
of chemistry would be helpful to this kind of engineering. Knowledge of differential equations
would be necessary for chemical engineering.
5
Civil is another kind of engineering. A civil engineer is "an engineer whose training or
occupation is in the design and construction especially of public works (such as roads or

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harbors)"(Merriam-Webster). Civil engineering is for the public. Differential equations would
still be useful for this kind of engineering.
Electrical engineering deserves a definition for this essay. Electrical engineering is "a
type of engineering that deals with the uses of electricity"(Merriam-Webster). An electrical
engineer deals with electricity. Differential equations are still part of electrical engineering.
The last branch of engineering explored in this essay will be mechanical. Mechanical
engineering is "a branch of engineering concerned primarily with the industrial application of
mechanics and with the production of tools, machinery, and their products"(Merriam-Webster).
This branch of engineering also would include differential equations.
In conclusion, differential equations are the leading fundamental to engineering and are
just equations involving derivatives. The partial differential equations are the ones where the
derivative is only taken as part of the equation. The applications of engineering are chemical,
civil, electrical and mechanical.
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Works Cited
“Engineering.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/engineering. Accessed 14 Sep. 2022.
University of Louisville . "ENGINEERING FUNDAMENTALS ."
engineering.louisville.edu.
engineering.louisville.edu/academics/areasofstudy/engineering-fundamentals/. Accessed 14 Sep.
2022.

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BYJU'S. "Differential Equations ." byjus.com. byjus.com/maths/differential-equation/.
Accessed 15 Sep. 2022.
BYJU'S . "Differentiation Formulas ." byjus.com.
byjus.com/maths/differentiation-formulas/. Accessed 16 Sep. 2022.
Green, Larry . "2.2: Classification of Differential Equations ." math.libretexts.org. 5 Sep.
2021. math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_
Differential_Equations/2%3A_First_Order_Differential_Equations/2.2%3A_Classificatio
n_of_Differential_Equations. Accessed 15 Sep. 2022.
WolframAlpha. "Differential Equations ." wolframalpha.com.
www.wolframalpha.com/examples/mathematics/differential-equations. Accessed 16 Sep. 2022.
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MIT. "The Definition of Differentiation ." web.mit.edu. 14 Oct. 1999.
web.mit.edu/wwmath/calculus/differentiation/definition.html#:~:text=The%20essence%2
0of%20calculus%20is,the%20function%20at%20a%20point. Accessed 16 Sep. 2022.
Khan Academy . "Introduction to partial derivatives ." khanacademy.org.
www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/partial-deri
vative-and-gradient-articles/a/introduction-to-partial-derivatives. Accessed 16 Sep. 2022.
SNHU , and Joe Cote . "Types of Engineering: Salary Potential, Outlook and Using Your
Degree ." snhu.edu. 10 Aug. 2022.

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www.snhu.edu/about-us/newsroom/stem/types-of-engineering#:~:text=In%20broad%20t
erms%2C%20engineering%20can,different%20skills%20and%20engineering%20educati on.
Accessed 15 Sep. 2022.
“Chemical engineering.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/chemical%20engineering. Accessed 15 Sep. 2022.
“Civil engineer.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/civil%20engineer. Accessed 15 Sep. 2022.
“Electrical engineering.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/electrical%20engineering. Accessed 15 Sep. 2022.
“Mechanical engineering.” Merriam-Webster.com Dictionary, Merriam-Webster,
https://www.merriam-webster.com/dictionary/mechanical%20engineering. Accessed 15 Sep.
2022.
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Conclusive Formulas

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“There are multiple formulas for calculating the fractal dimension of fractals, which are
geometric shapes that have detailed structure at very small scales:
D = log N/log S
This formula is used to calculate the fractal dimension of strictly self-similar fractals. In
this formula, N is the number of parts a fractal produces from each segment, and s is the
size of each new part compared to the original segment. The dimension is a measure of
how well fractals embed themselves into Euclidean space.
D = log (N)/ log (1/r)
This formula is also used to calculate fractal dimension, where N is the total number of
distinct copies similar to A and A is scaled down by a ratio of 1/r”(AI).
“
”(Cuemath).

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D= logN/logS may apply to graphs.
Steps:
1. Graph
2. Take Fractal of Graph

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Works Cited
Ohnemus , Alexander . “Critical Rationalist Physics.” ResearchGate.net , Ohnemus
University , 16 June 2024, dx.doi.org/10.13140/RG.2.2.21290.63689. Accessed 24 June 2024.
Weisstein, Eric W. “Trigonometry.” Mathworld.wolfram.com,
mathworld.wolfram.com/Trigonometry.html.
“Wolfram|Alpha Examples: Trigonometry.” Www.wolframalpha.com,
www.wolframalpha.com/examples/mathematics/trigonometry. Accessed 24 June 2024.
Weisstein, Eric W. “Fractal.” Mathworld.wolfram.com,
mathworld.wolfram.com/Fractal.html.
“Wolfram|Alpha Examples: Fractals.” Www.wolframalpha.com,
www.wolframalpha.com/examples/mathematics/applied-mathematics/fractals/. Accessed 24 June
2024.
Ohnemus , Alexander . “Fractal Vindication of CRT .” ResearchGate.net , Ohnemus
University , 20 May 2024, dx.doi.org/10.13140/RG.2.2.34117.67043. Accessed 24 June 2024.

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“Wolfram|Alpha Examples: Differential Equations.” Www.wolframalpha.com,
www.wolframalpha.com/examples/mathematics/differential-equations.
Ohnemus , Alexander . “Fundamentals and Applications of Engineering Thesis.”
ResearchGate.net , Ohnemus University , 15 Sept. 2022,
www.researchgate.net/publication/372503312_Fundamentals_and_Applications_of_Engineering
_Thesis_-. Accessed 24 June 2024.
www.google.com/search?q=fractal+formula&rlz=1CDGOYI_enUS1108US1108&oq=fra
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g4IARBFGCcYOxiABBiKBTIGCAIQRRg7MgYIAxBFGDwyBggEEEUYPDIGCAUQRRg8
MgYIBhBFGDkyDAgHEAAYQxiABBiKBTIHCAgQABiABDIHCAkQABiABNIBCDE3MD
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www.desmos.com/calculator.

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