This document provides an overview of fractals, beginning with their origins in mathematics and nature. It discusses how Benoit Mandelbrot introduced fractals as a way to describe irregular shapes in nature. Examples are given of fractals found in nature like trees, clouds, coastlines, and cauliflowers that exhibit self-similarity across scales. Applications of fractals are explored in fields like mathematics, science, engineering, art, music and finance due to their ability to model complex patterns.
Many computer graphics and Image Processing effects owe much of their realism to the study of fractals and noise. This short tutorial is based on over a decade of teaching and research interests, and will take a journey from the motion of a microscopic particle to the creation of imaginary planets.
Further resources at:
http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial
Many computer graphics and Image Processing effects owe much of their realism to the study of fractals and noise. This short tutorial is based on over a decade of teaching and research interests, and will take a journey from the motion of a microscopic particle to the creation of imaginary planets.
Further resources at:
http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial
Fractals are the mathematical explanation of our world. Knowledge of fractals is essential to everyone's experience of their world. Here, I have explained the concept of fractals.
A discussion on the theory behind fractals, several different examples and applications of fractals in modern day life. This discusses the Coastline Paradox, Image Compression and uses within Creative Media
Math is used in everything you see, including space. This presentation is about how mathematics were used in Kepler's Laws on Planetary Motion, plus how Gauss used those laws. This was made for The Cincinnati Observatory's annual ScopeOut event.
Fractals are the mathematical explanation of our world. Knowledge of fractals is essential to everyone's experience of their world. Here, I have explained the concept of fractals.
A discussion on the theory behind fractals, several different examples and applications of fractals in modern day life. This discusses the Coastline Paradox, Image Compression and uses within Creative Media
Math is used in everything you see, including space. This presentation is about how mathematics were used in Kepler's Laws on Planetary Motion, plus how Gauss used those laws. This was made for The Cincinnati Observatory's annual ScopeOut event.
TEDx WanChai Sat 23 Aug 2014
Form follows Flow
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The new Asynsis principle-Constructal law design paradigm: bridging western geometry and eastern philosophy.
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Revolution of Dignity in Ukraine: Visual Fractals of National Self-organizationОлена Семенець
The fractal nature of visual organization of socio-cultural space in Ukrainian cities during the Revolution of Dignity and external aggression (2013–2016) is analyzed. National symbols (the State Coat of Arms and the Flag, traditional folk embroidery) are considered as socio-cultural patterns that permeate all levels of the system organization. These socio-cultural fractals, i.e. self-similar objects, in which a part contains information about a whole object, are the features of powerful processes of social system self-organization.
I made this slide about fractals for one of my math course's presentation. I chose fractal as it has this beautiful pattern and different kinds of variation.
ps- There might be some mistake. Your corrections will be appreciated.
Fractals -A fractal is a natural phenomenon or a mathematical set .pdfLAMJM
Fractals -
A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that
displays at every scale. It is also known as expanding symmetry or evolving symmetry. If the
replication is exactly the same at every scale, it is called a self-similar pattern. An example of
this is the Menger Sponge.Fractals can also be nearly the same at different levels. This latter
pattern is illustrated in the small magnifications of the Mandelbrot set.Fractals also include the
idea of a detailed pattern that repeats itself.
Fractals are different from other geometric figures because of the way in which they scale.
Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the
new to the old side length) raised to the power of two (the dimension of the space the polygon
resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is
two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere
resides in). But if a fractal\'s one-dimensional lengths are all doubled, the spatial content of the
fractal scales by a power that is not necessarily an integer. This power is called the fractal
dimension of the fractal, and it usually exceeds the fractal\'s topological dimension.
As mathematical equations, fractals are usually nowhere differentiable. An infinite fractal curve
can be conceived of as winding through space differently from an ordinary line, still being a 1-
dimensional line yet having a fractal dimension indicating it also resembles a surface.
Fractal patterns have been modeled extensively, albeit within a range of scales rather than
infinitely, owing to the practical limits of physical time and space. Models may simulate
theoretical fractals or natural phenomena with fractal features. The outputs of the modeling
process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal
analysis. Some specific applications of fractals to technology are listed elsewhere. Images and
other outputs of modeling are normally referred to as being \"fractals\" even if they do not have
strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal
image that does not exhibit any fractal properties. Also, these may include calculation or display
artifacts which are not characteristics of true fractals.
Modeled fractals may be sounds,digital images, electrochemical patterns, circadian rhythms,etc.
Fractal patterns have been reconstructed in physical 3-dimensional spaceand virtually, often
called \"in silico\" modeling.Models of fractals are generally created using fractal-generating
software that implements techniques such as those outlined above.As one illustration, trees,
ferns, cells of the nervous system,blood and lung vasculature, and other branching patterns in
nature can be modeled on a computer by using recursive algorithms and L-systems
techniques.The recursive nature o.
Order, Chaos and the End of ReductionismJohn47Wind
The author presents a case against reductionism based on the emergence of chaos and order from underlying non-linear processes. Since all theories are mathematical, and based on an underlying premise of linearity, the author contends that there is no hope that science will succeed in creating a theory of everything that is complete. The controversial subject of life and evolution are explored, exposing the fallacy of a reductionist explanation, and offering a theory of order emerging from chaos as being the creative process of the universe, leading all the way up to consciousness. The essay concludes with the possibility that the three-dimensional universe is a fractal boundary that separates order and chaos in a higher dimension. The author discusses the work of Claude Shannon, Benoit Mandelbrot, Stephen Hawking, Carl Sagan, Albert Einstein, Erwin Schrodinger, Erik Verlinde, John Wheeler, Richard Maurice Bucke, Pierre Teilhard de Chardin, and others. This is a companion piece to the essay "Is Science Solving the Reality Riddle?"
Starting with a brief introduction of the digital culture impact in the design field, the lecture touches personal obsessions made explicit in some projects of the studio.
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This presentation is about the definition of Function, One to one, onto and bijective Functions defined in simple way. It also has examples of graphs of Functions.
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The presentation is about some information of Math softwares, and Apps that are helpful for teachers and students. It focuses on teaching Maths in different ways to enhance Math learning skills.
The presentation mainly focus on effective Math
Study skills, beginning with general study skills, and even when and how to begin your studies. It also tells ways to memorize formulae using mnemonics.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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2. FRACTALS???
For centuries, mathematicians rejected
complex figures, leaving them under a
single description: “formless”. For
centuries, geometry was unable to
describe trees, landscapes, clouds, and
coastlines. However, in the late 1970’s a
revolution of our perception of the world
was brought by the work of Benoit
Mandelbrot who introduced FRACTALS.
4. “Fractua” means
Irregular
Fractals are geometric figures like
circles, squares, triangles etc., but
having special properties. They are
usually associated with irregular
geometric objects, that look the same
no matter at what scale they are
viewed at.
A fractal is an object in which the
individual parts are similar to the
whole.
5. Fractals exhibit
self-similarity
Fractals have the property of self-
similarity, generated by iterations,
which means that various copies of
an object can be found in the original
object at smaller size scales.
The detail continues for many
magnifications -- like an endless
nesting of Russian dolls within dolls.
7. What is a Fractal?
A fractal is a rough or fragmented geometric
shape that can be subdivided into parts, each of
which is (at least approximately) a reduced-size
copy of the whole. The core ideas behind it are of
feedback and iteration. The creation of most
fractals involves applying some simple rule to a
set of geometric shapes or numbers and then
repeating the process on the result. This feedback
loop can result in very unexpected results, given
the simplicity of the rules followed for each
iteration.
Fractals have finite area but infinite perimeter.
8. Examples of Fractals
A cauliflower is a perfect example of a
fractal where each element is a
perfect recreation of the whole.
9. A naturally occurring Cauliflower
Fractal
Take a close look at a cauliflower:
Take a closer look at a single floret
(break one off near the base of your
cauliflower). It is a mini cauliflower with
its own little florets all arranged in spirals
around a centre.
10. Computer-generated
Fractal patterns
These days computer-generated
fractal patterns are everywhere. From
squiggly designs on computer art
posters to illustrations in the most
serious of physics journals, interest
continues to grow among scientists
and, rather surprisingly, artists and
designers.
27. The Sierpinski Triangle
Let's make a famous fractal called the
Sierpinski Triangle.
Step One Draw an equilateral triangle with
sides of 2 triangle lengths each.
Connect the midpoints of each side.
How many equilateral triangles do you now
have?
28. Cut out the triangle in the
center.
Step Two
Draw another equilateral triangle with sides
of 4 triangle lengths each. Connect the
midpoints of the sides and cut out the
triangle in the center as before.
30. The Sierpinski Triangle
Unlike the Koch Snowflake, which is
generated with infinite additions, the
Sierpinski triangle is created by infinite
removals. Each triangle is divided into four
smaller, upside down triangles. The center
of the four triangles is removed. As this
process is iterated an infinite number of
times, the total area of the set tends to
infinity as the size of each new triangle
goes to zero.
33. Theory of Fractals
Mandelbrot introduced and
developed the theory of fractals --
figures that were truly able to describe
these shapes. The theory was
continued to be used in a variety of
applications. Fractals’ importance is in
areas ranging from special TV effects
to economy and biology.
34. The term fractal was coined by
Benoit Mandelbrot in 1975 in his book
Fractals: Form, Chance, and
Dimension. In 1979, while studying
the Julia set, Mandelbrot discovered
what is now called the Mandelbrot
set and inspired a generation of
mathematicians and computer
programmers in the study of fractals
and fractal geometry.
Mandelbrot’s
discovery
35. The Mandelbrot Set
Named after Benoit Mandelbrot, The
Mandelbrot set is one of the most
famous fractals in existence. It was
born when Mandelbrot was playing
with the simple quadratic equation
z=z2+c. In this equation, both z and c
are complex numbers. In other words,
the Mandelbrot set is the set of all
complex c such that iteration z=z2+c
does not diverge.
40. The Julia set
The Julia set is another very famous
fractal, which happens to be very
closely related to the Mandelbrot set.
It was named after Gaston Julia, who
studied the iteration of polynomials
and rational functions during the early
twentieth century, making the Julia
set much older than the Mandelbrot
set.
41. Difference between the Julia set
and the Mandelbrot set
The main difference between the Julia set
and the Mandelbrot set is the way in which
the function is iterated. The Mandelbrot set
iterates z=z2+c with z always starting at 0
and varying the c value. The Julia set
iterates z=z2+c for a fixed c value and
varying z values. In other words, the
Mandelbrot set is in the parameter space,
or the c-plane, while the Julia set is in the
dynamical space, or the z-plane.
43. Lorenz Model
The Lorenz Model, named after
E. N. Lorenz in 1963, is a model for
the convection of thermal energy.
This model was the very first example
of another important point in chaos
and fractals, dissipative dynamical
systems, otherwise know as strange
attractors.
49. Objects in Nature
Many objects in nature aren’t formed of
squares or triangles, but of more
complicated geometric figures. e.g. trees,
ferns, clouds, mountains etc. are shaped
like fractals. Other examples include snow
flakes, crystals, lightning, river networks,
cauliflower or broccoli, and systems of
blood vessels and pulmonary vessels.
Coastlines may also be considered as
fractals in nature.
50. Similarity between fractals and
objects in nature.
One of the largest relationships with real-
life is the similarity between fractals and
objects in nature. The resemblance of
many fractals and their natural counter-
parts is so large that it cannot be
overlooked. Mathematical formulas are
used to model self similar natural forms.
The pattern is repeated at a large scale
and patterns evolve to mimic large scale
real world objects.
51. Fractals in Nature
As fractals are patterns that reveal greater
complexity as it is enlarged, they portray
the notion of worlds within worlds.
Trees and ferns are fractals in nature and
can be modeled on a computer by using a
recursive algorithm. This recursive nature
is obvious in these examples—a branch
from a tree or a frond from a fern is a
miniature replica of the whole: not identical,
but similar in nature. The connection
between fractals and leaves are currently
being used to determine how much carbon
is contained in trees.
65. Natural fractal pattern - air
displacing a vacuum formed by
pulling two glue-covered acrylic
sheets apart.
66. Fractal Geometry
Fractal geometry is a new language
used to describe, model and analyze
complex forms found in nature. Chaos
science uses this fractal geometry.
Fractal geometry and chaos theory
are providing us with a new way to
describe the world.
67. Fractal Geometry
While the classical Euclidean
geometry works with objects which
exist in integer dimensions, fractal
geometry deals with objects in non-
integer dimensions. Euclidean
geometry is a description for lines,
ellipses, circles, etc. Fractal
geometry, however, is described in
algorithms -- a set of instructions on
how to create a fractal.
68. Applications of fractals in science
Fractals have a variety of applications in science
because its property of self similarity exists
everywhere. They can be used to model plants,
blood vessels, nerves, explosions, clouds,
mountains, turbulence, etc. Fractal geometry
models natural objects more closely than does
other geometries.
Engineers have begun designing and constructing
fractals in order to solve practical engineering
problems. Fractals are also used in computer
graphics and even in composing music.
70. Application of fractals and chaos is
in music
Some music, including that of Back
and Mozart, can be stripped down so
that is contains as little as 1/64th of its
notes and still retain the essence of
the composer. Many new software
applications are and have been
developed which contain chaotic
filters, similar to those which change
the speed, or the pitch of music.
72. Special features of fractals
A fractal often has the following features:
It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in
traditional Euclidean geometric language.
It is self-similar (at least approximately or
stochastically).
It has a Hausdorff dimension which is greater than
its topological dimension (although this
requirement is not met by space-filling curves such
as the Hilbert curve).
It has a simple and recursive definition.
73. Application to biological analysis
Fractal geometry also has an application to
biological analysis. Fractal and chaos phenomena
specific to non-linear systems are widely observed
in biological systems. A study has established an
analytical method based on fractals and chaos
theory for two patterns: the dendrite pattern of
cells during development in the cerebellum and
the firing pattern of intercellular potential. Variation
in the development of the dendrite stage was
evaluated with a fractal dimension. The order in
many ion channels generating the firing pattern
was also evaluated with a fractal dimension,
enabling the high order seen there to be
quantized.
76. Real-Life Relevance And
Importance of Fractals and Fractal
Geometry
Fractals have and are being used in
many different ways. Both artist and
scientist are intrigued by the many
values of fractals.
Fractals are being used in applications
ranging from image compression to
finance. We are still only beginning to
realize the full importance and
usefulness of fractal geometry.
78. Fractals in Finance
Finance played a crucial role in the
development of fractal theory.
Fractals are used in finance to make
predictions as to the risk involved for
particular stocks.
79.
80. Why does it matter?
How is the stock market associated with a
fractal? Easily, if one looks at the market
price action taking place on the monthly,
weekly, daily and intra day charts where
you will see the structure has a similar
appearance. Followers of this approach
have determined that market prices are
highly random but with a trend. They claim
that stock market success will happen only
by following the trend.
81. Applications of fractals
One of the most useful applications of
fractals and fractal geometry is in image
compression. It is also one of the more
controversial ideas. The basic concept
behind fractal image compression is to
take an image and express it as an
iterated system of functions. The image
can be quickly displayed, and at any
magnification with infinite levels of fractal
detail. The largest problem behind this
idea is deriving the system of functions
which describe an image.
82. Fractals in Film Industry
One of the more trivial applications of
fractals is their visual effect. Not only do
fractals have a stunning aesthetic value,
that is, they are remarkably pleasing to the
eye, but they also have a way to trick the
mind. Fractals have been used
commercially in the film industry, in films
such as Star Wars and Star Trek. Fractal
images are used as an alternative to costly
elaborate sets to produce fantasy
landscapes.
83. Other Applications of Fractals
As described above, random fractals can be used
to describe many highly irregular real-world
objects. Other applications of fractals include:
Classification of histopathology slides in medicine
Fractal landscape or Coastline complexity
Enzyme/enzymology (Michaelis-Menten kinetics)
Generation of new music
Signal and image compression
Creation of digital photographic enlargements
Seismology
Fractal in soil mechanics
84. Computer and video game design, especially
computer graphics for organic environments and
as part of procedural generation
Fractography and fracture mechanics
Fractal antennas – Small size antennas using
fractal shapes
Small angle scattering theory of fractally rough
systems
T-shirts and other fashion
Generation of patterns for camouflage, such as
MARPAT
Digital sundial
Technical analysis of price series (see Elliott wave
principle)
85. Applications of Fractals in C.Sc.
fractal techniques for data analysis
fractals and databases, data mining
visualization and physical models
automatic object classification
fractal and multi-fractal texture
characterization
shape generation, rendering techniques
and image synthesis
2D, 3D fractal interpolation
image denoising and restoration
image indexing, thumbnail images
86. fractal still image and video compression,
wavelet and fractal transforms,
benchmarking, hardware
watermarking, comparison with other
techniques
biomedical applications
engineering (mechanical & materials,
automotive)
fractal and compilers, VLSI design
internet traffic characterization and
modeling
non classical applications