Fractals and symmetry are closely related concepts in mathematics. Fractals exhibit a repeating pattern that displays at every scale and often have non-integer fractal dimensions. Some key fractals include the Mandelbrot set, Julia set, and Koch snowflake. Symmetry refers to an object having identical corresponding parts on opposite sides of a dividing line or plane. There are several types of symmetry such as reflection, rotation, translation, and scale symmetry which fractals demonstrate. Benoit Mandelbrot coined the term "fractal" and studied their properties, finding their applications in fields like art, geography, and physics.
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement")[1] in everyday language refers to a sense of harmonious and beautiful proportion and balance.[2][3][a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.[4][b]
This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry.
In geometry[edit]
Main article: Symmetry (geometry)
The triskelion has 3-fold rotational symmetry.
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.[5] This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
An object has reflectional symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other.[6]
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape.[7]
An object has translational symmetry if it can be translated without changing its overall shape.[8]
An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.[9]
An object has scale symmetry if it does not change shape when it is expanded or contracted.[10] Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions.[11]
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.
Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement")[1] in everyday language refers to a sense of harmonious and beautiful proportion and balance.[2][3][a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.[4][b]
This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry.
In geometry[edit]
Main article: Symmetry (geometry)
The triskelion has 3-fold rotational symmetry.
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion.[5] This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
An object has reflectional symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other.[6]
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape.[7]
An object has translational symmetry if it can be translated without changing its overall shape.[8]
An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.[9]
An object has scale symmetry if it does not change shape when it is expanded or contracted.[10] Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions.[11]
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
3. Markus Reugels
• A photographer who showed
that beauty can exist in places
we don’t expect it to be.
• Most of his photographs are
close-ups of water droplets
and the water crown which
features a special geometric
figure called the crown is
formed from splashing water.
6. The First
• Is an imprecise sense of
harmony and beauty or
balance and proportion.
7. The Second
• Is a well-defined concept of
balance or patterned self-
similarity that can be
proved by geometry or
through physics.
8. Odd and Even Functions
Inverse Functions
Rotoreflection Glide Reflection Religious Symbols Mathematics
Rotation Scale/Fractals Logic
Reflection Geometry Helical
Translation
Social Interactions
Symmetry
Arts/Aesthetics
Passage through time
Science Music
Architecture
Spatial relationships
Knowledge
10. Symmetry in Geometry
• “The exact correspondence of form
and constituent configuration on
opposite sides of a dividing line or
plane or about a center or an axis”
(American Heritage® Dictionary of
the English Language 4th ed., 2009)
• In simpler terms, if you draw a
specific point, line or plane on an
object, the first side would have the
same correspondence to its
respective other side.
11. Reflection Symmetry
• Symmetry with respect
to an axis or a line.
• A line can be drawn of
the object such that
when one side is flipped
on the line, the object
formed is congruent to
the original object, vice
versa.
12. The location of the line matters
True Reflection Symmetry False Reflection Symmetry
13. Rotational Symmetry
• Symmetry with respect to the figure’s center
• An axis can be put on the object such that if the
figure is rotated on it, the original figure will appear
more than once
• The number of times the figure appears in one
complete rotation is called its order.
14. Figures and their order
Order 2 Order 4 Order 6 Order 5
Order 8 Order 3 Order 7
15. Other types of Symmetry
• Translational symmetry
– looks the same after a particular translation
• Glide reflection symmetry
– reflection in a line or plane combined with a translation along the line / in the plane,
results in the same object
• Rotoreflection symmetry
– rotation about an axis (3D)
• Helical symmetry
– rotational symmetry along with translation along the axis of rotation called the screw
axis
• Scale symmetry
– the new object has the same properties as the original if an object is expanded or
reduced in size
– present in most fractals
16. Symmetry in Math
• Symmetry is present in even • Symmetry is present in odd
functions – they are functions as well – they are
symmetrical along the y-axis symmetrical with respect to
the origin. They have order
2 rotational symmetry.
cos(θ) = cos(- θ) sin(-θ) = -sin( θ)
17. Symmetry in Math
• Functions and their inverses
exhibit reflection wrt the
line with the equation x = y
• f(f-1(x)) = f-1(f(x)) = x
ln(������ x) = xln(������) = x(1) = x
18. Time is symmetric in the sense that if it is
reversed the exact same events are
happening in reverse order thus making it
symmetric. Time can be reversed but it is
not possible in this universe because it
would violate the second law of
thermodynamics.
THIS WON’T APPEAR IN THE QUIZ
Passage of time
Perception of time is different from any
given object. The closer the objects
travels to the speed of light, the slower
the time in its system gets or he faster its
perception of time would be. This means
it could only be possible to have a reverse
perception of time on a specific system
but not a reverse perception on the entire
system.
24. Etymology
• Fractal came from the Latin
word fractus which means
“interrupted”, or “irregular”
• Fractals are generally self-
similar patterns and a
detailed example of scale
symmetry.
Julian Fractal
25. History
• Mathematics behind fractals
started in the early 17th cenury
when Gottfried Leibniz, a
mathematician and philosopher,
pondered recursive self-
similarity.
• His thinking was wrong since he
only considered a straight line to
be self-similar.
26. History
• In 1872, Karl Weiestrass
presented the first definition of a
function with a graph that can be
considered a fractal.
• Helge von Koch, in 1904,
developed an accurate geometric
definition by repeatedly trisecting
a straight line. This was later
known as the Koch curve.
27. History
• In 1915, Waclaw Sierpinski
costructed the Sierpinski Triangle.
• By 1918, Pierre Fatou ad Gaston
Julia, described fractal behaviour
associated with mapping complex
numbers. This also lead to ideas
about attractors and repellors an
eventually to the development of
the Julia Set.
28. Benoît Mandelbrot
• A mathematician who created
the Mandelbrot set from
studying the behavior of the
Julia Set.
• Coined the term “fractal”
Mandelbrot Set
29. What is a fractal?
• A fractal is a
mathematical set that
has a fractal dimension
that usually exceeds its
topological dimension.
And may fall between
integers.
Fibonacci word by Samuel Monnier
30. Iteration
• Iteration is the repetition of
an algorithm to achieve a
target result. Some basic
fractals follow simple
iterations to achieve the
correct figure.
First four iterations of the Koch Snowflake
31. Whut?
• Let’s look at the line on the right, when
it is divided by 2, the number of self-
similar pieces becomes 2. When
divided by 3, the number of self-similar
pieces becomes 3.
A formula is given to calculate the
dimension of a given object:
log ������)
(
log ������)
(
where N = number of self-similar pieces
������ = scaling factor
We can now substitute:
log 2
=1
log 2
33. Sierpinski Triangle
Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5
• Clue: Iteration 1 has an ������ of 1, Iteration 2 has an ������ of 2, Iteration 3 has
an ������ of 4 and so on.
• Answer:
log 3
= 1.584962500~1.58
log 2
That means that the Sierpinski triangle has a fractal dimension of about
1.58. How could that be? Mathematically, that is its dimension but our
eyes see an infinitely complex figure.
34. Types of Self-Similarity
Exact Self-similarity Quasi Self-similarity
• Identical at all scales • Approximates the same
• Example: Koch snowflake pattern at different scales
although the copy might be
distorted or in degenerate
form.
• Example: Mandelbrot’s Set
35. Types of Self-Similarity
Statistical Self-Similarity
• Repeats a pattern
stochastically so numerical
or statistical measures are
preserved across scales.
• Example: Koch Snowflake