SlideShare une entreprise Scribd logo
1  sur  20
Télécharger pour lire hors ligne
Fractals: A Brief Overview
Harlan J. Brothers, Director of Technology
The Country School Madison, CT



This presentation provides a broad and basic introduction to the
subject of fractal geometry.

My thanks to Michael Frame at Yale University for the use of
many of the photos and graphics that appear here. His
fascinating and comprehensive treatment of the subject can be
found at:

   http://classes.yale.edu/Fractals/ .
Familiar Symmetries


We commonly recognize when shapes demonstrate symmetry
under the three familiar transformations of reflection, rotation,
and translation.




      Reflection            Rotation             Translation
Scaling Symmetry

Fractals demonstrate a fourth type of symmetry; they possess
“self-similarity.”

Self-similar objects appear the same under magnification. They
are, in some fashion, composed of smaller copies of themselves.
This characteristic is often referred to as “scaling symmetry” or
“scale invariance.”




                         Sierpinski Gasket
Scaling Symmetry


Not all self-similarity, however, is of a fractal nature. Objects like
spirals and nested dolls that are self-similar around a single point
are NOT fractal.




               Not fractal                 Not fractal
Fractals

In the broadest sense, fractals can be divided into two
categories:

       objects that occur in Nature, and
       mathematical constructions.
Fractals in Nature


Natural objects exhibit scaling
symmetry, but only over a
limited range of scales. They
also tend to be “roughly” self-
similar, appearing more or less
the same at different scales of
measurement.          Sometimes
this means that they are
statistically self-similar; that is
to say, they have a distribution
of elements that is similar
under magnification.
Fractals in Nature




         Trees       Ferns
Fractals in Nature




                     Mountains
Fractals in Nature




  Coastline and snow fields of Norway   Waterfall
Fractals in Nature




                     Clouds
Fractals in Nature




        Bacterial colony         Lightening
       (courtesy E. Ben-Jacob)
Mathematical Constructions

In contrast to naturally occurring fractals, mathematical fractals
can possess an infinite range of scaling symmetry. The more
common constructions also tend to be exactly self-similar.




                          Koch Curve

 The Koch curve above is composed of exactly four copies of
 itself. Can you construct it from just two?
Mathematical Examples




Sierpinski Gasket

                                     Menger Sponge



                    Mandelbrot Set




      Cantor Comb                    Koch Snowflake
Scale Invariance


The fact that a fractal object is, in some sense, composed of
smaller copies of itself, has interesting implications. One of
these is that when we examine a fractal shape without a suitable
frame of reference, it is often impossible to tell the scale of
magnification at which it is being viewed.

For natural phenomena, this translates to uncertainty with
respect to the distance, extent, or size of the object. We end
with two examples of scale invariance.

What do the following two images look like to you?
Image 1
Image 2
Image 1   (another view)
Image 2   (another view)
Discussion

Depending on who you ask, the preceding images may look like
satellite or aerial photos, rock formations, or photomicrographs.

This simply illustrates the fact that certain natural processes, like
erosion or the formation ice crystals, follow patterns that can be
repeated at many scales of measurement. Without a frame of
reference, a photograph of a rock sitting one meter away can
effectively look the same as a boulder several meters away or a
cliff hundreds of meters distant.

With a knowledgeable eye, one sees a natural world that
abounds in fractal shapes.
Author Information

Harlan J. Brothers
Director of Technology
The Country School
Madison, CT 06443
Tel. (203) 421-3113
E-mail: harlan@thecountryschool.org

Contenu connexe

Tendances (20)

Fractals
FractalsFractals
Fractals
 
Fractal geometry
Fractal geometryFractal geometry
Fractal geometry
 
Mandelbrot
MandelbrotMandelbrot
Mandelbrot
 
Fractals
FractalsFractals
Fractals
 
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHI
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHIFRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHI
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHI
 
Fractals in physics
Fractals in physicsFractals in physics
Fractals in physics
 
Secrets of fractals dfs-yuc
Secrets of fractals dfs-yucSecrets of fractals dfs-yuc
Secrets of fractals dfs-yuc
 
Fractals
FractalsFractals
Fractals
 
Fractal Theory
Fractal TheoryFractal Theory
Fractal Theory
 
Linear Algebra Applications
Linear Algebra ApplicationsLinear Algebra Applications
Linear Algebra Applications
 
Fibonacci series and Golden ratio
Fibonacci  series and Golden ratioFibonacci  series and Golden ratio
Fibonacci series and Golden ratio
 
Applications of algebra and calculus
Applications of algebra and calculusApplications of algebra and calculus
Applications of algebra and calculus
 
Application of matrices in real life and matrix
Application of matrices in real life and matrixApplication of matrices in real life and matrix
Application of matrices in real life and matrix
 
Application of Matrices in real life
Application of Matrices in real lifeApplication of Matrices in real life
Application of Matrices in real life
 
FRACTALES
FRACTALESFRACTALES
FRACTALES
 
Graph theory
Graph theoryGraph theory
Graph theory
 
Maths Project Power Point Presentation
Maths Project Power Point PresentationMaths Project Power Point Presentation
Maths Project Power Point Presentation
 
Application of Matrix
Application of MatrixApplication of Matrix
Application of Matrix
 
application of complex numbers
application of complex numbersapplication of complex numbers
application of complex numbers
 
Application of Matrices in real life | Matrices application | The Matrices
Application of Matrices in real life | Matrices application | The MatricesApplication of Matrices in real life | Matrices application | The Matrices
Application of Matrices in real life | Matrices application | The Matrices
 

Similaire à Fractals

Fractals and symmetry by group 3
Fractals and symmetry by group 3Fractals and symmetry by group 3
Fractals and symmetry by group 3Leiko Ravelo
 
Fractals and symmetry group 3
Fractals and symmetry   group 3Fractals and symmetry   group 3
Fractals and symmetry group 3Leiko Ravelo
 
Emergence and Reduction in Physics
Emergence and Reduction in PhysicsEmergence and Reduction in Physics
Emergence and Reduction in PhysicsSebastian De Haro
 
Order, Chaos and the End of Reductionism
Order, Chaos and the End of ReductionismOrder, Chaos and the End of Reductionism
Order, Chaos and the End of ReductionismJohn47Wind
 
Project math in nature
Project math in natureProject math in nature
Project math in nature9562
 
Mathemativs in the modern World.pptx
Mathemativs in the modern World.pptxMathemativs in the modern World.pptx
Mathemativs in the modern World.pptxReginHayagan
 
Fractals And Chaos Theory
Fractals And Chaos TheoryFractals And Chaos Theory
Fractals And Chaos TheoryFNian
 
hidden dimension in nature
hidden dimension in naturehidden dimension in nature
hidden dimension in natureMilan Joshi
 
HIDDEN DIMENSIONS IN NATURE
HIDDEN DIMENSIONS IN NATUREHIDDEN DIMENSIONS IN NATURE
HIDDEN DIMENSIONS IN NATUREMilan Joshi
 
Hidden dimensions in nature
Hidden dimensions in natureHidden dimensions in nature
Hidden dimensions in natureMilan Joshi
 
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1zukun
 
Maths in nature (complete)
Maths in nature (complete)Maths in nature (complete)
Maths in nature (complete)Abhay Goyal
 
An Exploration of Fractal Geometry
An Exploration of Fractal GeometryAn Exploration of Fractal Geometry
An Exploration of Fractal GeometryM H
 
Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in natureJovin John
 
4p70chap1lect1
4p70chap1lect14p70chap1lect1
4p70chap1lect1Ennic_456
 

Similaire à Fractals (20)

Fractals and symmetry by group 3
Fractals and symmetry by group 3Fractals and symmetry by group 3
Fractals and symmetry by group 3
 
Fractals and symmetry group 3
Fractals and symmetry   group 3Fractals and symmetry   group 3
Fractals and symmetry group 3
 
Emergence and Reduction in Physics
Emergence and Reduction in PhysicsEmergence and Reduction in Physics
Emergence and Reduction in Physics
 
Order, Chaos and the End of Reductionism
Order, Chaos and the End of ReductionismOrder, Chaos and the End of Reductionism
Order, Chaos and the End of Reductionism
 
Project math in nature
Project math in natureProject math in nature
Project math in nature
 
Mathemativs in the modern World.pptx
Mathemativs in the modern World.pptxMathemativs in the modern World.pptx
Mathemativs in the modern World.pptx
 
Fractals And Chaos Theory
Fractals And Chaos TheoryFractals And Chaos Theory
Fractals And Chaos Theory
 
hidden dimension in nature
hidden dimension in naturehidden dimension in nature
hidden dimension in nature
 
HIDDEN DIMENSIONS IN NATURE
HIDDEN DIMENSIONS IN NATUREHIDDEN DIMENSIONS IN NATURE
HIDDEN DIMENSIONS IN NATURE
 
Hidden dimensions in nature
Hidden dimensions in natureHidden dimensions in nature
Hidden dimensions in nature
 
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1
MIT6.870 Grounding Object Recognition and Scene Understanding: lecture 1
 
Maths in nature (complete)
Maths in nature (complete)Maths in nature (complete)
Maths in nature (complete)
 
Fractals a research
Fractals a researchFractals a research
Fractals a research
 
An Exploration of Fractal Geometry
An Exploration of Fractal GeometryAn Exploration of Fractal Geometry
An Exploration of Fractal Geometry
 
Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in nature
 
4p70chap1lect1
4p70chap1lect14p70chap1lect1
4p70chap1lect1
 
Devy Essay
Devy EssayDevy Essay
Devy Essay
 
Applied Biochemistry
Applied BiochemistryApplied Biochemistry
Applied Biochemistry
 
BSc dissertation np
BSc dissertation npBSc dissertation np
BSc dissertation np
 
Mathematics in nature
Mathematics in natureMathematics in nature
Mathematics in nature
 

Plus de Maria Menendez

Plus de Maria Menendez (17)

Nº oro
Nº oroNº oro
Nº oro
 
Mujeres matem^^ticas
Mujeres matem^^ticasMujeres matem^^ticas
Mujeres matem^^ticas
 
Matematicas y el arte(1)
Matematicas y el arte(1)Matematicas y el arte(1)
Matematicas y el arte(1)
 
Número e
Número eNúmero e
Número e
 
Matemáticas en las series
Matemáticas en las seriesMatemáticas en las series
Matemáticas en las series
 
Astor y sergio
Astor y sergioAstor y sergio
Astor y sergio
 
Musica y matematicas
Musica y matematicasMusica y matematicas
Musica y matematicas
 
Matematicas y musica
Matematicas y musica Matematicas y musica
Matematicas y musica
 
Leonhard Euler
Leonhard Euler Leonhard Euler
Leonhard Euler
 
Fractales
Fractales Fractales
Fractales
 
Fibonacci
Fibonacci Fibonacci
Fibonacci
 
Mujeres matematicas lilia
Mujeres matematicas liliaMujeres matematicas lilia
Mujeres matematicas lilia
 
La música en las matemáticas
La música en las matemáticas La música en las matemáticas
La música en las matemáticas
 
La música en las matemáticas
La música en las matemáticasLa música en las matemáticas
La música en las matemáticas
 
Elementos postulados jeny
Elementos postulados jenyElementos postulados jeny
Elementos postulados jeny
 
La geometria
La geometriaLa geometria
La geometria
 
Presentación fractales
Presentación fractalesPresentación fractales
Presentación fractales
 

Dernier

How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17Celine George
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfMohonDas
 
Philosophy of Education and Educational Philosophy
Philosophy of Education  and Educational PhilosophyPhilosophy of Education  and Educational Philosophy
Philosophy of Education and Educational PhilosophyShuvankar Madhu
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice documentXsasf Sfdfasd
 
CapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapitolTechU
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 
Human-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesHuman-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesMohammad Hassany
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxKatherine Villaluna
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...CaraSkikne1
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphNetziValdelomar1
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsEugene Lysak
 
Quality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICEQuality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICESayali Powar
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 
3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptxmary850239
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationMJDuyan
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxDr. Santhosh Kumar. N
 
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptxSandy Millin
 

Dernier (20)

How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdf
 
Philosophy of Education and Educational Philosophy
Philosophy of Education  and Educational PhilosophyPhilosophy of Education  and Educational Philosophy
Philosophy of Education and Educational Philosophy
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice document
 
CapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptx
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 
Human-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming ClassesHuman-AI Co-Creation of Worked Examples for Programming Classes
Human-AI Co-Creation of Worked Examples for Programming Classes
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
Practical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptxPractical Research 1 Lesson 9 Scope and delimitation.pptx
Practical Research 1 Lesson 9 Scope and delimitation.pptx
 
5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...5 charts on South Africa as a source country for international student recrui...
5 charts on South Africa as a source country for international student recrui...
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a Paragraph
 
The Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George WellsThe Stolen Bacillus by Herbert George Wells
The Stolen Bacillus by Herbert George Wells
 
Quality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICEQuality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICE
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 
3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptx
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive Education
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptx
 
Prelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quizPrelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quiz
 
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
 

Fractals

  • 1. Fractals: A Brief Overview Harlan J. Brothers, Director of Technology The Country School Madison, CT This presentation provides a broad and basic introduction to the subject of fractal geometry. My thanks to Michael Frame at Yale University for the use of many of the photos and graphics that appear here. His fascinating and comprehensive treatment of the subject can be found at: http://classes.yale.edu/Fractals/ .
  • 2. Familiar Symmetries We commonly recognize when shapes demonstrate symmetry under the three familiar transformations of reflection, rotation, and translation. Reflection Rotation Translation
  • 3. Scaling Symmetry Fractals demonstrate a fourth type of symmetry; they possess “self-similarity.” Self-similar objects appear the same under magnification. They are, in some fashion, composed of smaller copies of themselves. This characteristic is often referred to as “scaling symmetry” or “scale invariance.” Sierpinski Gasket
  • 4. Scaling Symmetry Not all self-similarity, however, is of a fractal nature. Objects like spirals and nested dolls that are self-similar around a single point are NOT fractal. Not fractal Not fractal
  • 5. Fractals In the broadest sense, fractals can be divided into two categories: objects that occur in Nature, and mathematical constructions.
  • 6. Fractals in Nature Natural objects exhibit scaling symmetry, but only over a limited range of scales. They also tend to be “roughly” self- similar, appearing more or less the same at different scales of measurement. Sometimes this means that they are statistically self-similar; that is to say, they have a distribution of elements that is similar under magnification.
  • 7. Fractals in Nature Trees Ferns
  • 8. Fractals in Nature Mountains
  • 9. Fractals in Nature Coastline and snow fields of Norway Waterfall
  • 11. Fractals in Nature Bacterial colony Lightening (courtesy E. Ben-Jacob)
  • 12. Mathematical Constructions In contrast to naturally occurring fractals, mathematical fractals can possess an infinite range of scaling symmetry. The more common constructions also tend to be exactly self-similar. Koch Curve The Koch curve above is composed of exactly four copies of itself. Can you construct it from just two?
  • 13. Mathematical Examples Sierpinski Gasket Menger Sponge Mandelbrot Set Cantor Comb Koch Snowflake
  • 14. Scale Invariance The fact that a fractal object is, in some sense, composed of smaller copies of itself, has interesting implications. One of these is that when we examine a fractal shape without a suitable frame of reference, it is often impossible to tell the scale of magnification at which it is being viewed. For natural phenomena, this translates to uncertainty with respect to the distance, extent, or size of the object. We end with two examples of scale invariance. What do the following two images look like to you?
  • 17. Image 1 (another view)
  • 18. Image 2 (another view)
  • 19. Discussion Depending on who you ask, the preceding images may look like satellite or aerial photos, rock formations, or photomicrographs. This simply illustrates the fact that certain natural processes, like erosion or the formation ice crystals, follow patterns that can be repeated at many scales of measurement. Without a frame of reference, a photograph of a rock sitting one meter away can effectively look the same as a boulder several meters away or a cliff hundreds of meters distant. With a knowledgeable eye, one sees a natural world that abounds in fractal shapes.
  • 20. Author Information Harlan J. Brothers Director of Technology The Country School Madison, CT 06443 Tel. (203) 421-3113 E-mail: harlan@thecountryschool.org