This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Benoit Mandelbrot and have self-similar patterns seen at different scales. Examples of fractals in nature are given like coastlines, mountains, trees. The key fractals discussed are the Mandelbrot set and how it is defined mathematically. Applications of fractals mentioned include use in special effects in movies like Star Trek, modeling weather patterns, landscapes, human anatomy.
Fractal geometry is a new branch of geometry that was developed in the late 1970s and early 1980s. Unlike classical geometry which uses integer dimensions, fractal geometry uses fractional dimensions to describe objects. This allowed fractal geometry to more accurately model objects in nature which often have self-similar patterns at different scales. Fractal geometry has since found many applications and has fascinated many people due to the beautiful patterns it produces.
Sierpinski Triangle
This is the fractal we can get by taking the midpoints of each side of an equilateral triangle and connecting them. The iterations should be repeated an infinite number of times
Suppose the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat again:
COASTLINE PARADOX
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHIMILANJOSHIJI
This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Mandelbrot as a way to describe irregular patterns in nature. Some key areas where fractals are applied that are mentioned include astronomy (galaxies and Saturn's rings), biology (bacteria cultures, plants), data compression, diffusion, economy, special effects in movies like Star Trek, weather patterns, antennas, and understanding global warming. The document also provides mathematical definitions of fractals and the Mandelbrot set, and describes how to graph the Mandelbrot set. Examples of fractals discussed in more depth include the Sierpinski triangle, Koch curve, broccoli, veins, and the Lorenz
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.
Fractal introduction and applications modified versionAnkit Garg
This document discusses fractals and their properties. It provides examples of fractals commonly found in nature, such as trees, ferns, mountains and coastlines. These natural formations cannot be easily modeled using classical geometry. Fractals are defined by their self-similarity, meaning identical patterns repeat across different scales. Common fractals like the Sierpinski triangle and Koch curve are constructed through an iterative process of replacing shapes with smaller copies. Fractals can also be generated using an iterated function system by substituting shapes repeatedly according to transformation rules.
Fractals are irregular patterns that are self-similar across different scales. They are a branch of mathematics concerned with shapes found in nature that have non-integer dimensions. Some early contributors to fractal concepts included Leibiz, Cantor, and Hausdorff, but it was Mandelbrot who coined the term "fractal" and brought greater attention to the field. One of the most basic and important fractals is the Mandelbrot set, which is a set of complex numbers that fulfill certain recursive conditions. Fractals can also be classified based on their level of self-similarity, from exact to statistical self-similarity. Fractals are found throughout nature and have applications in fields like astrophysics
This document provides an overview of fractals, including a brief history, examples of specific fractals (Sierpinski triangle and Koch snowflake) and their properties (dimension, area, perimeter), and appearances and applications of fractals in technology and medicine. It outlines the presentation which includes introductions to self-similarity, dimension, early contributors to fractal geometry like Mandelbrot, examples of fractals and calculations of their dimensions, areas, and perimeters, as well as naturally occurring fractals and uses of fractals in technology and medicine.
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.
Fractal geometry is a new branch of geometry that was developed in the late 1970s and early 1980s. Unlike classical geometry which uses integer dimensions, fractal geometry uses fractional dimensions to describe objects. This allowed fractal geometry to more accurately model objects in nature which often have self-similar patterns at different scales. Fractal geometry has since found many applications and has fascinated many people due to the beautiful patterns it produces.
Sierpinski Triangle
This is the fractal we can get by taking the midpoints of each side of an equilateral triangle and connecting them. The iterations should be repeated an infinite number of times
Suppose the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast. Now cut the ruler in half and repeat the measurement, then repeat again:
COASTLINE PARADOX
FRACTAL GEOMETRY AND ITS APPLICATIONS BY MILAN A JOSHIMILANJOSHIJI
This document provides an introduction to fractal geometry and its applications. It discusses how fractals were discovered by Mandelbrot as a way to describe irregular patterns in nature. Some key areas where fractals are applied that are mentioned include astronomy (galaxies and Saturn's rings), biology (bacteria cultures, plants), data compression, diffusion, economy, special effects in movies like Star Trek, weather patterns, antennas, and understanding global warming. The document also provides mathematical definitions of fractals and the Mandelbrot set, and describes how to graph the Mandelbrot set. Examples of fractals discussed in more depth include the Sierpinski triangle, Koch curve, broccoli, veins, and the Lorenz
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.
Fractal introduction and applications modified versionAnkit Garg
This document discusses fractals and their properties. It provides examples of fractals commonly found in nature, such as trees, ferns, mountains and coastlines. These natural formations cannot be easily modeled using classical geometry. Fractals are defined by their self-similarity, meaning identical patterns repeat across different scales. Common fractals like the Sierpinski triangle and Koch curve are constructed through an iterative process of replacing shapes with smaller copies. Fractals can also be generated using an iterated function system by substituting shapes repeatedly according to transformation rules.
Fractals are irregular patterns that are self-similar across different scales. They are a branch of mathematics concerned with shapes found in nature that have non-integer dimensions. Some early contributors to fractal concepts included Leibiz, Cantor, and Hausdorff, but it was Mandelbrot who coined the term "fractal" and brought greater attention to the field. One of the most basic and important fractals is the Mandelbrot set, which is a set of complex numbers that fulfill certain recursive conditions. Fractals can also be classified based on their level of self-similarity, from exact to statistical self-similarity. Fractals are found throughout nature and have applications in fields like astrophysics
This document provides an overview of fractals, including a brief history, examples of specific fractals (Sierpinski triangle and Koch snowflake) and their properties (dimension, area, perimeter), and appearances and applications of fractals in technology and medicine. It outlines the presentation which includes introductions to self-similarity, dimension, early contributors to fractal geometry like Mandelbrot, examples of fractals and calculations of their dimensions, areas, and perimeters, as well as naturally occurring fractals and uses of fractals in technology and medicine.
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.
Fractals are mathematical objects that have detailed patterns at any scale. They are self-similar, meaning each part is similar to the whole. Benoit Mandelbrot introduced the term "fractal" in 1975 and studied their properties. Fractals have an initiator shape, a construction law to generate iterations, and a process of generation. Common fractals include the Koch snowflake, Sierpinski triangle, Hilbert curve, and Menger sponge. Fractals are important because patterns in nature like coastlines are often fractal, contradicting traditional geometry.
The document discusses fractals and the Mandelbrot set. It defines a fractal as an iterated pattern that displays self-similarity, like the Koch curve or Sierpinski triangle. The Mandelbrot set is the set of complex numbers where the sequence zn+1 = zn^2 + c remains bounded. While the Mandelbrot set itself is not a fractal, the border of the set has a Hausdorff dimension exceeding its topological dimension, making it a fractal.
This document discusses fractal geometry and its applications in materials science. It begins by providing background on fractals and how they were discovered to describe natural patterns. Fractals have fractional dimensions and self-similar patterns across different scales. Non-linear dynamics and chaos theory are then introduced to study irregular patterns in nature. Specific fractal objects like the Cantor set and Koch curve are described. The document outlines how fractal analysis can be used to characterize microstructures, surfaces, cracks and particles in materials using techniques like box counting to determine fractal dimension. Finally, the role of image processing in materials science images for quantitative microstructure analysis is briefly discussed.
Symmetry is the correspondence of form and configuration on opposite sides of a dividing line or plane. It is present in many areas including geometry, mathematics, science, and nature. Fractals are self-similar patterns that repeat at different scales and often have non-integer fractal dimensions. Benoît Mandelbrot coined the term fractal and studied fractals like the Mandelbrot set, which demonstrate iterative processes that lead to increasingly complex patterns appearing as one zooms in. Fractals are used in computer graphics, geography, art and other applications to model real-world irregular forms and processes.
This document provides an overview of fractals including their properties, examples found in nature, and their history. It discusses key fractal concepts such as recursion, self-similarity, and fractal dimension. Methods for generating geometric fractals like the Koch curve and Sierpinski triangle are presented. The document also demonstrates fractal software and discusses ways fractals can be incorporated into K-12 education.
This document provides an overview of fractals, including:
- Fractals demonstrate self-similarity and scaling symmetry where they appear the same at different levels of magnification.
- There are two categories of fractals - those occurring in nature like trees, ferns, mountains, clouds, and bacterial colonies, and mathematical constructions like the Koch curve, Sierpinski gasket, and Mandelbrot set.
- Natural fractals only exhibit self-similarity over a limited range of scales and tend to be roughly rather than exactly self-similar, while mathematical fractals can have perfect self-similarity over infinite scales.
This document discusses different definitions of dimension, focusing on topological dimension and Hausdorff dimension. It provides examples of fractals like the Sierpinski triangle and Koch curve that challenge classical dimension definitions. While topological dimension aligns with intuition, the Hausdorff dimension considers how content scales with measurement. The document explains how fractals like the Sierpinski triangle have topological dimension of 1 but area of 0, requiring a fractional Hausdorff dimension to adequately measure their content. This provides the conceptual basis for Mandelbrot's definition of fractals as sets where the Hausdorff dimension exceeds the topological dimension.
This document discusses chaos theory and fractals. It defines chaos as systems that are highly sensitive to initial conditions. It describes how Edward Lorenz discovered chaos through computer modeling of weather patterns. It explains key concepts like the butterfly effect and Lorenz attractor. It discusses pioneers in fractal geometry like Mandelbrot and how fractals are found throughout nature and can be used in various applications.
- Chaos theory is about understanding complex and nonlinear dynamic systems, not denying determinism or order. It recognizes that small changes can lead to large, unpredictable consequences.
- Fractals are geometric shapes that exhibit self-similarity, where parts of the shape resemble the whole. They are found throughout nature and can be modeled using mathematical equations.
- Pioneered by Mandelbrot, fractal geometry is useful for simulating and understanding natural phenomena like clouds, coastlines, and trees that appear irregular or chaotic but have underlying patterns. It has applications in fields like computer graphics, fluid mechanics, and telecommunications.
Master's Thesis: Closed formulae for distance functions involving ellipses.Gema R. Quintana
This thesis examines computing distances between ellipses and ellipsoids. Chapter 2 derives a closed-form formula for the minimum distance between two coplanar ellipses without calculating footpoints. Chapter 3 computes the closest approach of two arbitrary separated ellipses or ellipsoids over time. Future work includes using ellipses to check safety regions for robot motion and computing Hausdorff distances between ellipses and ellipsoids.
1) The document describes constructing a net for three square-based pyramids that can fold into a cube.
2) It then discusses Cavalieri's principle, which states that solids with the same perpendicular height and cross-sectional areas have the same volume.
3) Examples are given of using Cavalieri's principle to find the volumes of solids by relating them to more familiar solids of the same volume.
This document provides lecture notes on analytic geometry. It begins with an introduction discussing the goals of building an algebraic geometry framework for analytic situations by replacing topological abelian groups with condensed abelian groups. Condensed sets are defined as sheaves on the pro-étale site of the point, and behave like generalized topological spaces. The notes establish that quasiseparated condensed sets correspond to ind-compact Hausdorff spaces. This provides the needed abelian category structure to build an analytic geometry in parallel to algebraic geometry over schemes.
Fractals are geometric shapes that exhibit self-similarity and complex patterns that are repeated at different scales. They can be used to describe irregular shapes found in nature, such as trees, coastlines, and clouds. The document discusses how Benoit Mandelbrot introduced fractals and developed their theory. It provides examples of famous fractals like the Mandelbrot set and Julia set and explains how fractals appear throughout nature and can be modeled using computers. Fractals have applications in fields like mathematics, science, engineering, art and music.
This document discusses geometric modeling and representations. It introduces polygonal and polyhedral models for representing obstacle regions as the union or intersection of half-plane or half-space primitives. Logical predicates can also represent these models using logical ANDs and ORs. Transformations of geometric bodies are also introduced to enable modeling motion.
This document discusses the graphical method for solving linear programming problems (LPP). It describes the steps of the method which include plotting the constraints on a graph, identifying the feasible region, plotting the objective function, and finding the optimal point where it intersects the feasible region. The document provides an example problem and solution. It also discusses different cases that can occur with the optimal solution such as it being unique, unbounded, having multiple solutions, being infeasible, or there being a unique feasible point.
The document discusses different methods to solve assignment problems including enumeration, integer programming, transportation, and Hungarian methods. It provides examples of balanced and unbalanced minimization and maximization problems. The Hungarian method is described as having steps like row and column deduction, assigning zeros, and tick marking to find the optimal assignment with the minimum cost or maximum profit. A sample problem demonstrates converting a profit matrix to a relative cost matrix and using the Hungarian method to find the optimal solution.
This document discusses various applications of graph theory to other areas of mathematics and other fields. It provides new graph theoretical proofs of Fermat's Little Theorem and the Nielsen-Schreier Theorem. It also discusses applications to problems in DNA sequencing, computer network security, scheduling, map coloring, and mobile phone networks. Specific algorithms for finding minimum vertex covers, vertex colorings, and matchings in graphs are applied to solve problems in these areas.
NOTE:Download this file to preview as the Slideshare preview does not display it properly.
This is an introduction to Linear Programming and a few real world applications are included.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
This document provides an overview of linear programming. It discusses basic and basic feasible solutions, the geometric solution, definitions used in linear programming, and the simplex algorithm. It provides an example problem that is solved over multiple iterations using the simplex algorithm to find the optimal solution. Finally, it briefly discusses the primal dual relationship between linear programming problems.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
Fractals are mathematical objects that have detailed patterns at any scale. They are self-similar, meaning each part is similar to the whole. Benoit Mandelbrot introduced the term "fractal" in 1975 and studied their properties. Fractals have an initiator shape, a construction law to generate iterations, and a process of generation. Common fractals include the Koch snowflake, Sierpinski triangle, Hilbert curve, and Menger sponge. Fractals are important because patterns in nature like coastlines are often fractal, contradicting traditional geometry.
The document discusses fractals and the Mandelbrot set. It defines a fractal as an iterated pattern that displays self-similarity, like the Koch curve or Sierpinski triangle. The Mandelbrot set is the set of complex numbers where the sequence zn+1 = zn^2 + c remains bounded. While the Mandelbrot set itself is not a fractal, the border of the set has a Hausdorff dimension exceeding its topological dimension, making it a fractal.
This document discusses fractal geometry and its applications in materials science. It begins by providing background on fractals and how they were discovered to describe natural patterns. Fractals have fractional dimensions and self-similar patterns across different scales. Non-linear dynamics and chaos theory are then introduced to study irregular patterns in nature. Specific fractal objects like the Cantor set and Koch curve are described. The document outlines how fractal analysis can be used to characterize microstructures, surfaces, cracks and particles in materials using techniques like box counting to determine fractal dimension. Finally, the role of image processing in materials science images for quantitative microstructure analysis is briefly discussed.
Symmetry is the correspondence of form and configuration on opposite sides of a dividing line or plane. It is present in many areas including geometry, mathematics, science, and nature. Fractals are self-similar patterns that repeat at different scales and often have non-integer fractal dimensions. Benoît Mandelbrot coined the term fractal and studied fractals like the Mandelbrot set, which demonstrate iterative processes that lead to increasingly complex patterns appearing as one zooms in. Fractals are used in computer graphics, geography, art and other applications to model real-world irregular forms and processes.
This document provides an overview of fractals including their properties, examples found in nature, and their history. It discusses key fractal concepts such as recursion, self-similarity, and fractal dimension. Methods for generating geometric fractals like the Koch curve and Sierpinski triangle are presented. The document also demonstrates fractal software and discusses ways fractals can be incorporated into K-12 education.
This document provides an overview of fractals, including:
- Fractals demonstrate self-similarity and scaling symmetry where they appear the same at different levels of magnification.
- There are two categories of fractals - those occurring in nature like trees, ferns, mountains, clouds, and bacterial colonies, and mathematical constructions like the Koch curve, Sierpinski gasket, and Mandelbrot set.
- Natural fractals only exhibit self-similarity over a limited range of scales and tend to be roughly rather than exactly self-similar, while mathematical fractals can have perfect self-similarity over infinite scales.
This document discusses different definitions of dimension, focusing on topological dimension and Hausdorff dimension. It provides examples of fractals like the Sierpinski triangle and Koch curve that challenge classical dimension definitions. While topological dimension aligns with intuition, the Hausdorff dimension considers how content scales with measurement. The document explains how fractals like the Sierpinski triangle have topological dimension of 1 but area of 0, requiring a fractional Hausdorff dimension to adequately measure their content. This provides the conceptual basis for Mandelbrot's definition of fractals as sets where the Hausdorff dimension exceeds the topological dimension.
This document discusses chaos theory and fractals. It defines chaos as systems that are highly sensitive to initial conditions. It describes how Edward Lorenz discovered chaos through computer modeling of weather patterns. It explains key concepts like the butterfly effect and Lorenz attractor. It discusses pioneers in fractal geometry like Mandelbrot and how fractals are found throughout nature and can be used in various applications.
- Chaos theory is about understanding complex and nonlinear dynamic systems, not denying determinism or order. It recognizes that small changes can lead to large, unpredictable consequences.
- Fractals are geometric shapes that exhibit self-similarity, where parts of the shape resemble the whole. They are found throughout nature and can be modeled using mathematical equations.
- Pioneered by Mandelbrot, fractal geometry is useful for simulating and understanding natural phenomena like clouds, coastlines, and trees that appear irregular or chaotic but have underlying patterns. It has applications in fields like computer graphics, fluid mechanics, and telecommunications.
Master's Thesis: Closed formulae for distance functions involving ellipses.Gema R. Quintana
This thesis examines computing distances between ellipses and ellipsoids. Chapter 2 derives a closed-form formula for the minimum distance between two coplanar ellipses without calculating footpoints. Chapter 3 computes the closest approach of two arbitrary separated ellipses or ellipsoids over time. Future work includes using ellipses to check safety regions for robot motion and computing Hausdorff distances between ellipses and ellipsoids.
1) The document describes constructing a net for three square-based pyramids that can fold into a cube.
2) It then discusses Cavalieri's principle, which states that solids with the same perpendicular height and cross-sectional areas have the same volume.
3) Examples are given of using Cavalieri's principle to find the volumes of solids by relating them to more familiar solids of the same volume.
This document provides lecture notes on analytic geometry. It begins with an introduction discussing the goals of building an algebraic geometry framework for analytic situations by replacing topological abelian groups with condensed abelian groups. Condensed sets are defined as sheaves on the pro-étale site of the point, and behave like generalized topological spaces. The notes establish that quasiseparated condensed sets correspond to ind-compact Hausdorff spaces. This provides the needed abelian category structure to build an analytic geometry in parallel to algebraic geometry over schemes.
Fractals are geometric shapes that exhibit self-similarity and complex patterns that are repeated at different scales. They can be used to describe irregular shapes found in nature, such as trees, coastlines, and clouds. The document discusses how Benoit Mandelbrot introduced fractals and developed their theory. It provides examples of famous fractals like the Mandelbrot set and Julia set and explains how fractals appear throughout nature and can be modeled using computers. Fractals have applications in fields like mathematics, science, engineering, art and music.
This document discusses geometric modeling and representations. It introduces polygonal and polyhedral models for representing obstacle regions as the union or intersection of half-plane or half-space primitives. Logical predicates can also represent these models using logical ANDs and ORs. Transformations of geometric bodies are also introduced to enable modeling motion.
This document discusses the graphical method for solving linear programming problems (LPP). It describes the steps of the method which include plotting the constraints on a graph, identifying the feasible region, plotting the objective function, and finding the optimal point where it intersects the feasible region. The document provides an example problem and solution. It also discusses different cases that can occur with the optimal solution such as it being unique, unbounded, having multiple solutions, being infeasible, or there being a unique feasible point.
The document discusses different methods to solve assignment problems including enumeration, integer programming, transportation, and Hungarian methods. It provides examples of balanced and unbalanced minimization and maximization problems. The Hungarian method is described as having steps like row and column deduction, assigning zeros, and tick marking to find the optimal assignment with the minimum cost or maximum profit. A sample problem demonstrates converting a profit matrix to a relative cost matrix and using the Hungarian method to find the optimal solution.
This document discusses various applications of graph theory to other areas of mathematics and other fields. It provides new graph theoretical proofs of Fermat's Little Theorem and the Nielsen-Schreier Theorem. It also discusses applications to problems in DNA sequencing, computer network security, scheduling, map coloring, and mobile phone networks. Specific algorithms for finding minimum vertex covers, vertex colorings, and matchings in graphs are applied to solve problems in these areas.
NOTE:Download this file to preview as the Slideshare preview does not display it properly.
This is an introduction to Linear Programming and a few real world applications are included.
The document provides an outline of topics related to linear programming, including:
1) An introduction to linear programming models and examples of problems that can be solved using linear programming.
2) Developing linear programming models by determining objectives, constraints, and decision variables.
3) Graphical and simplex methods for solving linear programming problems.
4) Using a simplex tableau to iteratively solve a sample product mix problem to find the optimal solution.
This document provides an overview of linear programming. It discusses basic and basic feasible solutions, the geometric solution, definitions used in linear programming, and the simplex algorithm. It provides an example problem that is solved over multiple iterations using the simplex algorithm to find the optimal solution. Finally, it briefly discusses the primal dual relationship between linear programming problems.
This presentation is trying to explain the Linear Programming in operations research. There is a software called "Gipels" available on the internet which easily solves the LPP Problems along with the transportation problems. This presentation is co-developed with Sankeerth P & Aakansha Bajpai.
By:-
Aniruddh Tiwari
Linkedin :- http://in.linkedin.com/in/aniruddhtiwari
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
Linear programming - Model formulation, Graphical MethodJoseph Konnully
The document discusses linear programming, including an overview of the topic, model formulation, graphical solutions, and irregular problem types. It provides examples to demonstrate how to set up linear programming models for maximization and minimization problems, interpret feasible and optimal solution regions graphically, and address multiple optimal solutions, infeasible solutions, and unbounded solutions. The examples aid in understanding the key steps and components of linear programming models.
This document provides an overview of fractal geometry. It begins with a preface stating that the book aims to provide background on basic fractal geometry topics, assuming some prior geometry knowledge. It notes that fractals fascinatingly mix art and mathematics by demonstrating equations are more than numbers. The document then outlines the book's contents over four chapters: history of fractal geometry, calculating fractal dimension, specific famous fractals, and applications of fractals. The first chapter discusses the history from early contributors in the 17th-19th centuries to Mandelbrot coining the term "fractal" in 1975 to describe self-similar geometric shapes with non-integer dimensions.
This document discusses fractals and fractal geometry. It begins with an abstract that provides an overview of fractal research and applications. It then discusses the history and key concepts of fractals, including their self-similar and recursive properties. The document explains that fractals have non-integer dimensions due to their complex shapes that reveal new structures at increasing levels of magnification. It provides examples of calculating the fractal dimension using the similarity dimension formula. In summary, the document provides a high-level introduction to fractals, their characteristics, generation techniques, and applications in modeling natural phenomena.
Fractals are geometric shapes that exhibit self-similarity and complex patterns at every scale. Koch's snowflake is a famous fractal where the perimeter tends towards infinity as more iterations are done, even as the area approaches a limit. The Mandelbrot set is another well-known fractal that maps the behavior of values of c under a complex iterative function, resulting in diverse patterns when zooming in. Fractals are found throughout nature in shapes like clouds, coastlines, and Romanesco broccoli. They can also be generated through computer programs and used in applications like diagnosing skin cancer.
This document provides an overview of fractal geometry. It begins with an abstract that outlines how fractal patterns found in nature will be used to introduce the concept of fractals. It then provides a brief history of fractals, covering mathematicians like Georg Cantor and Benoit Mandelbrot who contributed to the discovery and study of fractals. The document goes on to examine key properties of fractals in depth, including recursion, self-similarity, iteration, and fractal dimension. It also provides examples of well-known fractals like the Sierpinski triangle and Mandelbrot set to illustrate these properties.
This document provides an introduction and overview of fractals. It begins by defining fractals as rough or fragmented geometric shapes that are self-similar and scale-independent. The first fractals were discovered by Gaston Julia in the early 20th century, and the term "fractal" was coined by Benoit Mandelbrot in 1975. Key properties of fractals are described, including self-similarity across scales, formation through iteration, and fractional dimension. Examples like the Koch snowflake and Sierpinski triangle are shown and their fractal dimensions are calculated. Real-world examples of fractals in nature and animation movies are also briefly mentioned.
This document discusses fractals and their applications in physics. It begins with an introduction to fractals and fractal dimension, then discusses some examples of fractals from nature and geometry. The history of fractals from 1883 to 1975 is outlined, followed by applications in engineering, medicine, astrophysics, and physics. Fractals have been used in antenna design, image processing, modeling turbulence and porous media, and describing rough surfaces. The document concludes that fractals are powerful tools for describing systems and solving problems, though they may not describe things perfectly.
Fractals in Physics discusses fractals and their applications in various fields of physics. Fractals are geometric shapes that exhibit self-similarity and have non-integer dimensions. Examples of fractals include the Sierpinski triangle and the Mandelbrot set. Fractals have been used successfully in engineering applications like chip cooling circuits and heat exchangers. They have also found applications in medicine for analyzing blood vessels and lungs. In astrophysics, fractals help explain the formation of stars. In physics, fractals can model turbulent flows, rough surfaces, antenna design, and image compression. Many scientists have found fractals to be useful tools for understanding and solving problems across various scientific domains.
Fractal theory describes objects that have self-similar patterns across different scales. Fractals are defined as geometric shapes that can be split into parts, each of which is a reduced-size copy of the whole. They often have non-integer dimensions and fine structures at arbitrarily small scales. Common examples include natural shapes like coastlines as well as mathematical constructs like the Cantor set and Koch curve. Fractals are created by recursive processes that repeat a simple pattern on decreasingly smaller scales. Their dimension can be calculated from how their size changes with scaling factor.
Correlation between averages times of random walks on an irregularly shaped o...Alexander Decker
This document discusses a study that investigated the correlation between the average times of random walks on irregularly shaped objects (like country maps) and the fractal dimensions of those objects. 20 country maps were analyzed by developing a program to estimate their random walk parameters and fractal dimensions. The program divided each map image into grids and calculated the boundary boxes and distance covered by random walkers on each grid. The slopes of the log-log graphs of these values against grid size represented the fractal dimension and random walk parameter respectively. The estimated fractal dimensions ranged from 1.116 to 1.212, while the random walk parameters ranged from 1.976 to 2.995. However, the correlation between the fractal dimensions and random
This document discusses computational modeling of the structures of biological macromolecules like proteins docking together. It focuses on using the geometry of molecular surfaces and conformal mappings between surfaces to predict docked configurations. Circle packing is proposed as a method to construct approximate conformal mappings between molecular surfaces by triangulating one surface, constructing a circle packing with the same triangulation on a sphere, and finding conformal transformations between the surfaces and standard metrics on spheres.
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
The document is a lesson plan for a unit titled "From Flatland to Spaceland" for 2nd year secondary students. The unit aims to move from two-dimensional shapes to three-dimensional bodies by exploring their main characteristics. It will use what is learned to describe everyday objects. The unit includes self-assessment. It is presented as a PDF and corresponds to the 2nd year secondary math curriculum. It is authored by Rafael Cabezuelo Vivo and is estimated to take 6 sessions plus a final task and self-assessment of content and skills.
This document describes a MATLAB application called FractalAnalyzer that is used to analyze earthquake clustering through multifractal seismicity analysis. It allows users to calculate fractal dimensions like correlation dimension and generalized fractal dimensions from earthquake data. These fractal dimensions can provide information about the heterogeneous and complex spatial and temporal clustering of earthquakes. The application includes tools for plotting graphs of the analysis and was demonstrated on a dataset from before the 2005 Kashmir earthquake.
The document discusses the Minkowski curve, a fractal curve first introduced by Hermann Minkowski. It has the property of self-similarity, where portions of the curve exactly replicate the whole curve at different scales. The construction of the Minkowski curve is based on a recursive procedure where at each step an 8-sided generator is applied to line segments, increasing the complexity. As the number of iterations increases, the length of the curve tends towards infinity, and its fractal dimension is calculated to be 1.5, demonstrating its self-similarity. Variations include starting with geometric shapes other than a straight line.
A MATLAB Computational Investigation of the Jordan Canonical Form of a Class ...IRJET Journal
This document discusses computational investigation of the Jordan canonical form for a class of zero-one matrices. It begins with background on Jordan canonical form and its relationship to directed graphs. The author then describes developing a MATLAB toolbox to calculate the Jordan canonical form for these matrices. The toolbox partitions the associated directed graph into cycles and chains to determine the Jordan form. It is found to be more accurate than MATLAB's built-in Jordan function for larger matrices over order 60.
Fractales bartolo luque - curso de introduccion sistemas complejosFundacion Sicomoro
¿Qué tienen en común los brócolis, las nubes y los cráteres meteoríticos? Todos exhiben fractalidad. Una nueva ciencia como la de los Sistemas Complejos, requería una nueva manera de caracterizar las formas: la geometría fractal. En esta charla aprenderemos qué es un fractal, dónde aparecen, dónde se usan y qué nos desvelan. Veremos que, en el fondo, la invarianza de escala, que va más allá de la geometría, es el concepto crucial.
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?"
This document summarizes a student paper about using the shortest path algorithm to interpolate contours in images. It discusses how the human visual system perceives 3D representations from 2D images and how extracting meaningful contours is challenging due to noise and discontinuities. The paper proposes using a modified Dijkstra's algorithm to find the shortest path in log-polar space, which maps circles in images to straight lines. This approach aims to identify simple, closed curves representing object contours while ignoring irrelevant edges.
This document is a project submitted for a Bachelor of Engineering degree that investigates computer simulations of fractal patterns through diffusion limited growth processes. The student developed three diffusion limited aggregation (DLA) models in an attempt to more realistically simulate diffusion limited growth processes with specific reference to electro-deposition. The models included growth of more than one cluster and complete coverage of the substrate. Patterns and data produced from the models allowed calculation of fractional dimensions between one and two as expected. However, unexpected results were also observed where the relationship between log of density and log of characteristic length deviated from mathematical predictability due to impingement effects.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
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ملزمة تشريح الجهاز الهيكلي (نظري 3)
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تتميز هذهِ الملزمة بعِدة مُميزات :
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5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
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This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
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hidden dimension in nature
1. HIDDEN DIMENSIONS OF NATURE(AN INTRODUCTION TO FRACTAL GEOMETRY AND ITS APPLICATIONS)ByMilan A. Joshi & Dr S.M.PadhyeDEPT OF MATHEMATICS SHRI RLT COLLEGE OF SCIENCE AKOLA.Email:- mlnjsh@gmail.com
2. ABSTRACT:- In this presentation we introduce the very basics of fractal geometry discovered by French/American mathematician Dr Benoit Mandelbrot and its applications. The most amazing thing about fractals is the variety of their applications. Besides theory, they were used to compress data in the Encarta Encyclopedia and to create realistic landscapes in several movies like Star Trek. The places where you can find fractals include almost every part of the universe, from bacteria cultures to galaxies to your body. In this paper, we have picked out the most important applications, trying to include them from as many areas of science and everyday life as possible. Here we list the area where fractals are applied. Astronomy: Galaxies, Rings of Saturn Bio / Chem. Bacteria cultures, Chemical Reactions, Human Anatomy, Molecules, Plants, Population Growth Other:Clouds, Coastlines and Borderlines ,Data Compression, Diffusion, Economy, Fractal ArtFractal Music, Landscapes, Newton's Method, Special Effects (Star Trek),Weather.
3. INTRODUCTION:- We are used to Euclidian geometry, where every thing is extremely regular for example straight lines , circles, triangles, spheres, cones , cylinders, and our regular calculus. We are always scared to study the patterns calling them monsters (Weirstrass nowhere differential function), pathological curve(Koch curve) and rejecting them all the time. But clouds are not spheres, mountains are not cones coast lines are straight lines ,barks are not regular ,but these patterns are in nature. Then Mandelbrot came up and say “Hey Guys” you can describe these patterns by mathematical formulas only it requires different kind of formulas. And he gave us a beautiful Mandelbrot set and fractal geometry.
5. History and Motivation The story begins with the young French mathematician Gaston Julia(1893 – 1978) who introduce the problem of iterated function (IFS)during world war I, which is just like a regular function except that it performs over and over again with each out put used as next input. Then he describes Julia sets. Few Julia sets are
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8. Dr Benoit Mandelbrot born(1924) at Warsaw (Poland) French /American Mathematician student of Gaston Julia has studied Julia's concept of iterated function and has done something that Julia could never do. He took a function f(z) = z2 + c ,for complex variable zand a complex parameter c and started to seethe Patterns emerging out in computer at IBM, (In 1980’s )and what he found is infinitely complex structure which he called as Mandelbrot set. The Mandelbrot set is visual representation of an iterated function on the complex plane
9. MATHEMATICAL DEFINITION (MANDELBROT SET) Mathematically Mandelbrot set is a set of all complex numbers c for which the orbit of 0 under iteration of the function z z2 + c, remains unbounded.
10. When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.
13. PROPERTIS OF MANDELBROT SET It is a compact set. It is contained in the closed disk of radius 2. It is connected. The area of Mandelbrot set is 1.50659177 ± 0.00000008 (Approximately) It is conjectured that the Mandelbrot set is locally connected. It is a fractal. It is a set of all points whose Julia sets are connected.
14. WHAT IS FRACTAL The word fractal is derived from a Latin word fractus means broken. It is defined to be geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern
15. Fractals’ properties Two of the most important properties of fractals are self-similarity and non-integer dimension. What does self-similarity mean? If you look carefully at a fern leaf, you will notice that every little leaf - part of the bigger one - has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same is with fractals: you can magnify them many times and after every step you will see the same shape, which is characteristic of that particular fractal. Classical geometry deals with objects of integer dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. However, many natural phenomena are better described using a dimension between two whole numbers. A fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves. The more the flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a "hilly fractal scene" will reach a dimension somewhere between two and three..
19. Types of Fractals There is an infinite variety of fractals and tons of ways to create them, from formulas to folding paper. We used the ways they are created to split them into several basic categories. Fractals not fitting into any categories were grouped together into nonstandard fractals. Please realize that you might not understand some concepts used in the descriptions if you haven't read the tutorial. However, concepts not discussed in the tutorial were not used in this section as well. Base-Motif FractalsDusts and ClustersFractal CanopiesIFS FractalsJulia SetsMandelbrot SetsNonstandard FractalsPaper-Folding Fractals Peano CurvesPlasma FractalsPythagoras TreesQuaternionsStar FractalsStrange AttractorsSweeps
20. FRACTAL DIMENSION Let’s calculate the fractal dimension. To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But why is this? You can think about it. We often say that a line has dimension 1 because there is only 1 way to move on a line. Similarly, the plane has dimension 2 because there are 2 directions in which to move. And space has dimension 3.
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22. With these three examples, you should see a clear pattern. If you take the magnification and raise it to the power of dimension... you will always get the number of shapes! Writing it as a formula, you get: N = rd Since we are trying to calculate the dimension, we should solve this equation for d. If you are familiar with logs, you should easily find that d = log N / log r .
23. DimensionTHE SIMPLEST METHOD One way to calculate fractal dimension is by taking advantage of self-similarity. For example, suppose you have a 1-dimensional line segment. If you look at it with the magnification of 2, you will see 2 identical line segments. Let’s use a variable D for dimension, r for magnification, and N for the number of identical shapes.
25. SIERPINSKI TRIANGLE D = log(N)/log(r) = log(3)/log(2) = 1.585. We get a value between 1 and 2.
26. VON-KOCH CURVE Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points. Karl Weierstrass had first demonstrated the existence of such a curve in 1872. The article by Sime Ungar provides a simple geometric proof. The length of the intermediate curve at the nth iteration of the construction is (4/3)^n, where n = 0 denotes the original straight line segment. Therefore the length of the Koch curve is infinite. Moreover, the length of the curve between any two points on the curve is also infinite since there is a copy of the Koch curve between any two points
29. APPLICATIONS:-Special Effects Computer graphics has been one of the earliest applications of fractals. Indeed, fractals can achieve realism, beauty, and require very small storage space because of easy compression. Very beautiful fractal landscapes were published as far back as in Mandelbrot’s Fractal Geometry of Nature. Although the first algorithms and ideas are owed to the discoverer of fractals himself, the artistic field of using fractals was started by Richard Voss, who generated the landscapes for Mandelbrot’s book. This sparked the imagination of many artists and producers of science fiction movies. A little later, Loren Carpenter generated a computer movie of a flight over a fractal landscape. He was immediately hired by Pixar,. Fractals were used in the movie Star Trek II: The Wrath of Khan, to generate the landscape of the Genesis planet and also in Return of the Jedi to create the geography of the moons of Endor and the Death Star outline. The success of fractal special effects in these movies lead to making fractals very popular. Today, numerous software allows anyone who only knows some information about computer graphics and fractals to create such art. For example, we ourselves were able to generate all landscapes throughout this website.
34. APPLICATION:-WEATHER Weather behaves very unpredictably. Sometimes, it changes very smoothly from day to day. Other times, however, it changes very rapidly. Although weather forecasts are often accurate, there is never an absolute chance of them being right. Using a different term, you can say that the weather behaves very chaotically. This should automatically tell you what we are getting too. Indeed, weather can create fractal patterns. This was discovered by Edward Lorenz, who was mathematically studying the weather patterns. Lorenz came up with three formulas that could model the changes of the weather. When these formulas are used to create a 3D strange attractor, they form the famous Lorenz Attractor
36. HEART BEAT Heart beat is not constant over time. It fluctuates and fluctuates lot . One of the most powerful application of fractal is in rhythms of heart something that Boston cardiologist Ary Goldberger has been studying in his entire professional life. Initially Galileo stated that normal heart beats like a metronome. But Ary and his colleagues has proved that this theory was wrong .Healthy heart beat has fractal architecture. It has a distinctive fractal pattern. HEART BEAT TIME SERIES
37. It definitely helps us to understand lot of things about pattern of heart and one day many cardiologist spot heat problem sooner
38. Application :- FRACTAL ANTEENA A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic radiation within a given total surface area or volume. Cohen use this concept of fractal antenna. And it is theoretically it is proved that fractal design is the only design which receives multiple signals.