The Beauty of Mathematics

... The number 0. The number 1. The number p, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis. The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828). The number i, imaginary unit of the complex numbers, w ...

... The number 0. The number 1. The number p, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis. The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828). The number i, imaginary unit of the complex numbers, w ...

Iceland

... May 15 – June 16, 2017 Five weeks in South Africa, learning about chaos theory and fractals, as well as South African history and culture. We will explore the ideas of fractal geometry and how it shows up in natural phenomena such turbulent fluids, cloud formations, and ecology. We will also see how ...

... May 15 – June 16, 2017 Five weeks in South Africa, learning about chaos theory and fractals, as well as South African history and culture. We will explore the ideas of fractal geometry and how it shows up in natural phenomena such turbulent fluids, cloud formations, and ecology. We will also see how ...

Cape Town, South Africa

... Five weeks in South Africa, learning about chaos theory and fractals, as well as South African history and culture. We will explore the ideas of fractal geometry and how it shows up in natural phenomena such turbulent fluids, cloud formations, and ecology. We will also see how it shows up African ar ...

... Five weeks in South Africa, learning about chaos theory and fractals, as well as South African history and culture. We will explore the ideas of fractal geometry and how it shows up in natural phenomena such turbulent fluids, cloud formations, and ecology. We will also see how it shows up African ar ...

Fractals

... "Pathological" "gallery of monsters" In 1875: Continuous non-differentiable functions, ie no tangent ...

... "Pathological" "gallery of monsters" In 1875: Continuous non-differentiable functions, ie no tangent ...

Chapter 7 - Ohlone College

... 1. a) Explain the difference between Euclidean, Remannian, and Lobachevskian geometry in reference to its parallel postulate and the sum of the angles of a triangle. b) Draw an equilateral triangle on the model for each of the above geometries. 2. Explain four techniques used by the artist to displa ...

... 1. a) Explain the difference between Euclidean, Remannian, and Lobachevskian geometry in reference to its parallel postulate and the sum of the angles of a triangle. b) Draw an equilateral triangle on the model for each of the above geometries. 2. Explain four techniques used by the artist to displa ...

Assignment • Hat Curve Fractal Handout

... "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." --Benoit Mandelbrot ...

... "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." --Benoit Mandelbrot ...

Assignment - FrancisLarreal

... mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. A fractal often has the following features: It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. ...

... mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. A fractal often has the following features: It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. ...

The Space-Filling Efficiency of Urban Form in Izmir

... urban form can not be fully described by Euclidean geometry, but rather be treated as fractals (Batty and Longley, 1987; Benguigui and Daoud, 1991; Batty and Xie, 1996; 1999; Shen 1997; 2002). ...

... urban form can not be fully described by Euclidean geometry, but rather be treated as fractals (Batty and Longley, 1987; Benguigui and Daoud, 1991; Batty and Xie, 1996; 1999; Shen 1997; 2002). ...

fractal geometry : an introduction

... capable of explaining the theory of Form. Topology is the study of qualitative properties of certain objects that are invariant under certain kind of transformations, especially those properties that are invariant under a certain kind of equivalence. In topology, all single island coastlines are of ...

... capable of explaining the theory of Form. Topology is the study of qualitative properties of certain objects that are invariant under certain kind of transformations, especially those properties that are invariant under a certain kind of equivalence. In topology, all single island coastlines are of ...

chaos - FSU High Energy Physics

... ► Mandelbrot realized you could break the periods with errors into smaller groups, some with errors, and some without. ► Mandelbrot then discovered that within any burst of errors, there would be smaller, error free periods. ► This relationship followed a scaling pattern, just as the cotton prices. ...

... ► Mandelbrot realized you could break the periods with errors into smaller groups, some with errors, and some without. ► Mandelbrot then discovered that within any burst of errors, there would be smaller, error free periods. ► This relationship followed a scaling pattern, just as the cotton prices. ...

Fractal Geometry

... • Clouds using circles??? • Leaves??? • Rocks??? • Humans and animals with rectangles and circles??? ...

... • Clouds using circles??? • Leaves??? • Rocks??? • Humans and animals with rectangles and circles??? ...

figure out the number of stitches needed (algebra) and

... The Practical Bit!! You are going to create a model for Gehrytown, a new development off the M6. •Each person will create one building. ...

... The Practical Bit!! You are going to create a model for Gehrytown, a new development off the M6. •Each person will create one building. ...

bioweek9

... • P. Prusinkiewicz : Computer Graphics group at the University of Regina • Lindenmayer Systems are – rewriting systems – also known as L-Systems ...

... • P. Prusinkiewicz : Computer Graphics group at the University of Regina • Lindenmayer Systems are – rewriting systems – also known as L-Systems ...

Fractal Project

... •Aristid Lindenmayer a biologist from Hungry developed and introduced the L-System fractal. •He discovered it from plants and algae. ...

... •Aristid Lindenmayer a biologist from Hungry developed and introduced the L-System fractal. •He discovered it from plants and algae. ...

Correlation between Averages Times of Random Walks

... Dimensions can be defined as the number of parameters or coordinates that are used for describing mathematical objects. For example, lines are defined as one-dimensional, squares as two-dimensional and cones as three-dimensional still, there are some objects such as clouds, trees veins, lungs, ocean ...

... Dimensions can be defined as the number of parameters or coordinates that are used for describing mathematical objects. For example, lines are defined as one-dimensional, squares as two-dimensional and cones as three-dimensional still, there are some objects such as clouds, trees veins, lungs, ocean ...

Sierpinski N-Gons - Grand Valley State University

... Introduction: The growing interest in chaos and fractal geometry has created a new field of mathematics that can by explored by faculty and undergraduates alike. Sierpinski triangles and Koch's curves have become common phrases in many mathematics departments across the country. In this paper we rev ...

... Introduction: The growing interest in chaos and fractal geometry has created a new field of mathematics that can by explored by faculty and undergraduates alike. Sierpinski triangles and Koch's curves have become common phrases in many mathematics departments across the country. In this paper we rev ...

pdf

... agreement between the simulated and actual images. For the stacked spheres we found D=1.6 in agreement with Sweet et.al., similar to the analytic value of the Sierpinski triangle D =1.58. In contrast the Sinai cube has significantly higher value of D = 1.8. The bottom left corner of Figure 2 we show ...

... agreement between the simulated and actual images. For the stacked spheres we found D=1.6 in agreement with Sweet et.al., similar to the analytic value of the Sierpinski triangle D =1.58. In contrast the Sinai cube has significantly higher value of D = 1.8. The bottom left corner of Figure 2 we show ...

fract2

... Infinity and Fractals… • Fractals are self-similar objects whose overall geometric form and structure repeat at various scales they provide us with a “glimpse” into the wonderful way in which nature and mathematics meet. • Fractals often arise when investigating numerical solutions of differential ...

... Infinity and Fractals… • Fractals are self-similar objects whose overall geometric form and structure repeat at various scales they provide us with a “glimpse” into the wonderful way in which nature and mathematics meet. • Fractals often arise when investigating numerical solutions of differential ...

What is the Relatedness of Mathematics and Art and why

... Another artists inspired by geometrical concepts are M. C. Escher. The concept of Escher can also be attributed as generative art, making artworks that can be generated from such artistic rules of drawing. One of his famous works and a technique of generating such art by Escher are shown in figure 5 ...

... Another artists inspired by geometrical concepts are M. C. Escher. The concept of Escher can also be attributed as generative art, making artworks that can be generated from such artistic rules of drawing. One of his famous works and a technique of generating such art by Escher are shown in figure 5 ...

Fibonacci Sequence and Fractal Spirals

... b. What’s the second number of the Fibonacci sequence? _______ Right above the square you just drew, draw another 1 x 1 square. ...

... b. What’s the second number of the Fibonacci sequence? _______ Right above the square you just drew, draw another 1 x 1 square. ...

Fractals and Dimension

... Fractals and Dimension Dimension We say that a smooth curve has dimension 1, a plane has dimension 2 and so on, but it is not so obvious at first what dimension we should ascribe to the Sierpinski gasket or the von Koch snowflake or even to a Cantor set. These are examples of fractals (the word is d ...

... Fractals and Dimension Dimension We say that a smooth curve has dimension 1, a plane has dimension 2 and so on, but it is not so obvious at first what dimension we should ascribe to the Sierpinski gasket or the von Koch snowflake or even to a Cantor set. These are examples of fractals (the word is d ...

Mathematics summer projects for undergraduate students

... Topic: Convex figures in the plane Supervisor: Dr Thomas Huettemann Description: A convex figure is one that has the following property: given two points in the figure, the line segment joining these two points is entirely inside the figure too. Some convex figures can "rotate" within a square whil ...

... Topic: Convex figures in the plane Supervisor: Dr Thomas Huettemann Description: A convex figure is one that has the following property: given two points in the figure, the line segment joining these two points is entirely inside the figure too. Some convex figures can "rotate" within a square whil ...

NOTICE from J - JamesGoulding.com

... teaching tool containing 14 tests for detecting hidden determinism in a seemingly random time series of up to 16,382 points provided by the user in an ASCII data file. Sample data files are included for model chaotic systems. When chaos is found, calculations such as the probability distribution, po ...

... teaching tool containing 14 tests for detecting hidden determinism in a seemingly random time series of up to 16,382 points provided by the user in an ASCII data file. Sample data files are included for model chaotic systems. When chaos is found, calculations such as the probability distribution, po ...

Math 8: SYMMETRY Professor M. Guterman Throughout history

... decorate their surroundings. In this course we will discuss the symmetries of wallpaper patters such as the student-drawn example to the left. Our approach illustrates the powerful theme in modern mathematics: we associate to each pattern a mathematical object called a group (consisting of the symme ...

... decorate their surroundings. In this course we will discuss the symmetries of wallpaper patters such as the student-drawn example to the left. Our approach illustrates the powerful theme in modern mathematics: we associate to each pattern a mathematical object called a group (consisting of the symme ...

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. 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 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.The mathematical roots of the idea of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century. The term ""fractal"" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning ""broken"" or ""fractured"", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as ""beautiful, damn hard, increasingly useful. That's fractals."" The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, and law.