Intersecting Two-Dimensional Fractals with Lines
... we will show that the intersection with the negative real axis has infinitely many components (Theorem 3.9). To prove this, we use the special structure of this fractal set. The idea is to find a contractive map around −1 which preserves the local structure. We expect results of similar type for the ...
... we will show that the intersection with the negative real axis has infinitely many components (Theorem 3.9). To prove this, we use the special structure of this fractal set. The idea is to find a contractive map around −1 which preserves the local structure. We expect results of similar type for the ...
File
... Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry ...
... Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry ...
Read the history below and answer the questions that follow
... that through a given point not on a line, there is one and only one line parallel to that line. NonEuclidian geometry provides the mathematical foundation for Einstein’s Theory of Relativity. The most recent development in geometry is fractal geometry. Fractal geometry was developed and popularized ...
... that through a given point not on a line, there is one and only one line parallel to that line. NonEuclidian geometry provides the mathematical foundation for Einstein’s Theory of Relativity. The most recent development in geometry is fractal geometry. Fractal geometry was developed and popularized ...
Aalborg Universitet Aesthetics and quality of numbers using the primety measure
... divided into exact or statistic self-similar objects, full or partial fractals, natural and mathematic fractals, etc. A common natural fractal is the coastline that was used as an example by the ‘father’ of fractal geometry (Mandelbrot, 1967). Common examples of mathematic fractals are the Cantor se ...
... divided into exact or statistic self-similar objects, full or partial fractals, natural and mathematic fractals, etc. A common natural fractal is the coastline that was used as an example by the ‘father’ of fractal geometry (Mandelbrot, 1967). Common examples of mathematic fractals are the Cantor se ...
Delineation and explanation of geochemical anomalies using fractal
... insight into the structure of the variance of the parameters (Muller et al., 2008). In PCA, the concentration dataset is divided into several subsets, represented by different factors. The elements in the subset are correlated with one another and largely independent of the elements in the other sub ...
... insight into the structure of the variance of the parameters (Muller et al., 2008). In PCA, the concentration dataset is divided into several subsets, represented by different factors. The elements in the subset are correlated with one another and largely independent of the elements in the other sub ...
Alg2 Notes 9.1.notebook
... feet. The ball rebounds to 70% of its previous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 10th bounce? ...
... feet. The ball rebounds to 70% of its previous height after each bounce. Graph the sequence and describe its pattern. How high does the ball bounce on its 10th bounce? ...
fractal geometry
... Many objects are difficult to categorize as one-, two-, or three-dimensional. Examples are: coastline, bark on a tree, mountain, or path followed by lightning. It’s possible to make realistic geometric models of natural shapes using fractal geometry. ...
... Many objects are difficult to categorize as one-, two-, or three-dimensional. Examples are: coastline, bark on a tree, mountain, or path followed by lightning. It’s possible to make realistic geometric models of natural shapes using fractal geometry. ...
Universality Laws for Randomized Dimension Reduction
... Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. The question is how large the embedding dimension ...
... Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. The question is how large the embedding dimension ...
Geometry
... be represented by a recursive definition When using a fractal to represent a physical object, some degree of randomness is usually added to make the image more realistic groups that have broken dimensions so that each one looks like an exact copy of the second (like the 8 Mandelbrot group in Mathema ...
... be represented by a recursive definition When using a fractal to represent a physical object, some degree of randomness is usually added to make the image more realistic groups that have broken dimensions so that each one looks like an exact copy of the second (like the 8 Mandelbrot group in Mathema ...
Review: Sequences and Series
... 15. A stack of oranges is completely arranged so the bottom layer consists of oranges in an equilateral triangle with n oranges on a side. The next layer to the bottom consists of an equilateral triangle of (n–1) oranges on a side. This pattern continues upward with one orange on the top. How many o ...
... 15. A stack of oranges is completely arranged so the bottom layer consists of oranges in an equilateral triangle with n oranges on a side. The next layer to the bottom consists of an equilateral triangle of (n–1) oranges on a side. This pattern continues upward with one orange on the top. How many o ...
Review: Sequences and Series
... 15. A stack of oranges is completely arranged so the bottom layer consists of oranges in an equilateral triangle with n oranges on a side. The next layer to the bottom consists of an equilateral triangle of (n–1) oranges on a side. This pattern continues upward with one orange on the top. How many o ...
... 15. A stack of oranges is completely arranged so the bottom layer consists of oranges in an equilateral triangle with n oranges on a side. The next layer to the bottom consists of an equilateral triangle of (n–1) oranges on a side. This pattern continues upward with one orange on the top. How many o ...
Dynamics of dinosaurs Dynamics of dinosaurs Infectious disease
... Mathematical models can: predict rate of spread, peak, etc., of epidemics predict effects of different disease control ...
... Mathematical models can: predict rate of spread, peak, etc., of epidemics predict effects of different disease control ...
(1), D.Grebenkov (2)
... This last result is not true in d=2 without some extra condition. But we are going to assume this condition anyhow to hold in any dimension since we will need it for other purposes. In dimension d it is well-known that sets of co-dimension greater or equal to 2 are not seen by Brownian motion. For ...
... This last result is not true in d=2 without some extra condition. But we are going to assume this condition anyhow to hold in any dimension since we will need it for other purposes. In dimension d it is well-known that sets of co-dimension greater or equal to 2 are not seen by Brownian motion. For ...
chapter 9
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
Fractals with a Special Look at Sierpinski’s Triangle
... • Generated using a linear transformation • start at the origin xn+1 = 0.5xn and yn+1=0.5yn xn+1 = 0.5xn + 0.5 and yn+1=0.5yn + 0.5 xn+1 = 0.5xn + 1 and yn+1=0.5yn ...
... • Generated using a linear transformation • start at the origin xn+1 = 0.5xn and yn+1=0.5yn xn+1 = 0.5xn + 0.5 and yn+1=0.5yn + 0.5 xn+1 = 0.5xn + 1 and yn+1=0.5yn ...
161_syllabus
... Circle Geometry Cartesian Coordinate System, Vector Geometry Angles in Coordinate Geometry, The Complex Plane Birkhos Axiomatic System for Analytic Geometry Review Midterm # 1 Euclidean Constructions Constructibility Background and History of Non-Euclidean Geometry Models of Hyperbolic Geometry Basi ...
... Circle Geometry Cartesian Coordinate System, Vector Geometry Angles in Coordinate Geometry, The Complex Plane Birkhos Axiomatic System for Analytic Geometry Review Midterm # 1 Euclidean Constructions Constructibility Background and History of Non-Euclidean Geometry Models of Hyperbolic Geometry Basi ...
Chapter 9 Slides
... He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles. ...
... He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles. ...
The hidden maths in great art by Marcus du Sautoy, www.bbc.com
... mathematician Archimedes – their descriptions had been lost over time, but they were uncovered through developments in drawing. One of Salvador Dalí’s most famous works, Crucifixion (Corpus Hypercubus) contains a hyper-dimensional solid, Jackson Pollock was unconsciously tapping into5 geometric stru ...
... mathematician Archimedes – their descriptions had been lost over time, but they were uncovered through developments in drawing. One of Salvador Dalí’s most famous works, Crucifixion (Corpus Hypercubus) contains a hyper-dimensional solid, Jackson Pollock was unconsciously tapping into5 geometric stru ...
Math 371 Modern Geometries Exam Info Winter 2013 The exam is
... The axioms for Hyperbolic Geometry- especially the fifth ...
... The axioms for Hyperbolic Geometry- especially the fifth ...
Fractals and Self
... And here is a quote by Thomasina, from Arcadia: “Each week I plot your equations dot for dot, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must b ...
... And here is a quote by Thomasina, from Arcadia: “Each week I plot your equations dot for dot, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must b ...
chapter 9
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
area - StFX
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
... line segment or triangle and perform some operation. Then repeat the process indefinitely (this is called iterating). Each iteration produces a more complicated object. • The fractal dimension D can be found by considering the scaling at each iteration, where r is the scaling amount and N is the num ...
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 ...
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 ...
Fractal
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.