Fractals By Amanda Lewis
What is a fractal? • A fractal is defined to be a rough or fragmented shape that can be broken up into smaller parts, which can be seen as a smaller copy of the original shape.
Benoit Mandelbrot • Born in Poland in November 1924 • Known as the “father of fractal geometry” • Coined the term “fractal” in 1975
How long is the coast of Britain? • Mandelbrot studied this question when he first discovered the idea of fractals in nature. • He concluded that in a sense, the coastline of Britain is essentially infinite. • By using smaller units of measurement, the length of the coast increases.
Maps of Britain
Sierpinski Triangle • Developed by WacLaw Sierpinski Let N be the number of triangles: N k = 3 k Let L denote the length of sides of each triangle: L k = (1/2) k = 2 -k Let A be the area of each triangle: 2 " N k = (3/4) k A k = L k
Koch curve • The length of any line segment can be described as being infinitely long. • To develop the Koch Curve: - Start out with a line segment. - Divide into three segments. - Replace middle segment with an equilateral triangle.
Koch Snowflake
N k Let define the number of sides after the kth step: N k = 4 k " 3 L k Let define the length of each side after the kth step: L k = 1 3 k P Let define the perimeter of k entire snowflake after the kth step: k = N k L k = 3(4 /3) k P
Cantor Point Set • Developed by Greg Cantor • How to develop the Cantor Fractal: Start out with one line segment. Divide that segment into three different parts. Remove the middle third. What is left is two line segments and four endpoints.
As we increase the amount of iterations, the length of the lines approaches zero: Steps # of line Length of line segments segments 1 1 1/3 2 4 1/9 3 8 1/27 n 2 n 3 " n
Fractal Dimension • Fractal dimension provides a way to measure how rough fractal curves are. D = log n log M Where n = number of pieces M= the magnification factor (how many times the fractal has been magnified) If the dimension is between 1 and 2, then it is a fractal
Dimension of the Koch Curve • After doing the process just one time, there is one line fragment that is divided up into four with an equilateral triangle in the middle. So, n = 4. Because these four pieces are 1/3 the length of the original line segment, we can say the magnification, M = 3. D = log(4) = 1.26185... log(3) Because this number has a dimension greater than 1, then the Koch curve is a fractal.
Julia Set • prisoner set: set of all complex numbers in the function ʼ s orbit that are bounded • escape set: the complex numbers that are unbounded in an orbit under a certain function.
Julia Set z 0 = 2 • An example of a prisoner is f ( z ) = z 2 " z + 1 under the function, z 0 = 1 + i z 1 = f(1+i) = (1+i) 2 - (1+i) +1 = i z 2 = f(i) = i 2 - i +1 = -i If we keep iterating, our values will switch back and forth between I and -I, so we call a prisoner. z 0 = 2
Julia Set • The Julia set is defined to be the boundary between the prisoner set and the escape set, under the function: f(z) = z 2 + c Where c represents a complex constant.
How are Fractals used in the real world? • Fractals have been observed in just about every living thing in nature from trees in the rain forest to our human bodies. • the Koch snowflake was used to make the antennas of our cell phones smaller while increasing the amount of frequencies they can receive. • Fractals are also used intensively in movies and video games…
Star Wars: Episode III • The idea of the fractal was taken and applied to a cylinder spiral shape of lava. They took the original shape, shrunk it down and reapplied it. They repeated this over and over again to get a extremely realistic huge ball of fire and lava
The End
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