Fractals Geometry Module 2 Lecture PDF

FRACTALS GEOMETRY My soul is spiraling frozen FRACTALS all around What are fractals? What are fractals? Consider the fern on the left. What property do the blade, the pinnae, and the pinnules share? They are similar in shape. What are...

FRACTALS GEOMETRY My soul is spiraling frozen FRACTALS all around What are fractals? What are fractals? Consider the fern on the left. What property do the blade, the pinnae, and the pinnules share? They are similar in shape. What are fractals? Fractals are never-ending patterns, and are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in a ongoing feedback loop. The word “fractal” was coined in 1980 by Belgian mathematician Benoit Mandelbrot (1924-2010). Mandelbrot chose the name fractal because it reminds him of the word “fraction”. This was after he realized that these self-similar shapes have the property of not being one-dimensional, or two-dimensional, or even three-dimensional, but instead, of fractional dimension. Where are fractals? Ice crystals Pinecone Romanesco Broccoli Where are fractals? 20 times magnification of Running electricity between two nails sunk in dendritic copper crystals a piece of wet pine forming Where are fractals? Leaves Tree branches Bubbles Fractal Dimension Given a line segment, to have a length ½ (s = ½) of the original, the line segment must be divided into two equal parts (n = 2). To achieve s = 1/3, the segment must be divided into 3 equal parts (n = 3). Reduction in size Dimensions s=1 s = 1/2 s = 1/3 1 n=1 n=2 n=3 2 3 Fractal Dimension To achieve squares with lengths ½ of the original (s = ½), 4 squares must be generated. Similarly, s = 1/3 requires 9 squares. Reduction in size Dimensions s=1 s = 1/2 s = 1/3 1 n=1 n=2 n=3 2 n=1 n=4 n=9 3 Fractal Dimension To achieve cubes with lengths ½ of the original (s = ½), 8 cubes must be generated. Similarly, s = 1/3 requires 27 cubes. Reduction in size Dimensions s=1 s = 1/2 s = 1/3 1 n=1 n=2 n=3 2 n=1 n=4 n=9 3 n=1 n=8 n = 27 Fractal Dimension Reduction in size Dimensions s=1 s = 1/2 s = 1/3 1 n=1 n=2 n=3 2 n=1 n=4 n=9 3 n=1 n=8 n = 27 Mandelbrot Set The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane. Mandelbrot Set Mandelbrot Set The region of the set centered around – 0.75 + 0.1i is sometimes known as the sea horse valley because the spiral shapes appearing in it resemble sea horse tails (Giffin, Munafo). SEA HORSE VALLEY Mandelbrot Set Similarly, the portion of the Mandelbrot set centered around 0.3 + 0i with size approximately 0.1 + 0.1i is know as elephant valley. ELEPHANT VALLEY Mandelbrot Set Mandelbrot Set Mandelbrot Set Koch Snowflake The Koch Snowflake was created by the Swedish Mathematician Niels Fabian Helge von Koch. In his paper entitled “Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire” he used the Koch Snowflske to show that it is possible to have figures that are continuous everywhere but differentiable nowhere.  Koch Edge Start with a line segment. Divide the line segment into 3 equal parts (s = 1/3), with middle part replaced with two linear segments at angles 600 and 1200, producing a figure with 4 line segments (n = 4). Repeat the steps to each line segment. Koch Edge Koch Snowflake  Cantor Set Georg Cantor (1845 – 1918) was the founder of set theory. He is also noted for studying one of the first fractal shapes. The Cantor set is formed by following the algorithm: (1) start with a line segment; (2) divide the line segment into thirds; (3) remove the middle third line segment, (4) iterate further.  Sierpinski Triangle Named after Polish mathematician Waclaw Sierpinski, the triangle is one of the simplest fractal shapes in existence. Fractal Applications Astrophysicists believe that the key to finding out how stars were formed is the fractal nature of interstellar gas, like smoke trail or clouds in the sky. Both are shaped by turbulence, giving them an irregular but repetitive pattern that would not be possible to describe without the help of fractals. Fractal Applications Biology – some systems are best described using fractals Self-similarity in Cerebral arteriovenous chromosomes & DNA malformation, 3D CT Scan Fractal Applications Fractal based devices Computer chip cooling Space-filling fractal devices for circuit high precision fluid mixing Practice Problem Using the base and motif, create a box fractal following the following steps. After which, solve for the fractal’s dimension. Creating the box fractal using the method of successive removals. 1. Draw a square. 2. For each side, divide them into three equal parts. 3. Remove the middle thirds, and replace it with 3 line segment, as indicated in the motif. 4. Repeat steps 2 – 3 for all line segments. base motif