Transcript Fractals
Fractals Siobhán Rafferty What Are Fractals? a set of points whose fractal dimension exceeds its topological dimension A “self-similar” geometrical shape that includes the same pattern, scaled down and rotated and repeated. The Koch Curve The Koch Curve is a famous example of a Fractal published by Niels Fabien Helge von Koch in 1906 Stage 0 is a straight line segment Stages 1 - infinity are produced by repeating stage1 along every line segment of the previous stage. The Koch Snowflake The perimeter of each stage is 1.33 x the perimeter of the pervious stage. When we repeat the stages to infinity the perimeter is infinite. Most geometrical shapes have an Area –Perimeter Relationship. This does not hold with Fractals An infinite perimeter encloses a finite area. Dimensions Fractals have non-integer dimensions that can be calculated using logarithms. If the length of the edges on a cube is multiplied by 2, 8 of the old cubes would fit into the new curve. Log8/Log2 = 3, a cube is 3 dimensional. Similarly for a fractal of size P, made of smaller units (size p), the number of units (N) that fits into the larger object is equal to the size ratio (P/p) raised to the power of d D = Log(N)/Log(P/p) Dimensions, an example. Each line in stage 1 is made up of lines 3cm long (P=3) There are 12 line segments Stage 2 has lines of length 1cm (p=1) It has 48 line segments (N = 48/12 = 4) d = log 4/ log 3 = 1.2619 Benoit Mandelbrot Born in 1924 and currently a mathematics professor at Yale University “The Mandelbrot Set” x²+c, where c is a complex number. X = 0² + c X = c² + c X = (c² + c)² + c 1 2 3 The Mandelbrot Set If it takes very few iterations for the iterations to become very large and tend to infinity then the value “c” is marked in red. Numbers are marked on the set following the light spectrum orange, yellow, green, blue, indigo, violet in order of those tending to infinity at different rates. The values shown in black do not escape to infinity The result is a fractal!