MODULEE 2.1-2.4 (FRACTALS)

These 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 an ongoing feedback loop.

Fractals

It is self-similar if its congruent to a uniform scaling of itself.

Pattern

FiIll in the blank: The word “fractal” was coined in ____ by Belgian mathematician ______________________ (1924-2010). __________ chose the name fractal because it reminds him of the word “__________”. 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.

1980, Benoit Mandelbrot, Mandelbrot, fraction

These are measures of an object’s size in one direction

Dimensions

It has a length, width and height, hence 3-dimensional.

Space figure

He said, “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

It is commonly known as fractal. Given some constant c E C, the forward iterations of a complex point x0 are generated

The Julia Set

It is the set of values c in the complex plane for which the orbit of critical point z=0

The Mandelbrot Set

It is named after the Polish mathematician Waclaw Sierpinski, this triangle is one of the simplest fractal shapes in existence.

The Sierpenski Triangle

It was created by the Swedish mathematician Niels Fabian Helge von Koch. He us this to show that it is possible to have figures that are continuous everywhere but differentiable nowhere.

The Koch Snowflake

It can be constructed by iterations

Koch Edge