fractals
FRACTAL GEOMETRY
INTRODUCTION
Fractals is a new branch of mathematics and art. Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid (i.e, Euclidian geometry: comprising of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc.) Fractal geometry offers almost unlimited ways of describing, measuring and predicting the natural phenomena.
“Why is geometry often described as ‘cold and dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. 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)
Many people are fascinated by the beautiful images termed fractals. Extending beyond the typical perception of mathematics as a body of complicated, boring formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. What makes fractals even more interesting is that they are the best existing mathematical descriptions of many natural forms. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.
Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.
A fractal is a never-ending pattern. Fractals 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. Driven by recursion, fractals are images of dynamic systems. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals.
PROPERTIES
Two of the most important properties of fractals are self-similarity and non-integer dimension.
What does self-similarity mean? If you look carefully at a fern leaf, you will notice
INTRODUCTION
Fractals is a new branch of mathematics and art. Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid (i.e, Euclidian geometry: comprising of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc.) Fractal geometry offers almost unlimited ways of describing, measuring and predicting the natural phenomena.
“Why is geometry often described as ‘cold and dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. 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)
Many people are fascinated by the beautiful images termed fractals. Extending beyond the typical perception of mathematics as a body of complicated, boring formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers. What makes fractals even more interesting is that they are the best existing mathematical descriptions of many natural forms. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.
Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.
A fractal is a never-ending pattern. Fractals 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. Driven by recursion, fractals are images of dynamic systems. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals.
PROPERTIES
Two of the most important properties of fractals are self-similarity and non-integer dimension.
What does self-similarity mean? If you look carefully at a fern leaf, you will notice