FRACTALS are beautiful, complex mathematical functions with hidden order created from quantum physics chaos theory.
Chaos theory is really about finding the underlying order in apparently random data. Each random algorithm creates mathematical shapes that follow a pattern when broken down into a single repeating shape.
Fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description of lines, ellipses, circles, etc. Fractal geometry is described in algorithims, which are a set of instructions on how to create a fractal.
A mathematical fractal is based on an equation that undergoes iteration (repetition), a form of feedback based on recursion (copying a previous version of itself).
The results of one iteration are used as the starting point for the next iteration.
Each solution (the answer to the first numbers put into the equation) gets put back into the equation, creating complex shapes with the property of self-similarity.
Fractals graphically portray the notion of 'worlds within worlds'.
Much like holograms that store the entire image in each part of the image, any part of a fractal can be repeatedly magnified, with each magnification resembling all or part of the original fractal. Fractals have finite areas but infinite perimeters; order dependent on randomness.
To get colorful images of the set of numbers, a color is assigned to each value of the number of executed iterations in the set of answers for the equation.
Click here to go to Math Forum and play with a fractal equation.
Click here to go to the Magic Sierpinski Triangle.
Back in the 1960's, a French mathematician named Benoit Mandelbrot started thinking about how to solve the problem of measuring coastlines. Measuring the distance in a straight line was not acurate. But how could an accurate measurement be taken without taking a ruler (or meter stick) and getting down on your hands and knees and crawling around the edge? The two lengths would be very different!
"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."
—Mandelbrot, in his introduction to The Fractal Geometry of Nature
Mandelbrot's work, which was entirely developed outside the main research channels, led to a modern information theory. He made up the word "fractal" because he thought these figures would look fractured or broken up. It wasn't until he had a computer that he could SEE a picture of what he'd been thinking about!
For all of the "good science" that fractals can do, it is the ART of the fractal that spellbinds us.
A mathematical fractal is based on an equation that undergoes iteration (repetition), a form of feedback based on recursion (copying a previous version of itself).
The results of one iteration are used as the starting point for the next iteration.
Each solution (the answer to the first numbers put into the equation) gets put back into the equation, creating complex shapes with the property of self-similarity.
Fractals graphically portray the notion of 'worlds within worlds'.
Much like holograms that store the entire image in each part of the image, any part of a fractal can be repeatedly magnified, with each magnification resembling all or part of the original fractal. Fractals have finite areas but infinite perimeters; order dependent on randomness.
To get colorful images of the set of numbers, a color is assigned to each value of the number of executed iterations in the set of answers for the equation.
Click here to go to Math Forum and play with a fractal equation.
Click here to go to the Magic Sierpinski Triangle.
Back in the 1960's, a French mathematician named Benoit Mandelbrot started thinking about how to solve the problem of measuring coastlines. Measuring the distance in a straight line was not acurate. But how could an accurate measurement be taken without taking a ruler (or meter stick) and getting down on your hands and knees and crawling around the edge? The two lengths would be very different!
"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."
—Mandelbrot, in his introduction to The Fractal Geometry of Nature
Mandelbrot's work, which was entirely developed outside the main research channels, led to a modern information theory. He made up the word "fractal" because he thought these figures would look fractured or broken up. It wasn't until he had a computer that he could SEE a picture of what he'd been thinking about!
For all of the "good science" that fractals can do, it is the ART of the fractal that spellbinds us.
These intricate geometric patterns are the result of positive feedback loops within the equation.
The basic shape repeats itself on every level to infinity.
Fractal equations are extremely sensitive and any tiny change will produce greatly different results.
The closer you look the more detail you see.
Below is the result of the basic fractal equation called the Mandelbrot set, the set itself appears in black, with the fractal boundary in a chosen color.
Using colors is not only visually appealing, but gives lots of extra information that don't show up in the colorless version. The shapes formed by the colors are not coincidence, but they are closely related to the set itself.
The Mandelbrot set has the property that for a value which is close to the border of the set it will take longer (that is, more iterations) to escape to infinity than for a value which is farther away from the border of the set. This means that the colors are actually showing where the border of the set is. The brighter the color, the closer the border of the set is.
Some say that the Universe is built in fractal scales made of pieces which occupy a piece of vital space and last a quantity of informative time.
Fractal equations create forms as life does. Fractal equations create self-similar “cells,” as life does. And in fact, they use the same term. Fractal equations are also called generator equations, as they are the mother cells that encode the information needed to repeat a living form. Finally a fractal equation will create a pattern of interconnected parts, which put together reproduce a whole, self-similar to their parts, as organisms do.
