![]() ![]() Outside the Set, all the values of ‘C’ cause the equation to go to infinity, and the colors are proportional to the speed at which they expand. These are the points inside the Mandelbrot Set, which we color black. ![]() Driven by recursion, fractals are images of dynamic systems the pictures of Chaos. They are created by repeating a simple process over and over in an ongoing feedback loop. Fractals are infinitely complex patterns that are self-similar across different scales. This is the fate of most starting values of ‘C.’ However, some values of ‘C’ do not get bigger, but instead get smaller, or alternate between a set of fixed values. Fractals: A fractal is a never-ending pattern. We are interested in what happens for different starting values of ‘C.’ Generally, when you square a number, it gets bigger, and then if you square the answer, it gets bigger still. We plug this back into the equation, as ‘Zold’ and calculate it again. Each complex number is actually a point in a 2-dimensional plane. We start by plugging a value for the variable ‘C’ into the simple equation below. The Mandelbrot Set is probably one of the most familiar fractal equations. The number of colored triangles increases by a factor of 3 each step, 1,3,9,27,81,243,729, etc.Īlgebraic fractals use a simple formula that repeats and repeats. However, because fractal patterns are found. ![]() Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.įractal patterns repeat themselves at different scales - this is called “self-similarity.” They can be found in branching (like the branches on a tree), through spirals (think of a nautilus shell), and geometric (like the Sierpinski Triangle which is made by repeatedly removing the middle triangle from the prior generation. Designs using mathematics and geometry are often though of as very rigid lines, triangles, squares, etc. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. The formula used for a particular image determines how each pixel in the. Fractal patterns are extremely familiar, since nature is full of fractals. In simple terms, a fractal is a graphical representation of a mathematical equation. Geometrically, they exist in between our familiar dimensions. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. ![]()
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