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Visual math: The art of fractals
Art very seldom follows rules; it is dynamic, rhythmic, substantial, sporadic, expressive. However, one can observe that fractals work a bit differently. Here, instead of a sense of freedom and composition, a computer uses formulas to create a repetitive, complex, self-similar image.
But what is a fractal?
While the exact definition seems to differ between sources—usually because of contention between what a self-similar image is—we can usually agree on some points:
Fractals are created through one of three means:
The first technique used creates escape-time fractals, such as the famous Mandelbrot set shown to the left and, in bigger detail, above. These are defined by a recurrence relation at each point in a space.
Iterated function systems, like that of the Koch snowflake, left, are created by a fixed geometric rule. For example, to make a Koch snowflake, you'd start with an equilateral triangle, then take the middle third of each segment, using two line segments to create an equilateral bump. Every iteration results in a growth to the perimeter by 4/3rds; the perimeter is infinite, but the area is finite.
Random fractals are generated by stochastic rather than deterministic processes. Fractal landscapes like the one shown to the left are made by this process.
Behind all these complexities, though, fractals are perfectly natural. Snow may be the best known example of a fractal, though things like lightning, vegetables like brocoli or cauliflower, or blood vessels are also considered as such. Even coastlines can be loosely considered a fractal.
Whether it they are created by our computers or crystallized water, fractals are a beautiful blend of a math and art.
But what is a fractal?
While the exact definition seems to differ between sources—usually because of contention between what a self-similar image is—we can usually agree on some points:
- It has a fine structure at arbitrarily small scales.
- It is too irregular to be described in traditional Euclidean geometric language.
- It has a simple and recursive definition.
Fractals are created through one of three means:The first technique used creates escape-time fractals, such as the famous Mandelbrot set shown to the left and, in bigger detail, above. These are defined by a recurrence relation at each point in a space.
Iterated function systems, like that of the Koch snowflake, left, are created by a fixed geometric rule. For example, to make a Koch snowflake, you'd start with an equilateral triangle, then take the middle third of each segment, using two line segments to create an equilateral bump. Every iteration results in a growth to the perimeter by 4/3rds; the perimeter is infinite, but the area is finite. Random fractals are generated by stochastic rather than deterministic processes. Fractal landscapes like the one shown to the left are made by this process.Behind all these complexities, though, fractals are perfectly natural. Snow may be the best known example of a fractal, though things like lightning, vegetables like brocoli or cauliflower, or blood vessels are also considered as such. Even coastlines can be loosely considered a fractal.
Whether it they are created by our computers or crystallized water, fractals are a beautiful blend of a math and art.
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