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In mathematics, a fractalis a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimensionstrictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set.
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 – the pictures of Chaos.
Oct 31, 2023 · You can create fractals with mathematical equations and algorithms, but there are also fractals in nature. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex.
Fractals are fundamental to understanding many chaotic systems, and have many applications in sciences. They are also beautiful and mesmerizing objects, as we have seen with the Mandelbrot and Julia sets. In this section, we collect examples of fractals in all scales in nature.
Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges.
Sep 12, 2020 · Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means through the rest of this section. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.
In mathematics, we call this property self-similarity, and shapes that have it are called fractals. They are some of the most beautiful and most bizarre objects in all of mathematics. To create our own fractals, we have to start with a simple pattern and then repeat it over and over again, at smaller scales.
