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The properties of a fractal are such that when one part of the fractal is subdivided into parts smaller than the scale of the fractal, the resulting parts are themselves fractals approaching similar properties to the fractal as the parts get smaller. Fractal geometry is the mathematical discipline that seeks to understand the geometry of fractals, that is, determine the geometric properties of fractals, based on their sizes or how they are scaled.[34]

For large-scale geometric patterns such as those seen in nature, the fractal dimensions are often interpreted to have an estimated value in the range of 2 to 3. It is conjectured that the largest dimension possible in nature is 4 (or D = 4). This conjecture is due to Benoit Mandelbrot (with an estimate of D = 2.65). Mandelbrot set is generally considered to be a model for a fractal exhibiting a self-similar geometry, as the original example of a fractal geometry discovered by Benoit Mandelbrot.[notes 2][38]

Fractal theory can be used to describe complex patterns in nature, art, and technology, and has also been applied in fields such as chemistry, physics, finance, linguistics, and engineering, where it has been used to describe such patterns as river networks or rainfall pattern. Self-similar structures are not only found in the natural world, but also on the microscopic level in materials science, architecture, and urban planning. Some examples of self-similar structures include clouds, snowflakes, snow-capped mountains, and ice crystals, to name only a few.[12][39]

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