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The first known name for a fractal was self-similarity. Later, under the influence of B. Mandelbrot, the word fractal came into use by mathematicians to describe infinitely self-similar geometric patterns or sets. The term fractal comes from the Latin fractus, meaning "broken" or "rough."[notes 3] Mandelbrot, in his 1983 paper The Fractal Geometry of Nature, has described three cases: “A set that is entirely irregular with non-repeating, distinct elements. A set that is self-similar but not repetitive or irregular. A set that can be decomposed into pre-fractal parts that are similar or identical, but belong to different order classes.”[notes 4] The first case, Mandelbrot shows, is easy to create: For example, a circle can be broken into repeating segments, such as each quarter. The second case is more interesting; the Koch curve is composed of smaller copies of itself, where the smaller copies of the Koch curve after a finite iteration are the original Koch curve (after r repeats), and subsequent iterations are copies of that after r2 repeats). The third case is subtle and subtle, but a form of decomposition that Mandelbrot calls “fractal.” When infinite iteration of a set is idealized to consist of a puncture (a singularity that does not form a curve), and such a set is infinitely smaller than the original set, one has a fractal. Many problems in number theory are shown to be described by the empty set, but, for a long time, the fractal set was in a sense ignored, as people have the apparent belief that an empty set is a much more trivial mathematical construct than a fractal set.
Hausdorff's work on dimension and Carathéodory's definition of dimension, together with the work of Mandelbrot and others, led to a redefinition of fractal dimension, and a precise definition of Hausdorff dimension took shape. d2c66b5586