What is fractal?

fractal D.J.[ˈfræktəl] K.K.[ˈfræktəl] n.(經典幾何學中沒有表示的)不規則碎片形 n.A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin frāctus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2] There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. They are useful in medicine, soil mechanics, seismology, and technical analysis.



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碎形通常被定義為「一個粗糙或零碎的幾何形狀，可以分成數個部分，且每一部分都（至少會大略）是整體縮小尺寸的形狀」[1]，此一性質稱為自相似。碎形思想的根源可以追溯到公元17世紀，而對碎形使用嚴格的數學處理則是開始於一個世紀後卡爾·魏爾施特拉斯、格奧爾格·康托爾和費利克斯·豪斯多夫對連續而不可微函數的研究。但是碎形（fractal）一詞直到1975年才由本華·曼德博創造出，來自拉丁文 frāctus，有「零碎」、「破裂」之意。一個數學意義上的碎形是基於一個不斷迭代的方程式，一種基於遞歸的反饋系統[2]。碎形有幾種類型，可以分別依據表現出的精確自相似性、半自相似性和統計自相似性來定義。