Fractals

Abstract

A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure on all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension. The prototypical example for a fractal is the length of a coastline measured with different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline paradox. The Mandelbrot set is a famous example of a fractal. A fractal is generally "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," a property called self-similarity. Roots of mathematical interest on fractals can be traced back to the late 19th Century; however, the term "fractal" was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."