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 at 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.
(Allouche and Shallit 2003, p. 407).
Taking five iterations gives the beautiful pattern illustrated above.
This fractal also appears on the cover of Allouche and Shallit (2003).
Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration. Then
Courtesy : Mathworld Wolfram